Notes On Vortex Based Mathematics

Notes On Vortex Based Mathematics

Contents Introduction Vortex Based Mathematics – A Summary Number Polarity Doubling Circuits, Reciprocals & Shearing Multiplication Table ("Multiples") Using Controls Circulation of Numbers & The Hourglass Family Number Groups & The Binary Triplet Emanations, Number Activation & Phasing Polar Number Pairs & The Coordinate System Nested Vortices, Negative Backdraft Counterspace & Spires Dividing By 7 & The Nexus Keys Scaling, Alternate Layouts & Other Number Bases Significance Technical Applications Resources Extension – Uncovering Familiar Mathematical Patterns Right Triangles The Golden Rectangle & The Importance of Angles Pascal's Triangle & The Fibonacci Sequence Sine Waves Extension – Uncovering Harmonics Moving From Number To Frequency Moving From Frequency To Number The Lambdoma Matrix Pythagorean Color Harmonics Extension – Making A 3D Grid Extension – The Flux Thruster Atom Pulsar & Its Geometry General Background Getting Started Synchronized Electricity Extension – "Advanced" Mathematics: Its Relationships To Energy VBM As A Wholistic Framework "Imaginary" (or "Complex") Numbers & Their Stereographic Projections Hopf Fibrations Fractals & "Chaos" Theory Attractors The Penrose Twistor Matrix Algebra Avenues For Further Exploration Addendum – Reorientation of Perspective Addendum – Light & Sound

Addendum – Hidden Form & Function Addendum – Binary Triplet Computing Addendum – Sine Waves Addendum – Changing Axes Addendum – Energy Dynamics Addendum – Coil Considerations Extra – Reference Key & Blank Templates Document Version – 2.5 This document will be periodically updated. Photo Credits: The beautiful picture on the cover is by BJ White. (Careful, the link leads to the original file which is a 6 MB large animation)! While many of the diagrams I have made or re-made from the images in Marko Rodin's books and articles related to his work (unless otherwise noted), there are probably a few that do not have proper credits (for example, a couple from here are in the Summary section, but are not labeled). Please let me know and I will update it or remove it.

Important Note To The Reader: What can one share but their opinions? Invariably, much of what follows will be colored by my interpretations. It will also be highly speculative in nature as I am still learning much, so please forgive me of any errors or redundancy. Many of these patterns were pointed out to me by others for which I am very grateful. I humbly offer my ideas on these matters for your consideration, and welcome any feedback in response. Let us work together to take it further and apply it towards constructive purposes (i.e.: that which is mutually beneficial towards ALL simultaneously).

Marko Rodin Marko Rodin is the creator of a system called "Vortex-Based Mathematics". The inspiration for it came about through a spiritual experience he had when he was 15, wherein he asked the universe "What is the secret of universal intelligence?". The answer he recieved was that it has to do with a person's name and the language in which it is spoken. He figured that the most important names would be those of God, so he searched many sacred texts for any references to the "names of God". He eventually came across the scripture of the Baha'i faith and within the original Persian texts1 he looked for number patterns using "Abjad numerals" (i.e.: when each Arabic letter is used to represent a number – see diagram below).

Table showing letter values (From Wikipedia) (As an aside, I have a feeling that perhaps the work of Bharati Krsna Tirthaji Maharaja, the creator of Vedic Mathematics, may have come about in a similar fashion except with the "Aryabhata numerals" of Sanskrit and using Vedic literature instead...however, I could be wrong. Either way, this too I feel is an important work that follows similar principles, although it may not be as explicitly stated sometimes).

1 There are several works that he continually makes references to: The Star Tablet, The Tablet of the Virgin, The Tablet of Wisdom, The Hidden Words, The Seven Valleys & The Four Valleys, and The Book of Certitude. See: http://bahai-library.com/compilations/bahai.scriptures/

Vortex Based Mathematics – A Summary Here is a general outline of Vortex-Based Mathematics as far as I understand it. This is system has its own particular vocabulary (sometimes calling something by more than one name, or varying the name to describe it more thoroughly). While the names of things within this work may sometimes seem exaggerative or even unnecessary, they are not arbitrary. I will try to define terms as best I can, and give any alternate names wherever possible. I have included many diagrams to make it easy to follow. The topics build upon one another, so please follow it in order. The diagram to the left is The Mathematical Fingerprint of God (also sometimes referred to as The Symbol of Enlightenment or The PetroglyphEquation for the Most Great Name of God). This figure is the crux of much of the work, and the means by which the number patterns necessary to be able to apply it are usually introduced. Each pattern has physical correspondences that I will attempt to elucidate after I describe them all. All the math patterns are shown by "casting out nines" – i.e.: reducing every multiple digit numbers into a single digit through addition (also known as Modular-9 Arithmetic, Decimal Parity or Digital Root; others might even use the terms Indigs or Fadic Sums for this type of action). Nearly everything will be referenced against the number 9. First, let us focus upon the infinity-like shape inside the circle. It begins in the middle and moves towards one, then two, and so on, following the line clockwise and then counterclockwise indefinitely, like two vortices connected at their apex. The pathway never breaks. This is called The Doubling Circuit because: *1 doubled is 2, *2 doubled is 4, *4 doubled is 8, *8 doubled is 16 (1 + 6 = 7), *16 doubled is 32 (3 + 2 = 5), and so on, ad infinitum. No matter how high you go (double digits, triple digits, etc.) it will always reduce back to its corresponding number on The Doubling Circuit. Because they are merely powers of 2 (e.g.: 1, 2, 4, 8, 16, 32, 64, 128, 256, etc.), one could add any combination of numbers from The Doubling Circuit together to form any other number of the Base-10 system. For example: 4 + 8 + 8 = 20, 1 + 4 + 8 + 8 = 21, 2 + 4 + 8 + 8 = 22, 1 + 2 + 4 + 8 + 8 = 23, etc.

You can also go through the circuit backwards to halve the numbers and get the powers of 10. Example of Halving (beginning at 1): *1 halved is 0.5 *0.5 halved is 0.25 (2 + 5 = 7), *0.25 halved is 0.125 (1 + 2 + 5 = 8), *0.125 halved is 0.0625 (6 + 2 + 5 = 13 & 1 +3 = 4), etc. Example of Powers of 10 (also beginning at 1): *1 X 1 = 1, *1 X 10 = 10 (located where 0.5 is because 5 X 2 = 10), *10 X 10 = 100 (located where 0.25 is because 25 X 4 = 100), *100 X 10 = 1,000 (located where 0.125 is because 125 X 8 = 1, 000), etc. The powers of 10 can be thought of as being "sorted out" according to the same pattern, as shown in the diagram to the left. For example, if you were to take it out one step further, all powers beginning with 1 are still on the 1's line, all of those beginning with 100 are still on the 100's line, etc.:

Now, let's look at the red figure made up of 3, 6, and 9 that looks like a triangle without a base. The 9 is self-similar. Any multiple of 9 reduces back to 9. For example: *9 X 1 = 9, *9 X 2 = 18 (1 + 8 = 9), *9 X 3 = 27 (2 + 7 = 9), *9 X 4 = 36 (3 + 6 = 9), etc.

This figure also has a doubling/halving pattern, but instead of cycling around like the doubling circuit, it sits there and pulsates, oscillating back and forth between 3 and 6 (with 9 as the axis between them): *3 doubled is 6, *6 doubled is 12 (1 + 2 =3), *12 doubled is 24 (2 + 4 = 6), *24 doubled is 48 (4 + 8 = 12 & 1 + 2 = 3), etc.

*6 halved is 3, *3 halved is 1.5 (1 + 5 = 6), *1.5 halved is 0.75 (7 + 5 = 12 & 1 + 2 = 3), *0.75 halved is 0.375 (3 + 7 + 5 = 15 & 1 + 5 = 6), etc. The oscillation of 3 and 6 can be shown in two other ways as well. One way is that all the lines of the symbol that shift towards the 3 add up to 3, while all those that shift towards the 6 add up to 6:

8 + 7 = 15 (1 + 5 = 6), 9 + 6 = 15 (1 + 5 = 6), 1 + 5 = 6, 2 + 4 = 6, 3+3=6

1 + 2 = 3, 9 +3 =12 (1 + 2 = 3), 8 + 4 = 12 (1 + 2 = 3), 7 + 5 = 12 (1 + 2 = 3), 6 + 6 = 12 (1 + 2 = 3)

Another way is by adding up adjacent numers within the doubling circuit:

1 + 2 = 3, 5 + 4 = 9, 8 + 7 = 15 (1 + 5 = 6)

4 + 2 = 6, 8 + 1 = 9, 7 + 5 = 12 (1 + 2 = 3)

The pulsation of the 3 and 6 (with an axis of 9) makes a sequence of 3, 9, 6, 6, 9, 3, 3, 9, 6, etc. The 3 is never next to the 6, always being separated by 9. They repel each other like the two ends of a magnet.

We are going to take this sequence (6, 9, 3, 3, 9, 6) and sandwhich it between two copies of the sequence that makes up the doubling circuit (1, 2, 4, 8, 7, 5), one going forwards ("positive" or doubling) and the other backwards ("negative" or halving), like this:

The Mathematical Fingerprint of God is not really a circle, but a simplified represenation of a sphere or tube torus1, and this grid will help you to see it in much more detail because it is the "map" or "skin" of this torus. By tiling it and then linking it up end-to-end, you can make toroidal rings of various sizes. The smallest one you can make (while still keeping all the necessary features intact for it to be a "true" torus – i.e.: an accurate reflection of The Mathematical Fingerprint of God) is a grid of 9 X 9. There are none smaller than this. The numbers are laid out in such a way that you never have more of one than any other. Once you know where just one number is on the grid, you know where every number is because they are all connected.

9 X 9 Grid

ABHA (1251) Torus2

Let us now explore some of the patterns within the torus. Although it may seem complex, it is simple so long as one keeps in mind the dynamic nature of the patterns, and uses The Mathematical Fingerprint of God as their guide. If something does not make sense, always reference it against the simple patterns contained with The Mathematical Fingerprint of God to find clarity. 1 A sphere can be thought of as a type of torus. 2 This is called the ABHA Torus because ABHA, the "most great name of God" in Baha'i scripture, was the inspiration behind it. Its pronunciation produces a compression ("AB") and decompression ("HA") in the breath when spoken. 1251 is the value of "ABHA" in Abjad numerals (A = 1, B = 2, H=5). Notice that it adds up to 9. He found this significant because 9 is considered a "holy number" in the Baha'i faith, with their icon even being a 9-pointed star. He reasoned that it must be like a mirror because it begins and ends in 1, so he placed a decimal point in the middle to form 12.51. If you add up each side of the decimal you get 3 and 6. It also gives the first 2 reciprocal number pairs, 1-1 and 2-5 (more about these number pairs below). He then drew a circle, dividing it into 9 equal parts (or 40 degree increments as 360 / 9 = 40), and labeled each segment with the numbers 1-9. From this, all the other patterns that make up The Mathematical Fingerprint of God started to unfold.

Number Polarity: The numbers on the surface of the torus alternate between "positive" and "negative", therefore there are really 18 numbers. We will represent this within our grid by white and black squares respectively. Also note that the number grid itself is outlined in red squares of 9. Although it is a mouth-full, this view of the surface of the torus is sometimes called The Diamond Crystal Grain Lattice Structure. Positive and negative polarities show up in other ways as well, and will be mentioned when applicable. Doubling Circuits, Reciprocals & Shearing: It is important to keep in mind that when we refer to the toroid, the term "Doubling Circuit" includes both of the positive (1-2-4-8-7-5) and negative (5-7-8-4-2-1) sequences that we sandwiched around the 6-9-3-3-9-6 (also called The Gap Space or Equipotential Major Groove) to form the grid that makes up the surface of it. These three lines of numbers wrap around the top of the torus, go in through the middle, curve around the side at a slight inclinaton to make an s-curve, and then come up around the other end from the bottom to meet themselves head-on, making a yin-yang-like shape. (See diagrams below).

Top View (Yin-Yang)

Side View (S-Curve)

The Doubling Circuit is marked in red and The Gap Space is in blue. There is a general principle that multiplying by any number is the same as dividing by its reciprocal on The Doubling Circuit. This means that particular numbers are always paired: 1 and 1, 2 and 5, 4 and 7, 8 and 8. Examples: 7 X 2 = 14 (1 + 4 = 5)

7/2 = 3.5 (3 + 5 = 8)

7/0.5 = 14 (1 + 4 = 5)

7 X 0.5 = 3.5 (3 + 5 = 8)

This is how reciprocals show up in The Mathematical Fingerprint of God: Step 1

Step 2

The first reciprocal pair is 1 and 1. Moving one away in each 1 X 1 = 1. direction from 5 and 2 gives us reciprocals 7 and 4. Moving one away in each direction gives us reciprocals 5 7 X 4 = 28 (2 + 8 = 10 and 2. & 1 + 0 = 1).

Step 3

Moving one away from both 4 and 7 we get the final reciprocal pair 8 and 8. 8 X 8 = 64 (6 + 4 = 10 & 1 + 0 = 1).

5 X 2 = 10 (1 + 0 = 1).

All Doubling Circuits must meet end-to-end as continuous or "unbroken" rings. You can use the reciprocals to check this. Adjacent Doubling Circuits (one positive and one negative) will move against one another to form reciprocals. This is called Shearing.

The reciprocals can be used to generate Base-10 as well. For example: Exponent Value etc.

Sum (generates numbers of The Doubling Circuit)

0.5^3

0.125

8

0.5^2

0.25

7

0.5^1

0.5

5

1

1

2^1

2

2

2^2

4

4

2^3

8

8

1

etc.

Multiplication Table ("Multiples"): Take the mutiplication series (or "multiples") of each number, and then reduce the products to a single digit by "casting out nines".

Regular Multiplication Table

Multiplication Table with 9's "casted out"

Here is a table showing just 1-9. Notice the symmetry:

9 X 9 Multiplication Table with 9's "casted out" (planes of symmetry are marked in red) Take the series of numbers for each digit and place it behind its corresponding number on The Mathematical Fingerprint of God, like this:

Notice that the series of each number is mirrored from the one across from it. For example: *2 is "positive", going up in units of two (2, 4, 6, 8, etc.) *7 is "negative", going down in units of two (7, 5, 3, 1, etc.) If you add up the digits in both sequences they all add up to 9. Generally speaking, everything to the left of 9 is "negative", while everything to the right of 9 is "positive" when looking at it in this way. 9, being the axis, polarizes all the other numbers because it is also like a mirror. Therefore, these groups of numbers (1-8, 2-7, and 4-5) are called Polar Number Pairs. Also, you might notice that each Polar Number Pair is made up of both an even and odd number: *1 is odd and 8 is even *2 is even and 7 is odd *3 is odd and 6 is even *4 is even and 5 is odd Using Controls: One can even assign different numbers to The Doubling Circuit and it will still function with all the proper patterns, with one exception. You cannot not reverse the numbers within it because the multiplication tables will no longer work (even though all the other patterns will still hold true)! Here are some examples: Correct

Control (Multiples of 2)

Control (Multiples of 4)

Incorrect

Backwards Doubling Circuit!

Backwards Doubling Circuit!

This is not to be confused with a reversing in the direction of travel through the circuit as in the case of halving, but an actual reversal of the numbers within it! Circulation of Numbers & The Hourglass:

The multiplication tables make vortices that circulate the numbers around inside of The Mathematical Fingerprint of God. For example, beginning at the top with numbers 5 and 4, we can follow each of them through the multiplication tables of the other numbers until we get to the bottom. To get to the next instance of 5 and 4, we have to cross the middle to get to the other side. Therefore, numbers turn around in the center point where the line of The Doubling Circuit seems to cross itself; this point is sometimes referred to as The Primal Point of Unity or The Decimal Point Singularity, amongst other names. Similiarly, each pole of the toroid has a rotation as well:

Northern Hemisphere (CCW – 3) *Rotation shown in magenta.

Southern Hemisphere (CW – 6) *Rotation shown in cyan.

This polar view of the torus with a center that looks pinched is sometimes referred to as The Sunflower Hologram, because of its similarity to the head of a sunflower, and the fact that every part of it contains the whole, just like a holographic image. The two vorticies that make up the poles meet in the middle of the torus to form a shape called The Hourglass with The Primal Point of Unity in the very center of it:

Cross-section of torus showing The Hourglass with number grids partially shown in red. Thus the entire toroid acts as a sort of pump, drawing numbers in at the top and ejecting them out of the bottom to move along the surface cyclically in a manner similar to the circulation of numbers inside The Mathematical Fingerprint of God:

Side-view of torus emphasizing vortex-like motion. Family Number Groups & The Binary Triplet: There are other groups of numbers that are related within The Mathematical Fingerprint of God that are hinted at by the interaction of the 3-6-9 and the 1-2-4-8-7-5 of The Doubling Circuit. For example, if we add 3 to 1 we get 4, 3 added to 4 makes 7, and so on ad infinitum. This is Forward Motion. By using 6 instead we get Backward Motion (e.g.: 1 + 6 = 7, 7 + 6 = 13 & 1 + 3 = 4, etc.).

Family Number Group 1 (1-4-7) Forward Motion: 1 + 3 = 4, 4 + 3 = 7, 7 + 3 = 10 (1 + 0 = 1), etc.

Family Number Group 2 (2-5-8) Forward Motion: 2 + 3 = 5, 5 + 3 = 8, 8 + 3 = 11 (1 + 1 = 2), etc.

Family Number Group 3 (3-6-9) Forward Motion: 3 + 3 = 6, 6 + 3 = 9, 9 + 3 = 12 (1 + 2 = 3), etc.

Backward Motion: 1 + 6 = 7, 7 + 6 = 13 (1 + 3 = 4), 13 + 6 = 19 (1 + 0 = 1), etc.

Backward Motion: 2 + 6 = 8, 8 + 6 = 14 (1 + 4 = 5), 14 + 6 = 20 (2 + 0 = 2), etc.

Backward Motion: 3 + 6 = 9, 9 + 6 = 15 (1 + 5 = 6), 15 + 6 = 21 (2 + 1 = 3), etc.

There is also oscillation between positive and negative numbers between these three groups. While the 3 and 6 are positive, the 9 is negative and vice versa. Similarly, while numbers of Family Number Group 1 are positive, the numbers of Family Number Group 2 are negative and vice versa. Because of this feature, if you were to follow The Doubling Circuit, you would oscillate between positive and negative numbers. This is called a Binary Triplet. (See diagram below).

The Family Number Groups are considered the "triplets", while the continual oscillation between positive and negative numbers is a "binary flip-flop", hence Binary Triplet.

Emanations, Number Activation & Phasing: You might have noticed in the very first diagram of The Mathematical Fingerprint of God that there are little arrows coming out of The Primal Point of Unity (see image to right). These are called Emanations and effect the numbers on the surface of the torus (Number Activation) in a particular order (called a Phasing, or less frequently, a Trinary Dwell Setting).

Here, the torus is open in the middle to show The Emanations radiating outward from center. The Emanations (black arrows) come out in thirds in all directions; this is called The Dandelion Puff Principle (because the rays look like a dandelion puff). They activate only the positive numbers of one Family Number Group (designated by red circles) at a time, so only 1/6th of the numbers on the torus are activated at once. This Phasing occurs in 4 cycles indefinitely: Cycle 01 – Positive numbers of Family Number Group 1, Cycle 02 – Positive numbers of Family Number Group 2, Cycle 03 – Positive numbers of Family Number Group 3, Cycle 04 – An interval of rest. Polar Number Pairs & The Coordinate System: The 3 Polar Number Pairs (of 1-8, 2-7, and 4-5) within The Doubling Circuit were mentioned briefly in relation to the multiplication tables. Now we will apply them in a different way.

If we look at the grid we will notice that each number of every red square is straight across from its pair mate with 9 as the axis between them.

It turns out that these three pairs of numbers are the axes of the coordinate system that forms the toroid itself. It is somewhat akin to a spherical coordinate system when shown in this view. The assignment of axes on this particular torus is as follows: 1-8 Polar Pair = Y-Axis (the "vertical") 4-5 Polar Pair = X-Axis (the "horizontal") 2-7 Polar Pair = Z-Axis (synonymous with The Emanations from The Primal Point of Unity – only one is shown here).

Here the Y-Axis (made by Polar Pair 1-8) is shown in gold, and the X-Axis (made by Polar Pair 4-5) is shown in green. Notice that if we take 9 as the center, going up is multiples of 1, going down is multiples of 8, going left is multiples of 4, and going right is multiples of 5. This can even account for how number lines are used in a regular Cartesian coordinate system because each Polar Number Pair is made up of a positive and negative number. For example, 1 goes up in units of 1 (positive), 8 goes down in units of 1 (negative), 4 goes up in units of 4 (positive), and 5 goes down in units of 4 (negative). The reason why 9 X 9 is the smallest grid you can make is because it keeps the rings that form the X and Y axes intact (as The Mathematical Fingerprint of God utilizes digits 1-9). You can check if you have full rings by looking for at least one mutliplication series for each axis because they make loops :

It is this characteristic that allows the numbers to circulate as well. On the torus the axes look like this:

Again, the Y-Axis is shown in gold, and the X-Axis is shown in green. The X-Axis can be thought of as any of the concentric rings you see when facing the pole of the equator. Technically, there are 2 Y axes shown here as they are just rings around the actual "tube" of the torus, like this:

Because the Z-Axis is literally coming out at you, it is synonymous with The Emanations from the center of the torus. However, their effects are detectable upon the numbers that make up its surface. This is called a Reticulation Pattern. To find the Z-Axis (made of Polar Pair 2-7 in this case) take this positive 1 for example. The numbers around it will make 2 and 7: 2 + 5 = 7, 1 + 6 = 7, 5 + 6 = 11 (1 + 1 = 2), 9 + 2 = 11 (1 + 1 = 2) It also creates a phenomenon referred to as Waves of 9, Shockwaves or Magic Squares. Adding up the squares around any number on the grid will give you 9 (even if the square you make does not have 9 at its center). For example: 8 + 8 + 3 + 9 + 6 + 5 + 2 + 4 = 45 (4 + 5 = 9) If you add the number in the center as well, you will get that number. For example: 45 + 7 = 52 (5 + 2 = 7) You can continue to go outward in 1-square increments in all directions like this and the pattern will still hold. If you map the Magic Squares to the torus they will make patterns:

2X2 Octadecagon / 18-Pointed Star

3X3 Dodecagon / 12-Pointed Star

Example:

Example:

5+2+1+1=9 This is obviously the smallest Magic Square you can make.

8 + 8 + 3 + 7 + 4 + 3 + 5 + 2 + 9 = 49 (4 + 9) = 13 (1 + 3) = 4 (the number in the middle)

4X4 Nonagon / 9-Pointed Star

6X6 Hexagon / Hexagram

Example:

Example:

7+4+3+8+5+2+9+7+1+1+6+5+ 2 + 5 + 6 + 1 = 72 (7 + 2) = 9 = 180 (1 + 8 + 0) = 9

9X9 Square

12 X 12 Triangle

Example:

Example:

= 424 = 10 (1 + 0) = 1 (the number in the middle) = 720 (7 + 2 + 0) = 9 As demonstrated in the examples above, you could also think of the number patterns inside the Magic Square in this way: If the length of each side is an odd number of tiles, it will add up to the number in the center. If the length of each side is an even number of tiles, it will add up to 9. One can keep increasing the size of the Magic Square until the entire surface of the toroid becomes one giant Magic Square.

Nested Vortices, Negative Backdraft Counterspace & Spires: Just as the torus is a giant vortex, its surface is also made up of smaller vortices created by the Phasing of The Emanations. They are often likened unto "dimples on a golfball" or "sunspots". Notice that each red square has the Doubling Circuit sequence (1-2-4-8-7-5) inside of it. Following it shows how the Nested Vortices turn. The 9's in the center of every red square are the opposite polarity of the Nested Vortice that makes it up. For example, a +9 makes a negative Nested Vortice that flows inward (marked by the cyan arrow), while a -9 makes a positve Nested Vortice that flows outward (marked by the yellow arrow).

They make Nested Vortice Circuits along the surface of the torus like this:

Positive Nested Vortices are white, while their outward flows are marked by the magenta arrows. Negative Nested Vortices are black, while their inward flows are marked by the cyan arrows. This motion towards the center is known as Negative Backdraft Counterspace.

Northern Hemisphere *Rotation shown in red. Although it may look like it is rotating clockwise because of the Nested Vortices, it is actually rotating counterclockwise. The Nested Vortices can be assigned numbers because they follow the same pattern as The Doubling Circuits:

Negative Nested Vortices are like large negative tiles, while positive Nested Vortices are like large positive tiles (designated by black and white respectively). If you follow the rotation of the torus in red, you can see their similarity to The Doubling Circuit. They even wrap around the torus and meet each other in the same way to make the Yin-Yang shape. But there are some other patterns to look out for also: When moving towards the center, the negative Nested Vortice Circuits count down (9, 8, 7, etc.), while the positive Nested Vortice Circuits count up (1, 2, 3, etc.).

The numbers of each Nested Vortice Circuit come in sets of 4 shown in the diagram to the right by the magenta and cyan crosses. This is called Quadrature.

Linking up the adjacent circuits of Nested Vortices by connecting positive and negative vortices in alternating sequences, makes a logarithmic or equiangular spiral coming out of the center, usually refered to as a Spire. There are 4 of them on this particular torus which are shown in orange, yellow, green and pink.

If you were to follow the numbers of the Nested Vortices by going through one of the Spires, you would find that on every third one they make another Doubling Circuit pattern. Here only the green Spire is shown to highlight this feature.

Dividing By 7 & The Nexus Key: This may seem a bit abstract, but you can use the Polar Number Pairs to give discrete number values to repeating decimals. Taking division by 7 for example: 1/7 = 0.142857... (4) 2/7 = 0.285714... (8) 3/7 = 0.428571... (3) 4/7 = 0.571428... (7) 5/7 = 0.714285... (2) 6/7 = 0.857142... (6) 7/7 = 1 8/7 = 1.142857... (5) 9/7 = 1.285714... (9) Because 2 is the Polar Pair of 7, we can look at the corresponding numbers that result from division by 2. They are mirrored just like the multiplication tables. 1/2 = 0.5 2/2 = 1 3/2 = 1.5 (1 + 5 = 6) 4/2 = 2 5/2 = 2.5 (2 + 5 = 7) 6/2 = 3 7/2 = 3.5 (3 + 5 = 8) 8/2 = 4 9/2 = 4.5 (4 + 5 = 9) If 8/2 = 4 then 8/7 must equal a 5 (because this is the polar number of 4). Therefore, the cluser of numbers after the decimal (142857) must equal a 4 because we have to add them to the 1 before the decimal point to equal 5. Further, if this cluster of numbers equals a 4, then 1/7 equals 4. We can continue this process to assign whole numbers to all of the repeating decimals generated by dividing by 7 (these are shown in parentheses next to the repeating decimals listed above). Both division by 7 and the entire skin of the torus is represented by a symbol called The Nexus Key. It is not quite the same as The Mathematical Figerprint of God.

Here is The Nexus Key in red as it exists on the toroid. If you could see the whole thing it would repeat like a palindrome.

The Nexus Key 1-6-5-2-9-7-4-3-8-8-3-4-7-9-2-5-6-1

If you look carefully, there is another sequence of the same type running in the other direction as well. It is mirrored in both direction and polarity. This is the Antithesis Nexus Key:

Antithesis Nexus Key 8-3-4-7-9-2-5-6-1-1-6-2-5-2-9-7-4-3-8

Here is the Antithesis Nexus Key in blue. It too would run in palindrome if you could see it continuing throughout the toroid.

Each axis also has its own symbol called a Nexus Key Domain Schematic or Spin Angle Waveform:

1-6-5-2-9-7-4-3-8-8-3-4-7-9-2-5-6-1 8-3-4-7-9-2-5-6-1-1-6-5-2-9-7-4-3-8

2-3-1-4-9-5-8-6-7-7-6-8-5-9-4-1-3-2 7-6-8-5-9-4-1-3-2-2-3-1-4-9-5-8-6-7

4-6-2-8-9-1-7-3-5-5-3-7-1-9-8-2-6-4 5-3-7-1-9-8-2-6-4-4-6-2-8-9-1-7-3-5

1-8 Polar Pair = Y-Axis or the "vertical" (in this case)

2-7 Polar Pair = Z-Axis or The Emanations (in this case)

4-5 Polar Pair = X-Axis or the "horizontal" (in this case)

Scaling, Alternate Layouts & Other Number Bases: Vortex-Based Mathematics is sometimes referred to as Polarized Fractal Geometry because the toroid oscillates and is self-similar in nature. No matter how large the grid (and its associated toroid) becomes, it will always have the same features (although there can be many variations). It is much like how The Doubling Circuit continually reduces back to 1-9. This is why it is called a Whole Number Fractal.

Zoom of toroid showing how its grid can be used for higher numbers. The single digit that these numbers reduce to by "casting out nines" is in the parentheses next to them. While you can make the dimensions of the torus longer in one axis (e.g.: a 9 X 18 tile is possible), the grid of 9 X 9 scales easily according to an Inverse Square Law:

One Quanta Features:

Features:

Features:

Features:

*3 Nested Vortice Circuits

*6 Nested Vortice Circuits

*9 Nested Vortice Circuits (9 Groups of 18

*12 Nested Vortice Circuits (12 Groups of 24

(3 Groups of 6 Vortices Each – 18 Vortices Total)

(6 Groups of 12 Vortices Each – 72 Vortices Total)

Vortices Each – 162 Vortices Total)

Vortices Each – 288 Vortices Total)

*6 Doubling Circuits

*12 Doubling Circuits

*18 Doubling Circuits

*24 Doubling Circuits

*1 Spire

*2 Spires

*3 Spires

*4 Spires

The 3, 6, and 9 are all assoicated with particular shapes:

While we have gone through the square/diamond-shaped tiling, there are alternate layouts (sometimes called Carpets) based upon both the triangle and the hexagon. This is suiting as these are the only 3 regular polygons that can completely tile a plane by themselves. One can make tori with surfaces based upon these other tilings too.

Examples

Triangular Layout

Hexagonal Layout

Even though they overlap to some degree, I have included all the major features so that you can see how they function: The Doubling Circuit is in red, The Gap Space is in blue, The Y-Axis is in gold (it is the "vertical" even though it seems horizontal in the hexagon tiling), a Nested Vortice is shown in cyan, and The Nexus Key is in pink. If you look carefully you can even see the Shearing and its Reciprocals. You can derive each layout from one another as well:

Split the diamond tiles in half to make it into the Combine the triangular tiles in groups of six to make triangular carpet. The number in the square becomes the hexagonal carpet. Adding together the numbers the number of the left-most triangle inside of it. of the triangles (and "casting out nines") gives you the number that makes up that hexagon. Here is an example of what they would both look like on a toroid:

Top-view of toroid showing both triangular and hexagonal carpets. There are other specific base systems that naturally form tori. Base-10 is the first and smallest base to generate a true torus because it is the first prime to the second power; each consecutive base follows the same algorithm. However, we won't cover them here. Examples:

Base-26 (5^2) Significance: So what exactly is the significance of all of this?

Base-50 (7^2)

It unites all branches of mathematics because it can do all mathematical functions simultaneously. Furthermore, the coordinate system it contains allows you to model everything as a vortex in great detail, from the structure of atoms to galaxies. The vortice is ubiquitous throughout nature. By "casting out nines" you take all things back to their archetypal form. All numbers can only interact with each other in certain patterns, and therefore imply an underlying geometry. It is this geometry that describes how all energy naturally flows, the numbers acting like pathways. It is the science of harmonics. All physical things grow from the center out, and unfold according to doubling/halving (e.g.: cell division, nuclear fission, etc.) in a diagonal or spiral pathway around that center. The reversing of the numbers within The Doubling Circuit destroys the multiplication tables and is thus synonymous with decay. The 3 and 6 are the pulsation of Magnetism and the 1-2-4-8-7-5 is the flow of Electricity, curving around it to create a boundary through its spin that also allows motions to continue indefinitely (for this reason it is sometimes called a Bounded Infinity and/or Spin Continuum). The motion towards the center of the toroid made by the Negative Backdraft Counterspace is Gravity. The Hourglass in the middle is a blackhole-whitehole pair (such as that which resides inside every Active Galactic Nucleus). The Primal Point of Unity is a singularity and is not solid (i.e.: it does not eventually condense into "nothingness"). The blackhole causes compression (or "implosion") because it is a negative inward vortex (the Northern Hemisphere of the torus symbolized by the number 3). The whitehole causes decompression (or "explosion") because it is a positive outward vortex (the Southern Hemisphere of the torus symbolized by number 6). The equator of the toroid is the turning point. Matter expands and cools as it is released from the whitehole on the bottom, and then begins contracting and heating as it passes the equator going into the blackhole on the top. All matter is continually recycled in this manner. The Emanations from the center of the vortex are the cause of its rotation, and are the energy that literally controls everything (being synonymous with God's Will). It is symbolized by the number 9. This energy is called by many names throughout the work, I believe to accentuate some particular aspect of it: Name

Property Emphasized

“Radiant Energy”

for its characteristic of radiating linearly in all directions from the center point like the radii of a sphere; probably also for its similarity to the energy spoken of by Nikola Tesla (not to be confused with radioactivity or the ambient heat within the atmosphere)

“Inertial Aether” / “Quasi-Mass Energy”

for its characteristic of deflecting matter and/or becoming the spin axis for all mass; it does not bend and does not decay, therefore it "penetrates everything and nothing can resist it"

“Tachyons”

for its ability to travel faster than light, such as the capability that sound is said to have when it travels through a blackhole; some might call it "scalar" or "longitudinal"

“Aetheron Flux” / “Monopole”

for its dynamic nature and the fact that Magnetism is always coupled to it

“Orgone” / “Prana” / “Qi”

for its ability to trigger cell regeneration, and thus its assocation with the various names of the Life Force both ancient and modern (Orgone being the term used by Wilhelm Reich for the energy that sustains life)

The Phasing is what causes time, which occurs in frames. This is how it was known that the Electric and Magnetic moments do not occur simultaneously. There is 1 Aetheron (represented by the 9 of Family Number Group 3) for every 2 parts Magnetism (represented by the 3 and 6 of Family Number Group 3). They move in opposite directions, with the Magnetism being the after-effect of the Aetherons.

Technical Applications: He often states that his goals are to have “inexhaustible energy”, to “end all diseases”, “produce unlimited food”, to “travel anywhere in the universe”, and to “utilize the full potential of the brain”. It seems that he, his associates, and many independent researchers have caried the work far enough to see how each of these might be possible. The one device he is probably most known for, and the proof of concept for all other devices, is The Flux Thruster Atom Pulsar (or “Rodin Coil” for short).

(Images of coil from test by JLN Labs) It is a bifilar coil (i.e.: has two windings) wrapped in a way that mimics the patterns of the torus. It is this shape which gives it several interesting properties: *It is a room-temperature superconductor; electricity passes through it without resistance because the pathway made for it by the wires keeps it from interfering with itself.

*The magnetic field is spread out all along the surface, extending outward in a rotating spiral configuration, rather than contained inside the coil (for this reason it is sometimes called a Field Generator Coil). However, the center is not necessarily inactive. The flux is pinched within it and thrown out of one end; he often compares this action to a “nozzle”.

It has been suggested that this could be used to make: *a propulsion device (a reactionless drive), *a fusion reactor (through torodial pinch), *a dynamo (using the spin of the field to rotate magnetic motors), etc. It also has already been demonstrated to form a magnetic monopole (i.e.: magnetizing a bar of iron with only one magnetic polarity). He states that it should be energized with Pulsed DC, but that ideally, one would not even use wire as a conductor. They have made a toroid out of glass that would use a plasma instead. (See image).

There are other designs and ideas that are being pursued as well: *A computer processor of potentially unlimited speed (because the transistors do not heat up and one can use less of them to build it), *A type of programming language based on the Binary Triplet (Combinational Explosion Tree), said to have use in Data Compression and Artificial Intelligence, *Communications antennas, *Cables for long-distance transmission of DC electricity, *3D speakers, *Ceiling fans (perhaps of a vortexial design like that of Victor Schauberger's Klimator or Raymond Avedon's Thermal Equalizer – unless of course he is referring only to the motor it runs off of), etc. There are even several biological applications: *A system called Biophysical Harmonics that can be used to heal brain damage through sound (an off-shoot of the original idea of intelligence being dependent upon a person's name and the language in which it is spoken – the different tones of the voice activiating particular centers within the brain).

He feels that the mathematics can be used to model DNA to: *develop the understanding that "evolution is not haphazard, random trial and error", *understand cellular communication more deeply, *control it for genetic engineering purposes (particularly for the repair of genetic damage), etc. To explain this, he uses the analogy of DNA being like a barber pole. The red and blue stripes being like the strands that make up the DNA, and the white stripe being like the Major Grove, the space where all its major functions take place (e.g.: chemical bonding, cleavage, receptor site signaling, etc.). The phosphate groups that make up the backbone of each strand have a negative electric charge that generates a magnetic field in the Major Groove. It is responsible for DNA's helical structure, and all the energy interactions within it. He calls it the Bioetheric Template and states that it is analogous to the concept of a Morphogenetic Field as described by Rupert Sheldrake. Each strand of the DNA moves in opposite directions like The Doubling Circuit, and its numbers are the various purines and pyrimidines that make up the nucleic acids. The Major Groove of the DNA is The Gap Space made up of 3-9-6.

The Doubling Circuits (in red and blue) & The Gap Space (in white).

Nested Vortices

Examples demonstrating The Quantum Mechanical State of DNA Sequencing.

Resources: His website(s) http://markorodin.com/1.5/ http://www.rense.com/RodinAerodynamics.htm It contains all his endorsement papers, audio interviews, and other information. His books Aerodynamicsss - Point Energy Creation Physics The Rodin Glossary His video presentations http://www.youtube.com/TheUMMCorg The websites of several persons he works with Jamie Buturff https://www.spiritualresults.com/ http://www.youtube.com/user/jamiebuturff Randy Powell http://www.theabhakingdom.com/The_Gateway.html http://www.youtube.com/user/theabhakingdom Other Researchers http://www.youtube.com/user/2tombarnett http://www.youtube.com/user/rwg42985 http://www.youtube.com/user/GregorArturo85 http://www.youtube.com/user/jackscholze Study Groups & Forums http://vbm369.ning.com/ http://vortexspace.org/dashboard.action http://forum.davidicke.com/showthread.php?t=61370 http://www.abovetopsecret.com/forum/thread651297/pg1 http://concen.org/forum/showthread.php?tid=12786 http://www.thunderbolts.info/forum/phpBB3/viewtopic.php?f=8&t=1073 Similar Works Based off of the Luo Shu Squares of the I Ching http://web.mac.com/paulmartynsmith/iWeb/IChingmath/Home.html http://loshumatrix.webs.com/ http://www.youtube.com/user/theleeburton http://the-magic-square.blogspot.com/ Based off of Buckminster Fuller's work with Indigs (short for "Integrated Digits") http://treeincarnation.com/articles/Number.htm http://www.rwgrayprojects.com/synergetics/s12/p2000.html This list of links is by no means exhaustive, but is included to give you some resources to explore if this subject is of interest to you.

Extension – Uncovering Familiar Mathematical Patterns We have already seen how The Diamond Crystal Grain Lattice Structure can operate somewhat like a Cartesian plane in some instances, and how the Spires are equiangular-logarithmic spirals. Here we will show that there are other familiar patterns that show up within the grid and torus as well. Right Triangles: One can make right triangles on the grid with sides of 6, 12, or 24 squares.

Example of a 6 X 6 X 6 triangle (in green) and a 12 X 12 X 12 triangle (in yellow). By adding up the numbers in the squares and then "casting out nines", you can use the "Pythagorean Theorem" (a^2 + b^2 = c^2) on them. It will always produce 9 = 9. Examples: 6 X 6 X 6 Triangle Above

12 X 12 X 12 Triangle Above

Side A = 8 + 3 + 4 + 7 + 9 + 2 = 33 (3 + 3 = 6) Side B = 2 + 1 + 5 + 7 + 8 + 4 = 27 (2 + 7 = 9) Side C = 4 + 3 + 2 + 1 + 9 + 8 = 27 (2 + 7 = 9)

Side A = 8 + 3 + 4 + 7 + 9 + 2 + 5 + 6 + 1 + 1 + 6 + 5 = 57 (5 + 7 = 12 & 1 + 2 = 3) Side B = 5 + 7 + 8 + 4 + 2 + 1 + 5 + 7 + 8 + 4 + 2 + 1 = 54 (5 + 4 = 9) Side C = 1 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 9 + 8 = 63 (6 + 3 = 9)

6^2 = 36 (3 + 6 = 9) 9^2 = 81 (8 + 1 = 9) 9^2 = 81 (8 + 1 = 9)

3^2 = 9 9^2 = 81 (8 + 1 = 9) 9^2 = 81 (8 + 1 = 9)

9 + 9 = 18 (1 + 8 = 9) 9=9

9 + 9 = 18 (1 + 8 = 9) 9=9

The Golden Rectangle & The Importance of Angles: One can produce Golden Rectangles on the grid by linking up Polar Number Pairs so that each corner is one number, while the center of the rectangle is the other. The only exception being 9, as it has no pair.

Here is a Golden Rectangle made by connecting 9's. While readily noticable on the Triangular and Hexagonal Carpets, The Golden Rectangle is only noticeable on the diamond layout if your tiles are made up of 2 Golden Triangles.

Golden Triangle

60 Degree Equiangular Triangle This is a perfectly balanced tile.

Golden Triangle

The shapes that make up the surface of the toroid are elastic in a sense, becoming skewed depending upon where you are on the surface. In the case of the diamond layout, the tiles start off like the shape to the left and become like the one on the right as you approach The Primal Point of Unity. This is why they are sometimes referred to as Astrophysical Shapeshifters. (See diagram below or animation here).

Top-view of toroid showing stretching of tiles.

Pascal's Triangle & The Fibonacci Sequence: Although it was present in many cultures before its properties were systematized by the French mathematician Blaise Pascal in 1653, this mathematical figure often bears his name. Pascal's Triangle is a table of numbers generated like this:

Pascal's Triangle The sides of the triangle are all 1's. Adding up the numbers next to each other in one line gives you the number below them (just follow the red arrows). (Example: 1 + 1 = 2, 2 + 1 = 3, 3 + 1 = 4, 3 + 3 = 6, etc.). You can continue this process indenfinitely to make a larger triangle.

Adding up the numbers horizontally (cyan arrows) and "casting out nines" gives you the numbers of The Doubling Circuit. (Example: 1, 1 + 1 = 2, 1 + 2 + 1 = 4, 1 + 3 + 3 + 1 = 8, etc.). Adding up the numbers diagonally (magenta arrows) gives you the Fibonacci Sequence (1, 1, 2, 3, 5, 8, etc.). The Fibonacci Sequence (named after the mathematician Leonardo of Pisa because he popularized it through a book he wrote in 1202), is a string of digits generated by adding each number to the one that comes before it in the sequence starting with 1. (Example: 1 + 1 = 2, 2 + 1 = 3, 3 + 2 = 5, 5 + 3 = 8, etc.). While there are many fascinating patterns within both Pascal's Triangle and the Fibonacci Sequence, we are only going to cover one particular aspect. If you take the Fibonacci Sequence and "cast out nines" on every number that has more than one digit, you will eventually get a repeating sequence of 24-digits. Example:

Fibonacci Sequence

Sums

1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 ...etc.

1 1 2 3 5 8 1+3=4 2+1=3 3+4=7 5 + 5 = 10 & 1 + 0 = 1 8 + 9 = 17 & 1 + 7 = 8 1+4+4=9 2+3+3=8 3 + 7 + 7 = 17 & 1 + 7 = 8 6+1+0=7 9 + 8 + 7 = 24 & 2 + 4 = 6 1 + 5 + 9 + 7 = 22 & 2 + 2 = 4 2 + 5 + 8 + 4 = 19 & 1 + 9 = 10 & 1 + 0 = 1 4 + 1 + 8 + 1 = 14 & 1 + 4 = 5 6 + 7 + 6 + 5 = 24 & 2 + 4 = 6 1 + 0 + 9 + 4 + 7 = 21 & 2 + 1 = 3 1 + 7 + 7 + 1 + 1 = 17 & 1 + 7 = 8 2 + 8 + 6 + 5 + 7 = 28 & 2 + 8 = 10 & 1 + 0 = 1 4 + 6 + 3 + 6 + 8 = 27 & 2 + 7 = 9 7 + 5 + 0 + 2 + 5 = 19 & 1 + 9 = 10 & 1 + 0 = 1 ...etc.

24-digit sequence: 1-1-2-3-5-8-4-3-7-1-8-9-8-8-7-6-4-1-5-6-3-8-1-9 By taking this string of numbers and plotting it out along the circumfrence of a circle in 15 degree increments, you get another torus!

24-digit sequence mapped onto a circle. The Doubling Circuit is in red (forward - CCW) and blue (reverse - CW), and The Gap Space is in green. Notice that they make hexagons. Also, the Polar Number Pairs are connected by yellow lines.

It can also be viewed in another way when we focus in on The Family Number Groups:

Family Number Group 3 (3-6-9) is in red. There are two of them, making a hexagram. Family Number Group 1 (1-4-7) is in green and Family Number Group 2 (2-5-8) is in orange. They also make a hexagram. There is an oscillation from one of these hexagrams to the other, with the 1-8 Polar Number Pair (marked in yellow) acting as the axis between them. There are also variations of the number sequence that can be made in a similar way. For example, by beginning our adding on the number 3 instead of 2, we get a new sequence that generates the same sort of pattern: Modified Fibonacci

Sums

1 3 4 (1 + 3 = 4) 7 (4 + 3 = 7) 11 (7 + 4 = 11) 18 (11 + 7 = 18) 29 (18 + 11 = 29) 47 (29 + 18 = 47) 76 (47 + 29 = 76) 123 (76 + 47 = 123) 199 (123 + 76 = 199) 322 (199 + 123 = 322) 521 (322 + 199 = 521) ...etc.

1 3 4 7 1+1=2 1+8=9 2 + 9 = 11 & 1 + 1 = 2 4 + 7 = 11 & 1 + 1 = 2 7 + 6 = 13 & 1 + 3 = 4 1+2+3=6 1 + 9 + 9 = 19 & 1 + 9 = 10 & 1 + 0 = 1 3+2+2=7 5+1+2=8 ...etc.

New 24-digit sequence: 1-3-4-7-2-9-2-2-4-6-1-7-8-6-5-2-7-9-7-7-5-3-8-2 You can keep doing this with different numbers, and they will continuously fall into particular groups. These are called Fibonacci Families, and there are only 5 of them: *1, 2 (This is the one made by the original Fibonacci Sequence, the one we have already diagrammed.) *1, 3 (This is the example given in the table directly above.) *1, 4 (Associated Number Sequence: 1-4-5-9-5-5-1-6-7-4-2-6-8-5-4-9-4-4-8-3-2-5-7-3) *3, 3 (Associated Number Sequence: 3-3-6-9-6-6-3-9-3-3-6-9-6-6-3-9-3-3-6-9-6-6-3-9) *9, 9 (Associated Number Sequence: 9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9-9)

Sine Waves: As a flat projection, The Doubling Circuit makes a sine wave.

The 1 is also The Decimal Point Singularity, while the 3 dots on each end signify that the numbers continue in both directions indefinitely like a regular number-line. Here is the same diagram with the numbers reduced by "casting out nines" to make the patterns more evident:

Note the oscillation of the 3 and 6 at the crest and trough of the waves. Adding up adjacent black numbers inside the wave will give you the red number on the wave between them. Example: 1 + 2 = 3, 2 + 4 = 6, 4 + 8 = 12 (1 + 2 =3), 8 + 7 = 15 (1 + 5 = 6), etc. You might have also noticed that 9 is the only number that does not show up in the two diagrams above. Remember, since this is a flat projection, the wave is really a spiral. All the numbers spin around the path made by the 9 and continue on in the direction in which it points. 9 is both an axis and a vector1. Another animation depicting how we might visualize the sine wave is shown here.

1 A vector is a line that has both direction and magnitude (or length). They are used to mathematically describe a "force" or action. For example, a car moving north (the direction) at 35MPH (the magnitude) can be described as a "vector" in a geometric space.

Extension – Uncovering Harmonics Definitions: *A tone or pitch is a sound wave moving at a particular rate of vibration, or frequency. It can be represented by a note (usually a letter – e.g.: A, B, C, D, E, F, G) or by the Solfege (usually a syllable – e.g.: do, re, me, fa, so/sol, la, si/ti, etc.). If we describe them in terms of combinations of sine waves instead, each of these is called a partial. *A chord is when multiple tones are simultaneously occuring. *An interval is the ratio between two or more of these frequencies. They can be thought of as proportions. *A scale is a set of tones in series. The type of scale it is changes depending upon the intervals made between the notes within it. Moving From Number To Frequency: We can use the number patterns in The Mathematical Fingerprint of God (doubling and triplets) to generate a table of harmonic frequencies. Step 1 – Beginning from 1, make a column of doubles and a row of triples:

1 X 2 = 2, 2 X 2 = 4, 4 X 2 = 8, etc. 1 X 3 = 3, 3 X 3 = 9, 9 X 3 = 27, etc. Step 2 – Multiply every number in the column by every number in the row to get all the numbers in the table:

2 X 3 = 6, 2 X 9 = 18, etc. 4 X 3 = 12, 4 X 9 = 36, etc. Step 3 – Continue this action until at least an 8 X 8 grid is filled to get this type of table:

This gives you the frequency of every note in every octave (for a musical scale utilizing a standard pitch of A = 432 Hz with intervals that are whole number ratios1). The red notes on the bottom show how the order of the columns follows the "circle of fifths" of a Pythagorean chromatic scale. The blue square is the speed of light (to less than 1% deviation) which is equal to 432^2. Moving From Frequency To Number: We can go in the other direction and derive a number pattern from a set of frequencies. Note

Interval

Solfege

Harmonic

Frequency

Sum

C

Unison

do

1st partial

256 Hz

4

C

Octave

do

2nd partial

512 Hz

8

G

Perfect Fifth

sol

3rd partial

768 Hz

3

C

Octave

do

4th partial

1,024 Hz

7

E

Major Third

mi

5th partial

1,280 Hz

2

G

Perfect Fifth

sol

6th partial

1,536 Hz

6

Bb-

Minor Seventh

si

7th partial

1,792 Hz

1

C

Octave

do

8th partial

2,048 Hz

5

D

Major Second

re

9th partial

2,304 Hz

9

E

Major Third

me

10th partial

2,560 Hz

4

F#-

Augmented Fourth

fa

11th partial

2,816 Hz

8

G

Perfect Fifth

sol

12th partial

3,072 Hz

3

A-

Minor Sixth

la

13th partial

3,328 Hz

7

Bb-

Minor Seventh

si

14th partial

3,584 Hz

2

B

Major Seventh

si

15th partial

3,840 Hz

6

C

Octave

do

16th partial

4,096 Hz

1

1 Standing waves can only be generated in whole number multiples (this is called a "harmonic series"), or stated another way, by dividing their medium into partial wavelengths that are whole numbers. There are no standing waves of a fractional wavelength in Nature as they cannot be sustained.

Beginning with C = 256 Hz (the "fundamental"), when we "cast out nines" on this number and every sucessive frequency until the numbers repeat (its first eight "overtones"), we get a string of digits (4-8-3-7-2-6-1-5-9), highlighted in the right-most column above. Notice that it contains every digit of 1-9 at least once. This is The Harmonic Number Sequence, and it can be diagramed by a symbol called The Ennead:

The Ennead

Divisions in degrees

Corresponding musical notes

It is interesting that "casting out nines" on the degrees going clockwise gives you the sequence over again. Example: 4 + 0 = 4, 8 + 0 = 8, 1 + 2 + 0 = 3, 1 + 6 + 0 = 7, etc. Note that the 0 at the top is also 360 degrees (and 3 + 6 + 0 = 9). The Ennead is not only one 9-pointed star, but 3 nested ones:

Here are the 3 stars ordered from largest (yellow) to smallest (magenta). It can also be thought of as 3 nested triangles whose areas increase in size logarithmically:

Triangles are shown in red, green, and blue. They continually make the 3-9-6 pattern of Family Number Group 3. While The Ennead may look similar to The Mathematical Fingerprint of God, it is actually a polarview of the toroid. The triangles are becoming bigger or smaller because they are spinning along an axis, which gives it this tunnel-like effect. The Harmonic Number Sequence is the X-Axis of the toroid (the "horizontal" made up of Polar Number Pair 4-5). Therefore, you can also derive it from multiples of 4 and 5. For example: 4X1=4

5X1=5

4X2=8

5 X 2 = 10 (1 + 0 = 1)

4 X 3 = 12 (1 + 2 = 3)

5 X 3 = 15 (1 + 5 = 6)

4 X 4 = 16 (1 + 6 = 7)

5 X 4 = 20 (2 + 0 = 2)

4 X 5 = 20 (2 + 0 = 2)

5 X 5 = 25 (2 + 5 = 7)

4 X 6 = 24 (2 + 4 = 6)

5 X 6 = 30 (3 + 0 = 3)

4 X 7 = 28 (2 + 8 = 10 & 1 + 0 = 1)

5 X 7 = 35 (3 + 5 = 8)

4 X 8 = 32 (3 + 2 = 5)

5 X 8 = 40 (4 + 0 = 4)

4 X 9 = 36 (3 + 6 = 9)

5 X 9 = 45 (4 + 5 = 9)

Notice that the multiples of 4 are The Harmonic Number Sequence moving forwards, while the multiples of 5 are it moving backwards.

Top-view of torus showing how the The Harmonic Number Sequence is the X-Axis. The corresponding musical notes are also shown, along with two 9-pointed star patterns. These star patterns are logarithmic spirals that demonstate the harmonics of vortex motion. They show how energy is drawn into or away from The Primal Point of Unity in the center of the toroid to recycle matter. If you were to trace one of them out it is easy to see how it spirals:

Beginning on the red 4 to the right, follow the arrows to touch each of the numbers within The Harmonic Number Sequence. Notice that when you finish the star you come right back to 4 again. It is "unicursal" (i.e.: can be made in one motion). Also note that we have flows going in both directions simultaneously. We just traced out a star counterclockwise, but the sequence runs clockwise as well. The following may be somewhat complicated, but is included for sake of completeness.

On this particular torus: *There are 4 star patterns for each of the 4 Spires. (2 of them control compression, and the other 2 control decompression along one of the poles of the toroid; each of these is 180 degrees away from its mate, and 90 degrees from each of the others). *There are 4 repetions of The Harmonic Number Sequence (4 repetitions X 9 digits that make up the sequence = 36 digits total). There are thus 36 pathways of energy on the entire torus (9 of which control compression and 9 that control decompression, making a total of 18 on each pole; 18 pathways X 2 hemispheres = 36 pathways total). These pathways are actually the lines that make up the star patterns (4 stars X 9 lines = 36 pathways). Musically, each sequence represents a fundamental tone and the first 8 overtones (spanning 4 octaves). 4 sequences X 4 octaves = 16 octaves of overtones in each Spire and for each of the 18 pathways on each pole. 16 octaves X 18 pathways = 288 octaves of energy expressed in the entire toroid, with 144 octaves to complete one cycle of compression or decompression. The Lambdoma Matrix: Pythagoras (an ancient Greek philosopher who studied mathematics and music) is said to have made a table of ratios based upon a combination of the Harmonic Series and a sacred symbol called a Tetractys. It is called a "lambdoma" for its similarity to the Greek letter lambda, Λ.

The Lambdoma Matrix The left side (marked in red) represents a overtone series (i.e.: the ratios of overtone wavelengths to the fundamental tone wavelength), whereas the right side (marked in blue) is the undertone series (i.e.: the overtone series inverted). Stated more simply, doubling a frequency produces the same note one octave higher, while halving the frequency produces the same note one octave lower. (See diagram below). The Harmonic Number Sequence (X-Axis of the toroid) is equivalent to the left side of The

Lamdoma Matrix above.

From C to C on a piano is either doubling or halving depending on which direction you move. Having C as 256 Hz (and by extension A as 432 Hz, depending upon what "intonation" or tuning you are using) is sometimes used in science ("Scientific Pitch") because of its recognition of conserving the pattern of doubling in frequency from octave to octave. The current, arbitrarily chosen "Concert Standard Pitch" is C = 261+ Hz / A = 440+ Hz. Is this a reflection of how energy moves (like all the harmonic tables given above)? This may seem insignificant, some might even state that these rules are unecessary and/or wholly dependent upon musical taste, but all manifest things are of motion, of vibration. Sound is thus form, and can not only affect deep changes in people2, but literally help create or destroy physical structures as well, even on the most subtle of levels. What effect might a constant dissonance have upon your body and your behavior?3 Pythagorean Color Harmonics: Pythagoras is also have said to have assigned a color to each musical note. *C – Red *D – Orange *E – Yellow *F – Green *G – Blue *A – Indigo *B – Violet 2 There are many disciplines (Music Therapy, Brainwave Entrainment, etc.) that provide evidence of sounds in general, and especially music (whether playing or just listening), having profound effects on a whole slew of bodily processes, both immediate and long-term. Examples: http://www.emedexpert.com/tips/music.shtml 3 It is said that the Pythagoreans believed all things could be expressed in number (including "abstract concepts such as justice"). Perhaps thinking of things like emotions in terms of vibration (which can be designated by a particular number, also known as its frequency) can make this point of view more tangible?

If we color code the tones we used in The Harmonic Number Sequence earlier, we get can then color the grid that makes up our torus in the same fashion.

Top-view of torus showing all positive tiles of The Harmonic Number Sequence colored according to Pythagorean Color Harmonics. 4 – C, Red (First & Second Octave – Fundamental & First Overtone) 3 – G, Blue (Second Octave – Second Overtone) 7 – C, Red (Third Octave – Third Overtone) 2 – E, Yellow (Third Octave – Fourth Overtone) 6 – G, Blue (Third Octave – Fifth Overtone) 1 – Bb, Violet (Third Octave – Sixth Overtone) 5 – C, Red (Fourth Octave – Seventh Overtone) 9 – D, Orange (Fourth Octave – Eighth Overtone)

Extension – Making A 3D Grid Marko Rodin, working in tandem with Greg Volk, have extended The Diamond Grain Crystal Lattice Structure into 3-dimensions. There is a masterful presentation of this by Russ Gries. I highly recommend you watch it. A detailed paper about it can also be found here. This is only going to cover the introductory aspects of it in as simple a way as possible. Step 1 – Take a cube and cut it diagonally both ways to make 6 pyramids with square bases.

A cube cut diagonally both ways with pyramid formation highlighted in pink. Step 2 – Attach each of these 6 pyramids to another cube of equal volume so that their bases meet each square face of the cube. This makes another shape called a rhombic dodecahedron.

5 pyramids connecting to a cube.

Rhombic Dodecahedron

One pyramid is not shown so as to make the cube on the inside easy to see. If you were to split each one of the diamond shaped faces into triangles you could see the cube inside of it. (See diagram below).

Rhombic dodecahedron with 1 pyramid removed to show how cube is contained inside it. Why is it called a "rhombic dodecahedron"? Because its faces are made of rhombi and it has 12 of them like a dodecahedron.

(Image from Wolfram Mathworld)

You can see how its faces are also Equiangular Triangles just like The Diamond Grain Crystal Lattice Structure.

These rhombic dodecahedra can be stacked to fill space perfectly, and each one is equivalent to a tile in 3-dimensions (i.e.: each has a number inside it). The smallest stack you can make is a block of 9 X 9 X 9, similar to the grid of 9 X 9 in 2-dimensions. However, there are 3 times as many negative numbers as positive numbers inside of it. By facing the block directly (so that you are only looking at one face), you can use it as a normal grid, or you can move through it as if it were many grids layered on top of one another. You can easily tell which Polar Number Pairs are associated with each axis by finding a 9 and looking at how the other numbers orient around it. For example:

On this grid, Polar Number Pair 2-7 are the Y-axis, Polar Number Pair 4-5 are the X-axis, and Polar Number Pair 1-8 are the Z-Axis.

Extension – The Flux Thruster Atom Pulsar & Its Geometry WARNING: This work is potentially dangerous. Please use proper safety precautions when using electrical equipment, and get a thorough understanding before attempting to experiment with it.

General Background: Vortex Based Mathematics is referred to as Omni-Fourth Dimensional. What does this mean? Let us parse it: *Physical matter is electrical, its action is represented by The Doubling Circuit (and the Family Number Groups 1 and 2). This is 3-dimensional. *Energy is magnetic, its action is represented by the pulsation of 3 and 6 in Family Number Group 3. This is 4-dimensional. *It is The Emanations that makes both of these possible, and its action is represented by the selfsimilar number 9 in Family Number Group 3. This is "omni"-dimensional because it is ever-present, controling all things from within and without. There are many interesting correlations between the math and Electromagnetism. For example, on The Mathematical Fingerprint of God, each of the "magnetic numbers" (3, 6 and 9) are 120 degrees away from each other (120, 240, and 360 degrees respectively). This is the same as the 3-phases of a rotating magnetic field. Also, The Doubling Circuit (being the "electrical numbers") embodies patterns similar to the "stepping-up" (doubling) and "stepping-down" (halving) of an electrical transformer. However, in this section of the document we are going to cover specifically how to wind a "Rodin Coil" and how it embodies the geometry of the toroid. Like we did with the above term, let's parse the full, original name of the "Rodin coil", The Flux Thruster Atom Pulsar Electrical Venturi Space-Time Implosion Field Generator Coil. It may seem excessive, but each term contributes to the overall meaning in some way and is very important: *"Flux Thruster" – As described much earlier in the Technical Applications section, the magnetic field goes in one end, is pinched by the The Primal Point of Unity in the center (usually referred to as the Aeth Coalescence in this case), and thrust out of the other end to make what is called a Centroid Nozzle. It is therefore a "thruster" of magnetic "flux". To be more specific, and it cannot be emphasized enough, The Emanations are a real energy and this device demonstrates their existence. They are radiated from the Aeth Coalescence, imparting spin to the magnetic field (thus creating a Torsion B-Field). Notice that there is no mechanical input which causes this phenomena, but only the geometry of the wire. When the device is constructed and powered in the right way, this energy does two more things in addition to the "pinching" described above: It makes Nested Vortices of magnetism along the surface. These keep it from heating up as they act as little heat sinks, pumping the heat generated by the electricity away from the wires to keep them cool. Therefore, one can make "computer chips that cool themselves". The electricity also acts as its own insulation (a phenomenon called Harmonic Shear), allowing you to have two exposed wires

next to each other without shorting out. When all of these features are present it is called Synchronized Electricity. We will talk more about this later. *"Atom" – The geometry of the coil literally makes it a macroscopic model of a sub-atomic particle or atom. How so? On the subatomic level, we will use the electron as an example. Electrons have spin (angular momentum) which allows them to act somewhat like little magnets. This property is actually being utilized to create devices in a branch of electrical engineering called Spintronics. The electron is a little toroid. Like the Torsion B-Field in the coil, The Emanations are responsible for its spin, hence Marko is said to have "discovered the source of the non-decaying spin of the electron". Atoms function in a similar manner. We know atoms are not always symmetrical on both ends because "parity is not conserved" (meaning that when polarized, an atom will eject small particles from one end). Their electrons don't make circular orbits, but tunnel through the "nucleus" (The Primal Point of Unity) in logarithmic spirals like the Spires. Although they might seem complex, they are harmonic and predictable with the math (being called an Interferometry Numerical Pattern when used in this sense). The "nucleus" is like an antenna, and the electrons do not make "jumps", but changes in frequency. Although this is not mentioned, we could also show that the planets do the same thing on a much larger scale (in the sense of both time and space), being emitted from the sun in egg-shaped waves (or circuits) that produce a logarithmic spiral (1, 2). Just as through harmonics, Johannes Kepler approximated the structure of the solar system by nesting Platonic Solids, Robert J. Moon used a similar model to describe the structure of the atom. *"Pulsar" – As we just saw, harmonic principles apply to every level of scale. Pulsars, which "are highly magnetized, rotating neutron stars that emit a beam of electromagnetic radiation", follow the same patterns as evident by their description. In two words, we've jumped from atomic to stellar sized objects that the properties of this device (and its associated mathematics) are able to describe. One of the most intriguing examples given of this has to do with the similarity of red giant stars and red blood cells. Red blood cells are toridial. They contain chemicals ("metalloproteins") called "hemoglobin" that help transport oxygen in the bloodstream. Part of what makes up these chemicals is a substance called a "heme group". (See diagrams below).

Heme Group of Hemoglobin Molecule Contains: Iron is in yellow. Nitrogen is in blue. Carbon is in white. Oxygen is in red. There is a balance of 4, 5, and 6-fold symmetries created by their bonding.

Central core of Red Giant Star showing its various "combustion shells". From center to periphery: Iron (Fe) Silicon (Si) Oxygen (O) Neon (Ne) Carbon (C) Helium (He) Hydrogen (H)

(Images adapted form ones in The Rodin Glossary)

The elements within the "heme group" follow a particular pattern from the center to the periphery, while the elements within a red giant star do also, being stratified into layers due to the pressure differences throughout it. Notice that iron centers both of them. What if stellar evolution and biological evolution are one and the same, unfolding in the same sort of sequences only at different pressures and sizes? The interconnectedness of ourselves to our living environment seems to point out that we are like cells in a larger being. *"Electrical Venturi" – Various descriptions are given as to the properties that an "ideal conductor" for this device would have. They should have the least acute incline as possible in relation to the toroid (Longest Mean Free Path of Least Resistance), and should be fatter on the outside than on the inside (Changing Aspect Ratio). Why? The first feature creates an angular relationship that keeps the electricity from interfering with itself and causing resistance. The squeezing of the conductor specified by the second feature helps the electricity "flow" better in a sense, increasing in amperage while decreasing in voltage, much like how water flowing through a pipe will increase in speed and decrease in pressure when going through a constricted section of that pipe. This is known as the Venturi Effect, and hence it is an Electrical Venturi. *"Space-Time Implosion" – The Primal Point of Unity is the singularity of a blackhole-whitehole pair, the blackhole half of it causing the "implosion" of "Space-Time". Alternately, it is a

"coalescence" of "Aether". It was mentioned in the Significance section closer to the beginning of this document that one of these resides inside the "Active Galactic Nuclei" of every galaxy. In a very literal sense, this device makes a miniture version of one to some extent. To give a more complete description, the Aeth Coalescence itself is actually produced by two interpenetrating tetrahedral-shaped fields of energy whose apices are 180 degrees from one another. They rotate in opposite directions in a harmonic ratio at the speed of light in the center of the toroid. Together they make up The Unit and their motion is called a Cosmic Eggbeater.

The Unit spins along The First 9-Axis (in red) while the two vortices that make up The Hourglass are formed perpendicular to it, NOT on the same axis.

Here is a cross-section of the torus showing the Aeth Coalescence created in the center point of The Unit. The Emanations are the red arrows moving in all directions from it radially.

Not only does it spin along its own axis, but the The black lines on the vortices that make up The entire Unit itself also rotates around to form The Hourglass show the general path that matter Second 9-Axis (in blue). This IS the axis on travels upon (going in on the top and coming out which the vortices of The Hourglass spin. on the bottom). (Images adapted form ones in The Rodin Glossary)

The center of mass in a tetrahedron is called a "centroid" (in pink), and when these points merge inside The Unit in its star tetrahedron configuration they make the Aeth Coalescence (in red) which produces the Centroid Nozzle action. Rotation of interpenetrating tetrahedrons is shown in blue. The faces of these tetrahedron are equiangular (60 degrees). The angles of the tetrahedra specify the directions in which The Emanations are emitted from the center, and how the surface of the toroid crystallizes. It is hexagonally polarized.

The Unit is the center of all matter on every level of scale. It emits a Fundamental Tone created by the frequency (i.e.: the rate of spin) of the two tetrahedons. In turn, this creates a series of overtones and undertones that create a spiral path for matter to circulate on through the toroid (like The Ennead star patterns made by The Harmonic Number Sequence). One tetrahedron produces positive Emanations that give this matter spin and momentum (positive Nested Vortices), while the other produces negative Emanations that create Gravity (negative Nested Vortices or Negative Backdraft Counterspace). There is an infinite number of variations to the toroid, as there are an infinite number of wavelengths the energy that makes up The Unit can take. It can ratchet to form different geometries when you look at it from one of the axes of spin (like the stars generated by the Magic Squares). It can also move through space to make very complex patterns with its movement, or oscillate in place in different ways to do the same. It is compared to a "kaleidoscope" for the endless number of forms it can make, but the way in which it makes the toroid skin through its dynamic motions make me think of it as a multi-dimensional Spirograph. There is a main Unit that integrates the energies of all other Units of the same system. The rotation of all Units is continually sustained by Love. Pure Love is the only energy that can allow Creation to take form because it is absolute coherence. How can a thing exist without coherence? For example, to put it within a personally relevant context, how do the trillions of cells that make up your body function as ONE whole? *"Field Generator Coil" – This part of the name has to do with the fact that the magnetic field is on the outside of the coil, rather than wholly contained inside of it like a traditionally wound toroid coil. Getting Started: Now, we are going to go through the winding of a coil step-by-step and explain how each feature relates to the mathematics. There are many resources on the internet that can guide one in making a well-designed "Rodin Coil" if these instructions alone are insufficient. While there are an increasing number of variants out there, we are only going to focus on the original design. The features this particular design has are very important. Step 1 – We need something to act as a support for wrapping our coil. It should be a ring-shape, and to get the full effect of the magnetism, it should also have a hollow core. There is an easily attainable item that meets all of these criteria, the toy doughnuts made for children. (See picture).

A child's toy made of rings. (Image from Kidloo)

Step 2 – Taking the largest ring from the set, we must prepare it for winding. It is necessary to remove any raised lettering or designs upon its surface. Scrape them off with a razor. (Please be careful). If the surface is too slick, you can also lightly scruff it up with sandpaper to give it some traction so as to hold the wire upon it properly. If the plastic is too flimsy, you can cut it in half, pack it with foam, and then put it back together with glue to make it more sturdy. You do not want it to deform while you are wrapping your wire around it as that will alter the geometry of the coil. Step 3 – We aren't quite done preparing it for wrapping just yet. We need to separate the surface of our ring into evenly spaced increments, 36 of them to be exact. Taking a marker and protractor, make a line every 10 degrees along the equator of the ring. Within each one of these points we are going to place a pin to act as a jig for our wire.

(Image from Alex Petty)

Step 4 – Now we are ready for the wire. Any thickness is fine, so long as it is not too thick (e.g.: cannot allow for multiple wraps around the ring) or too thin (e.g.: it breaks easily). If the wire you have is on a spool, make sure that the spool is small enough to fit into the hole of the ring for ease of wraping. If the spool is too big, you can always remove the wire from the spool and wrap it on any skinny object (e.g.: popsicle stick, pencil, pen, etc.) to feed it through the hole instead. This coil is going to have 2 wires (making it a "bifilar coil"). We have divided it up into 36 points (like the 36-digits that make up The Harmonic Number Sequence) because this is the X-axis (the "horizontal") of our toroid, and each of the wires is going to be wrapped in a star pattern along the surface of the ring in a way that is similar to The Ennead, except with 12-points instead of 9.

Here is a picture of a completed coil to give you an idea of what they look like before we describe how to make the windings. While it looks like only one wire wrapped into a spiral-like shape, it is really two wires that are right next to each other. (Image from Alex Petty)

(First Wire)

(Second Wire)

To begin, take one wire and wrap some of it around the pin corresponding to the number 1 on the ring. This is our starting point for this wire.

Now, going over the top of the ring and into the hole, come out on the other side on the number 180 degrees from the 1, in this case, 16. This is one "turn". Then, go over the top of the ring and back into the hole towards the number 180 degrees from 16, which is 31. Continue this process until you have come back to the number 1 again. This is one "wrap" or "winding". When it is done, it should look something like this:

One wrap. He has cleverly extended the pins as a reference for what numbers he is winding around. If you decide to do the same, be careful not to let any of them come out until you are finished as that would ruin your windings. You could also use different colored pins to differentiate them instead. (Image from Alex Petty)

While the wire should not be so taught as to snap it or bend the plastic, it cannot be too slack or it will not duplicate the necessary geometry. You can use a little bit of tape or glue to hold it in place if necessary. Continue making wraps in the same way for both wires. You can make as many as you would like, but with two very important stipulations: *The number of wraps for both wires should be equal to each other. (Perhaps aim for a number divisible by 3, 6, or 9). *You cannot fill the entire ring with wraps. You have to leave a space of equal width to all the wraps of one wire in the areas corresponding to all the numbers not touched by the above two star patterns. (See diagrams below).

Picture on the left is a close-up of the coil showing the space as equal to the width of the wraps of one wire. Picture on the right shows the numbers associated with them. So, what is the significance of these number patterns used to make the coil? While the 36-digits point out its relationship to The Harmonic Number Sequence and the energy pathways made by The Ennead stars, it does not seem to be the X-axis alone. Maybe, like the 36 X 36 tile (One Quanta), these stars are an embodiment of the energies of the entire toroid in great detail if we know how to look at them? "Casting out nines" on them makes their patterns more transparent:

This is a positive Family Number Group 1 and The Doubling Circuit.

This is a positive Family Number Group 2 and The Doubling Circuit.

This is a positive Family Number Group 3 and The Gap Space.

When you are winding a wire, you move through the digits that make up the Family Number Groups repeatedly, always ending up 90 degrees from the previous one.

Comparison of "Rodin coil" to ABHA Torus. The picture to the left shows the wires color-coded to their corresponding star patterns. He has the bottom of the coil facing himself because the wires are running clockwise. The picture in the middle shows how they would show up on the torus. They create Shearing like adjacent Doubling Circuits moving in opposite directions. The picture on the right shows Nested Vortices (in blue and yellow) and Spires (in magenta). This is what happens in the magnetic field when the coil is activated properly! Synchronized Electricity: WARNING: This information has not yet been tested, and is merely a guess at what the proper activation of the coil would be. Please exercise caution if attempting to implement the ideas herein. I do not know how big of an effect it might have. It is very important to note that the device really should NOT be used with AC as the oscillating magnetic field could generate microwaves. The fact that the magnetic field is exposed may also create interference. Keep sensitive equipment properly shielded. The similarity of the description for Synchronized Electricity (given in The Rodin Glossary) and The Binary Triplet leads me to believe that proper activation might be quite simple to achieve with a regular "Rodin coil" made to the specifications given above. It has to have The Gap Space and a hollow core! Before we begin, find out which end of the coil is where the flux is being expelled from (the windings should be running clockwise on that side because it is the Southern Hemisphere). If you are unsure, run a little bit of current through it. Once found, place this end downward and keep clear of it when attempting to energize the device in the event that a jet of energy is released from it upon

activation. Power it from a distance if possible and begin only with low voltages. Generally, the process would be as follows: *Keeping in mind that wire 1 is a positive Family Number Group 1 and wire 2 is a positive Family Number Group 2, we should pulse DC into the first wire. (This is the activation of Family Number Group 1).

*Then, on the next cycle as the first pulse is coming back, pulse the second wire. (This is the activation of Family Number Group 2).

*Stop right there! Don't put any more pulses in just yet. I believe that the combination of this sort of twisting and oscillating action will lead to Nested Vortices being produced in the magnetic field on the third cycle, with effects being most evident within The Gap Space. Having a conductor here will interfere with it, which is why you don't put wraps all the way around the ring. (This is the activation of Family Number Group 3). The Nested Vortices are sometimes called Vector Spatial Interstices because they are generated by "Vectors" (i.e.: the Positive and Negative Emanations) in The Gap Space between The Doubling Circuits. *At this point, the fourth cycle, we must give it an interval of rest as the magnetic field collapses in preparation of the next pulse that will start the entire process all over again. You might have to play with the timing of it (e.g.: using a microcontroller as a switch between your wires and DC power supply), but it is likely that it will achieve resonance even if you are off by a little bit. The conductors are meant to be perfect loops in any torus that is formed in Nature (being the Bounded Infinity / Spin Continuum that allows motions to continue forever). But in order to energize the one we have made with our coil, they have to be open so that they can be connected to our power source. In a sense we are working backwards, using the electricity and magnetism to make a toroidal space for the Aetheron Emanations to show up, rather than building it from the center out with The Unit and its Cosmic Eggbeater action powered by Pure Love as Nature does. Despite this, I have a strong feeling that it is quite likely that it will begin to sustain itself after a while if all of this is done correctly (meaning that the field will no longer require a power source). I also have a feeling that it will change the atmosphere around the device, possibly even having an effect at great distances.

It seems that even when it is not energized, the geometry of the coil might even allow it to act as a Shape Power device (like Pyramids, Genesa Crystals, BioSignatures, etc.). Ultimately however, it is an entity rather than "just a device" (and even the field it generates is a living energy – e.g.: Plasmoid). ALL things are alive and intelligent.

Extension – "Advanced" Mathematics: Its Relationships To Energy VBM As A Wholistic Framework: I find that Vortex Based Mathematics is a very elegant framework. I would even venture to say that it literally contains and unifies all branches of all currently known mathematics. However, the approach is so different that it may not be readily apparent as to how. While it seems to be only arithmetic patterns based on numerology-like reasoning, these only serve as a means to introduce a more fundamental order which can be approached in "higher" dimensions as well. As Rodin's associate, Alastair Couper writes: "Marko's original vision was essentially a perception of a four dimensional sphere, which becomes a complicated toroidal structure when projected into three dimensions. He also perceived a mapping of an energy flow on the surface of this projected toroid." I will try to make the following topics as approachable and friendly as possible. Even if it does not make sense immediately, please keep reading. Feel free to skip over parts and come back to them later. "Imaginary" (or "Complex") Numbers & Their Stereographic Projections: A good introduction to imaginary numbers can be found here and here; a video that specifically covers this topic can be found here (click the link and open up video number 5 in the playlist that comes up; if it is inconvenient to do so, there is a text explanation here). In the above resources it is pointed out that imaginary numbers "'rotate' numbers, just like negatives make a 'mirror image' of a number". Here is a graphic from one of those websites describing this action:

(Graphic from BetterExplained)

Would it be possible to use The Diamond Grain Crystal Lattice Structure as a "complex plane" (i.e.: like a Cartesian plane but for plotting out "imaginary" numbers)?

Regular grid showing X and Y axes as multiples of 4-5 (green) and 1-8 (gold) respectively.

Y-Axis (gold) as "imaginary" dimension and X-Axis (green) as "real" dimension. Powers of imaginary numbers shown in red; they are mirrored like the Polar Number Pairs.

In the video they demonstrate that you can make a "stereographic projection" on this "complex plane" by placing the pole of a sphere on the origin.

"Projecting" through the north pole of the sphere onto the complex plane under it. Each point on the plane then becomes one on the surface of the sphere. (Graphics from Dimensions)

This is used to describe that a sphere can be thought of as a "complex projective line". This means that it can be 1-dimensional (the blue line along the surface of the yellow sphere in the picture to the left), in addition to being either 2-dimensional (the surface of a sphere as a flat plane) or 3dimensional (a sphere as a geometrical "solid"), because it only takes one complex number to describe it...much like how by knowing one number on The Diamond Lattice you know all of them. Is it possible to describe every number on the toroid at once with a loxodrome-like curve, similar to that in the box above on the right? Have we might already even have done this in a simple way without knowing it?

This is a view of both poles of the torus simultaneously. I have made a continuous red line that shows how one would tunnel through every number along its surface. It makes a curvy "S" shape like the two connected spirals at the top. (Note that these small spirals are just for illustration and do not have the proper number of turns). The line where the two spirals meet would tunnel through all the numbers along the equator, as indicated by the grey line. The center of both spirals is The Primal Point of Unity.

As we already know, the X axes, being rings, make concentric circles when viewed at the pole, such as in the diagram to the right.

It is this feature that leads to little kinks in the curve as you hop from ring to ring when making the turns of the spiral. (They can be seen as two little staggared lines on each pole in the diagram on the previous page). The picture to the left is a zoom of that segment... Wait a second, do these two lines coincide with The Nexus Key and its Antithesis? I have highlighted them in pink and filled in some of the numbers to show their patterns. The Nexus Key 1-6-5-2-9-7-4-3-8-8-3-4-7-9-2-5-6-1 The Antithesis Nexus Key 8-3-4-7-9-2-5-6-1-1-6-5-2-9-7-4-3-8 I feel that The Nexus Key and its Antithesis, as their name implies, are "keys" in the sense of a map. You look at the key on a map to figure out what its symbols mean; at a glance, you get all the information it contains. In the same way, either one of these sequences can give you all the patterns of the toroid (because it passes through both The Doubling Circuit and The Gap Space, running in the other direction as them or "counter-diagonally"). By taking the whole sequence, we get all 18 numbers that make up the positive and negative tiles. By taking half of the sequence and following each end towards the 9 in the middle of it, we get each of the Polar Number Pairs. The sequences either count up or down; it is this "breathing" action that the palindromes make which can also give us the polarity of the tiles (and by extension, every feature that can be derived from such: The Emanations and their Phasing, the Nested Vortices and their circuits, the Spires, etc.). It also gives you at least one of the axes of the coordinate system (because The Nexus Key is the same as the Nexus Key Domain Schematic for Polar Number Pair 1-8). But that is not all. These are also "keys" in the sense of those that unlock gateways or open doors. The toroid is a link between dimensions, in more ways than one.

Of course, all this talk of complex numbers is conjecture on my part. The Rodin Glossary mentions them on page 35, in a section entitled Complex Planes and Fermat's Last Theorem. It states: "Periodic functions, such as sines and cosines, were studied by Poincare on a 'complex plane' rather than on a 'number line' as did Fourier. Poincare examined the complex plane, which contains real numbers on the horizontal axis and imaginary numbers on a vertical axis. This allowed Poincare to model periodcity along multiple axes, and to place elements of the matrix into algebraic groups with infinitely many possible variations. Poincare extended this concept of periodic functions on a complex plane to complicated functions called modular forms. Goro Shimura's conjecture delt with the folding of a complex plane: 'If we 'fold' the complex plane as a Torus, then this surface will hold all the solutions to elliptic equations over the rational numbers, these in turn arising from the equations of Diophantus.' What proved to be important later was that if a solution to Fermat's equation (x^n + y^n = z^n) existed, then the solution would have to lie on that Torus." I have inserted links to all the topics mentioned within the quote, and highlighted what I feel to be the biggest clue as to understanding it, but we won't decipher its meaning just yet. Hopf Fibrations: It is at this point in the document that we get to the heart of the quotation by Alastair Couper that was mentioned earlier. A video that specifically covers this topic can be found here (click the link and open up video number 7 in the playlist that comes up; if it is inconvenient to do so, there is a text explanation here). I recommend the video however as it is very visual and makes it easy to understand. This video is a continuation of the topic of the previous section. Without getting too deep into the mathematics, it basically demonstrates that by using "stereographic projection" again, we can turn a 4-dimensional sphere (i.e.: one made up of many intersecting "imaginary" dimensions) into a series of rings in 3-dimensions. To make it simple, think of the jump from a sphere in 3-D (e.g.: a ball), to a respresentation of one in 2-D (i.e.: a circle). This is quite similar.

The picture to the left is a 4-D sphere made by the two intersecting complex planes, but when we only look at one at a time, it seems to make a circle like in the picture to the right. This circle is really a ring in 3-D. (Graphics from Dimensions)

These rings can be rotated along an axis or linked together like a chain (without intersecting one another) to form a torus. This is another representation of the 4-D sphere just expressed in a different way. In the mathematics of topology, this shape is called a Hopf Fibration (after Heinz Hopf, the mathematician who discovered it). Each ring is referred to as a "fibration" because because they are like "fibers" that weave together to make the "fabric" of a 4-D sphere. (See diagrams below).

Top-view of Hopf Fibration

Side-view of Hopf Fibration

(Graphics from The Hopf Mapping)

It is interesting to note that in order for the rings of the Hopf Fibration to be non-overlaping (or "conformal") they becomed tapered as they near the center, with only a gradual incline in relation to the ring in the middle. These features reflect almost exactly the descriptions of what an "ideal conductor" would be like structurally within Rodin's Flux Thruster Atom Pulsar as mentioned earlier. As for the Hopf Fibration, have we already seen something akin to this? Indeed, we have! You might have noticed that The Sunflower Hologram is made out of 36 circles, in a very similar fashion to how the "fibrations" make up the Hopf structure:

The Sunflower Hologram with one circle highlighted in purple. It is made up of 36 circles because it is One Quanta. Fractals & "Chaos" Theory: There is a beatifully simple (and quite short) introduction to these topics available here. We have only briefly touched upon the fractal nature of the toroid so far, even though the unfolding of the same patterns over and over on every level of scale makes it plain. With the insight provided by the above resources, let us see if we can uncover more patterns together. In the Alastair Couper article it is noted that: "All that is required to generate fractal behavior is two things: nonlinearity and iteration; which is to say take a nonlinear process and feed back some form of its output to its input. The numerological binary doubling sequence as used by Marko is indeed nonlinear and iterative, so it should be no surprise when he shows us the repeating fractal patterns of the Sunflower Map." We know that The Doubling Circuit is non-linear. We can see this in its cyclical motion and the fact that the sequence that makes it up is not sequential (i.e.: it is 1-2-4-8-7-5, not 1-2-4-5-7-8). One "iteration" would be equivalent to one complete pass through The Doubling Circuit. It scales in units of 64 (i.e.: 1 becomes 64 when you come back to that spot on The Doubling Circuit again). It continues on: "It is shown that when these sequences are laid out in a grid, and various groupings are made using the horizontal addition, a fractal effect occurs where the same sequence of numbers appears at higher and higher levels of groupings. One is immediately reminded of the various examples of period doubling leading to chaos that are seen in all sciences these days. These bifurcation maps, such as that produced by the much studied logistics equation, show a repeating pattern at all levels of magnification, much as does the Sunflower map." What exactly all this means is greatly clarified by the link given at the beginning of this section. To give a very general outline, a "bifircation map" is a graph that shows the behavior of something complex (i.e.: with many variables) over time. We can use this to describe things like population growth; it follows a particular pattern (described by a "logistics equation"). We can't predict it exactly because it is so complex, but we can state general trends based on what we do know.

This is a bifircation map. (Image is from Wikipedia, but I have added the blue text) Again, in the case of population growth, if the rate of growth is a particular value we will get a "steady state" (i.e.: a line that does not change, such as the one to the left of the diagram). If we increase that growth rate, something interesting happens. It "bifircates" (or splits in two). This means that the population will periodically fluctuate between two values (e.g.: being a certain amount one year, a different amount the next, going back to the first, and so on). This is called a "period", and this process, a "period-doubling bifircation". If you keep increasing it, after 8 divisions ("period 4") you get "chaos" (i.e.: an infinite number of possible states). It is at this point that the system becomes completely unpredicatble... Isn't interesting that when you get to 8 (after 4 doublings) on The Doubling Circuit, it is at that point that the numbers become double digits and you have to "cast out nines" to get the 7 and 5 that make up the rest of the sequence. Is the "digital doubling" (the kind we do with the numbers) the same as the "analog doubling" (the kind that complex systems do, which is expressed in the "bifircation map") one and the same? Are we seeing order amidst the seeming "chaos" by "casting out nines"? There also is said to be a "period-halving bifircation" that shows the movement from "chaos" to order:

To the left is "period-halving bifircation", and to the right is "period-doubling bifircation". (Image from Wikipedia)

Another interesting property of the "bifircation map" is that it too is a fractal, containing a copy of itself within the complexity:

(Image from Fractal Wisdom)

Let's relate all of this to energy... To quote the Alastair Couper article again: "So white noise is the signature of infinite information, rather than zero information. And the fractal road of bifurcations is a map (perhaps one of many) of the territory. It is the virtue of the fractal approach to ungainly systems that high orders of complexity can often be collapsed to a very simple model which mimics the overall characteristics, not in the sense of a linearized approximation, but in the manner in which noise and order (or information) are transmitted by the system." There are so many intresting things one could probably link this to, most notably the use of statistics in various scientific disciplines such as Thermodynamics and Quantum Physics. But lets keep it simple. Generally speaking, "noise" is often treated as something undesirable. This is understandable. In communications, these signals are damped out for sake of clarity (e.g.: when sending messages) and safety (e.g.: "noise" can generate standing waves in equipment that can break it if their oscillations produce resonance; amplitude is power). However, what if Nature controls "noise" in a very specific way to produce matter? What if all matter is a complex series of standing waves arising out of an omnipresent backgroud eternally seething with infinite energy and information (or "noise")? All things are motion. Even that seemingly still bit of matter has rhythms that make up its form and exists in an environment full of ongoing processes. Perhaps we already even have a thorough understanding of some of this within our sciences (e.g.: "zero point energy", "virtual particles", various "aether" models, etc.)? But more importantly, what if this infinite information is already present inside of you in some form, just like how a copy of the entire "bifircation map" is inside itself like a hologram?

Before continuing on to the next section, I would like to share one more idea. It is interesting to note that there is a specific type of "noise" (called 1/f, "flicker", or "pink noise") that seems to occur in many different types of systems. There have also been applications developed that utilize "noise" in some sort of constructive manner. Truly it may be said that there is no "chaos" in the sense of disorder, even if we may not fully understand a thing's cause. Attractors: I again point to this article here as describing the topics relevant to this section. Like the "bifircation map" that was just described, there is another way to describe complex processes, with a figure called an "attractor". We won't go too deeply into them, concerning ourselves only with a particular one called a Lorenz Attractor. Stated very simply, it is a double spiral that depicts an oscillation between two states of a complex system (much like the first "bifircation", or split of the "bifircation map").

(Image from the website of Josep Cayuela)

Wow, that even looks like The Doubling Circuit doesn't it? It is similar in more than just looks. One of its features is that the values along its curve never repeat; in a sense, the numbers of The Doubling Circuit do the same because they are becoming exponentially larger as you continuously spiral along it.

Picture showing three repetitions of the The Doubling Circuit. First Pass: 1 – 2 – 4 – 8 – 16 – 32 – 64 Second Pass: 128 – 256 – 512 – 1,024 – 2,048 Third Pass: 4,096 – 8,192 – 16,384 – 32,768 – 65,536 – 131,072 ...etc. It also has a periodic oscillation between two states because all the numbers to the right are positive and all those to the left are negative (the 9 acting as a mirror between them). Alternately, one could also think of the "flip-flop" of The Binary Triplet. There is another type of "attractor" that is related to The Doubling Circuit; it is called a Torus Attractor. The type of motion it describes is a movement away from the center of a torus in all directions, along the outside of its surface, and back into its center again repeatedly.

(Images from Fractal Wisdom)

This is equivalent to the pumping action described before. (See diagram below).

I wonder if perhaps it could also be described as a increase in size with every successive circulation, somewhat like this:

Cross-section view of nested tori that are increasing size. Red arrows describe circulation. With each pass you jump to the next highest torus. It is stated in The Rodin Glossary that the geometry is specifically related to a Torus Attractor, and these are the ways in which I think it might show up. The Penrose Twistor: There is another 4-D geometry that has the shape of a toroid called a Penrose Twistor (Roger Penrose being the mathematician whom formulated it, and a "twistor" being a type of mathematical object). It is part of a larger physics theory used to merge quantum physics and relativity. Summaries are given here and here; while highly mathematical summaries can be found here and here (PDF link).

Picture of Twistor geometry. (Image from UniverseReview)

We aren't going to go into exactly what the theory consists of (although the first link given above is a very good and fairly simple introduction if you are interested). We are only going to talk about a quote within one of the endorsement papers for Rodin's work by Tom Bearden: "In the late 70s and early 80s, Bill Tiller, Frank Golden and I worked on curl-free A-potential antennas, and Golden built dozens of curl-free A-field coil antenna variants. One of the most interesting variants he built was quite similar to Ramsay's buildup of the Rodin coil. Simply, he built a coil embodiment of the diagrammatic geometry for a 'twistor' that was shown by Roger Penrose. That coil antenna exhibited about what Ramsey and Rodin are reporting, and dramatically extended the communication range of a small CB radio from, say, its nominal 1/4 mile to 20 miles or more." Matrix Algebra: I believe that within one of the video presentations on this work it is said that it is directly relevant to several branches of mathematics, some of which include Surface Topology and Matrix Algebra. We have already applied Surface Topology to some extent by taking the grid (a plane) and wrapping it up into a toroid (although we did not use the name "Surface Topology" to describe this deformation). Alternatively, we might also think of it as folding up a plane into a sphere, making two twists in it (one one its north pole and the other on the south pole), and then connecting them inside the center of the sphere to form The Hourglass. As for how Matrix Algebra shows up within it, my guess is that it has to do with objects called Rings, which are a part of something called Ring Theory, funnily enough. Here is a helpful glossary of terms should you choose to explore this branch of mathematics.

Avenues For Further Exploration: Some may have been struck by the fact that at several points within this document I make analogies between The Diamond Grain Crystal Lattice Structure and the Cartesian Coordinate System, as it might be stated that they are not the same. While I agree to some extent, I feel that the similarities they do possess make this analogy more useful than not. I encourage everyone to play with it in the same manner, and look for patterns related to every type of math you may already know. Whatever level your education it does not matter; our understanding can only grow. There are many more topics to cover and be uncovered, but alas, we conclude this document for now. Thank you very much for reading! And happy calculating! :D

Addendum – Reorientation of Perspective I think that there are several other ways that one can look at The Mathematical Fingerprint of God.

Original Fingerprint of God

Highlighting of patterns.

It seems as if it is a side-view of the toroid, but from the inside of it. The rotations of the poles given by The Doubling Circuit (shown as little black arrows going clockwise and counterclockwise in the picture to the right) are backwards from what it would be if you were to look directly at the pole (such as in the view given by The Sunflower Hologram). Maybe we could change our perspective of it like this:

Your fingerprint is your identifier. It is unique to you. The Creator leaves its fingerprint on ALL the things IT Creates. Once you are aware of ITS method of manifesting Reality, your potential to constructively Co-Create with IT is boundless. You are an individualized expression of IT.

Addendum – Light & Sound

(Images derived from ones in The Rodin Glossary)

At first, I thought these might be related with the Color Subtractive (CYMK) and Color Additive (RGB) Models of Light by the fact that it contains both white and black as shown in the diagram on the right. While I still find this a valid idea that requires further exploration (perhaps in a way similar to this), maybe there is an easier way to approach it. Upon closer inspection, the section where these diagrams are extracted from states that it is a color wheel (starting from the bottom-left and going counterclockwise on the diagram to the left: red, orange, yellow, green, blue, violet). We know that The Seed of Life pattern (i.e.: the overlapping circles that make up the color wheel itself) is actually a polar-view of the torus (with the center point being The Primal Point of Unity). Therefore, these "white" and "black" arrows are also showing The Second 9-Axis that centers The Hourglass. The color wheel contains interesting patterns based on triplets that help us to take this further:

The Primary & Secondary Colors make a hexagram.

The Tertiary Colors also make a Taking them together gives one hexagram 90 degrees from the a 12-pointed star. one on the left.

(Images derived from interactive applet at Tiger Color).

It seems like the Primary and Secondary triangles are flat projections of the male and female tetrahedra that make up The Unit. Like how the Tertiary colors are made from mixing together the Primary and Secondary colors, this second hexagram shown in the middle diagram would be the result of their union. It might also be said that the two oscillate back and forth to generate the 12-pointed star of the rightmost diagram. We can relate this to many things (e.g.: the the oscillations of the two hexagrams that make up the Fibonacci Sequence toroid, the 12-point star made by the 3 X 3 Magic Squares, and the 12-point stars that make up each winding of our coil). Since the relationships of the tetrahedra (with equiangular triangle faces of 60 degrees) determine the way in which The Emanations are emitted from the Aeth Coalescence to generate the hexagonally-polarized skin of our toroid, we simultaneously obtain the angular relationships for all tiles that make up the surface of the torus as well. Another way to look at this would be through The Ennead, as we know that it too is a polar-view of the toroid:

Pole-view of torus with The Ennead (and its associated angles), and The Harmonic Number Sequence (with its associated musical notes). As the above picture demonstrates, these colors and angles have a direct association to Music. Because there is a 40 octave difference in Light and Sound, we can assign colors directly to their associated notes in a way that reflects how energy moves, similar to what has been done in Marcotone Color Theory. Further, we know that this gives us a key for describing matter as well. Much of the history of chemistry is rich in the Wisdom of the Sacred Science of Harmonics for example:

*"The Periodic Law" (of the Periodic Table) originally being known as "The Law of Octaves". *The original names of different atomic shells (s - "sharp", p - "principle", d - "diffuse", f "fundamental") being a reflection of a musical approach to spectral emissions. This is probably because of the recognition (at least intuitively) that the internal structure of the atom follows harmonic patterns: "The Nobel prize-winning physicist Niels Bohr found that, in the first four shells of the hydrogen atom, the single electron moved at exactly proportional speeds: at 2160 kilohertz per second in the first shell; at 1080, 720, and 540 kHz/s respectively in the second, third, and fourth shells. In musical terms, the electron in the second shell creates the octave below that in the first shell; in the third shell, we hear the fifth below that lower octave, and in the fourth shell, the fourth below the fifth – a natural occurrence of the undertone series. The mathematician H.E. Huntley, discussing the golden section in his book The Divine Proportion, describes the energy exchange between several levels of electron shells and suggests that this exchange, rising and falling in different amounts from one shell to another, occurs in a pattern that follows the ratio of Fibonacci numbers. Huntley affirms the proportion of the musical minor sixth, or the ratio 5:8, in the quantum leaps between energy levels of simple atoms of a hydrogen gas. C.N. Banwell, writing in a highschool physics text hook, describes spectral analysis of energy levels in simple electron organization using terms like 'fundamental,' 'overtones,' 'combination frequencies,' etc. Donald H. Andrews, chemistry professor at Johns Hopkins University in the mid-20th century, writes enthusiastically about the musicality of atoms." - From pg. 94 of The Musical Order of the World by Siglind Bruhn *This leads to the correspondence between crystallization and musical intervals: "...crystals accrete as a lattice of atoms and molecules which coordinate themselves according to their specific axes, producing the physical shape of the crystal, so that the sub-atomic harmonic ratios are reflected in the larger, visible dimension of the crystal itself. Over 100 years ago the Berlin crystallographer C. H. Weiss showed that the angles in the crystals and the proportions between their sides and planes could be represented as musical relationships. This was taken further by Victor Goldschmidt, another German scientist, who isolated specific matrices, the exact measurements of which showed that the principles of musical harmony are fundamental to crystalline growth. In the 1940s specific crystals were isolated which follow precise tonalities – such as E sharp major, or E flat minor – and show along their coordinate axes different motives for polyphony and counterpoint. These came to be called the 'crystal tuning forks'." - From pg. 120 of Rhythms of Vision by Lawrence Blair The use of Music for describing Reality is not limited to minerals or crystals however. It shows up in the dimensions of plants, the biological rhythms of human beings, the motions of the planets, and can be extended to describe the cycles of all things.

ALL physical things are Light crystallized into form through musical patterns. Everything is orderly and sustained by Pure Love. We must extend consonant vibrations to others through virtuous actions that resonante with the coherence in their Soul, lest we produce dissonant chords that threaten our very existence. "The intensity of a force is precisely proportionate to the number of units vibrating at that particular pitch. For instance, let fear assail one man, and according to its intensity will be the effect; but let a crowd of men experience fear, see the result in the augmentation of fear, though its source be relatively insignificant. A curtain in a theater, for example, takes fire; one or two persons, cowardly at heart, become afraid through the dominance of the purely animal instinct of bodily preservation; there is actually no real danger, but these two or three persons are sufficient to arouse the unreasoning dread which lies latent in every breast, with perhaps a very few remarkable exceptions. The fire burns nobody; but blind fear, which is extremely contagious among people mutually sympathetic, by reason of the rapidity with which etheric waves transmit all feeling, occasions a terrible panic, during which many severe accidents and many instances of fierce cruelty occur, all because of this sympathetic transfer of feeling starting from one or two augmented or intensified fear-centers, each person being a center emanating the feeling of fear. Were there no counteracting centers of influence in an audience, radiating contrary feelings, the result of a panic would be the total bodily extinction of a very large percentage of the assembled multitude. Thus the human race is immersed in forces whose intensity is vast in proportion to the number of EGOS adding each its quota to the already intense vibration, tending either to love or hate, kindness or cruelty, timidity or bravery. Those who intensify the force of cruelty in the place where they reside, may be strengthening a murders hand to strike the deadly blow in a distant land. This result is brought about through the agency of etheric waves, which transmit forces with undiminished intensity even to uncalculated distances. This phenomenon may be termed transympathetic. They who feel that force called love, which on higher planes is known as sympathy, thrill with waves of force which are already strong, augmenting them or increasing their intensity. They who indulge such sentiments and encourage such forces may stop the falling hand on evil sped. In order to protect ourselves effectually from becoming the dispensers or propagators of deadly force, we must consciously and deliberately relate ourselves by resolute determination, to awaken within us such centers only as are concordantly sympathetic with all force radiating in the interest of universal goodwill, thereby aiding the establishment of universal brotherhood. All ye who feel a longing for a better life or nobler existence draw to yourselves streams of force which they alone feel who have attuned their bodies to the higher harmonies." - John Ernst Worrell Keely (from Amplitude of Force)

Addendum – Hidden Form & Function "Ageless Wisdom is not susceptible to the mutations of time, nor is it primarily a product of man's thinking. It is written by God upon the face of Nature, and is always there for men and women of all epochs to read, if they can." (Paraphrasing a belief of the B.O.T.A.) I enjoy looking for correspondences between various works as I feel that all things are related in some manner that is mutually beneficial. Every person's experience is unique, and therefore every point of view is valuable in some way. For example, the things that are considered "enigmas" in one field may already have been resolved in another. By having the patience to understand another's experience, we change our own percpetions for the better. We can look at the same things with "new eyes" to see things we have never seen before, even though the thing itself did not seem to change. It was "hidden in plain sight" all along. In an attempt to bring harmony to all the seemingly different points of view, let us find the common ground that is sturdy enough for every person to stand on together simultaneously.

The Sunflower Hologram Possible Correspondences:

Polar Coordinates (2-D) (Image from Wikipedia)

The Hourglass Possible Correspondences:

Hyperbolic Cones (3-D) / Pseudospheres (4-D), Dini Surfaces (3-D) & Breather Pseudosphere (4-D?) (Images from WikipediaRU, Wikipedia and 3D-Xplormath)

ABHA Torus Possible Correspondences:

Catenoid (3-D) / Helicoid (3-D) & Clifford Torus (4-D) (Images from Wikipedia and theory.org)

The Ennead

Trigonometry Unit Circle (Image from DUDEfree)

While all of this is only showing similarities in structure, maybe some will find use of it. It seems that form and function are intrinsically connected (at least to some degree) because in order to carry out a particular task, a thing must have (or the capacity to develop) the features that will allow it to do so.

Addendum – Binary Triplet Computing To quote Marko Rodin's website: "State of the art manufacturers of conventional processors have no idea how to prevent heat buildup. Rodin knows how to incorporate, in the conductors, a crucial gap space that creates underpinning nested vortices that are responsible for super-efficiency. One of Rodin's great abilities is to create microprocessor electrical circuitry in which the conductors touch, yet do not short-circuit. He is able to do this as a result of what he calls harmonic shear which creates a natural wall of insulation without requiring any special materials. This natural electrical insulating shear is created by the harmonic phasing activation sequence of electricity. With the Rodin Solution, Marko Rodin is able to navigate on all axes of a Rodin Coil Torus, thus resolving the obstacles to creating artificial intelligence by being able to compute multi-dimensionally. Rodin also adds a new factor of polarity to the binary code by using his binary triplet code which is based on the fact that all numbers begin and end at a point. The basis of the binary triplet is Rodin's binary combinational explosion tree which enables Rodin to map this process through the event horizon of a torus and into the vortex-well singularity where it inverts. No mathematics, other than Rodin's can calculate while inverting, since all existing branches of mathematics self-destruct before emerging on the other side of the toroid. The Rodin Solution harnesses a heretofore unavailable mathematical skill, or language, that takes advantage of number patterns' six different self-referencing axis configurations over the surface topology of the Rodin Coil's toroidal matrix, thus enabling the creation of new revolutionary artificial intelligence hardware and software." While I am not familiar with the details, I could attempt a guess at what all this means. It requires a bit of explaining, but I will try to make it as simple as possible so that everyone can follow along: *First, let's work our way backwards. It is referred to as a Binary Combinational Explosion Tree. A "combinational explosion" is a computer science term for when a mathematical process is prohibitively complex to the point where it takes an astronomical amount of calculation (or an "explosion" of "combinations") to compute, even on the fastest of computers. For example, in data compression, if you were to try to encode or decode information as a fractal it would take an exorbitantly long amount of time to do so. *Secondly, to state it plainly, the main limitation of computer processing power is the fact that the chips heat up. Computer companies have reached the upper limit on the number of transistors (i.e.: the little electrical pathways that do the calculating in the processor) that they can stick on a chip. At 1Ghz your computer processor is a little microwave. Imagine the cooling systems of the large supercomputers? *Thirdly, for very advanced applications such as computer intelligence, you have another thing to consider: It has to be able to handle many complex operations simultaneously (or "compute multidimensionally"). Think of how complex even the simplest of actions is for you to perform. To put it in the context of computers: What would you have to do to build a computer that could decide what action is the most efficient to take next, by referencing an internally generated holographic image

that predicts what types of outcomes that action might have, based upon knowledge gathered in all previously experienced scenarios which are brought to recall instantaneously upon any sensory input that evokes even the slightest association with said experiences? We do this all the time when we use our imagination to anticipate potentials by reflecting upon memories from our past. While at the same time, we are accomplishing many other tasks throughout our day! How do you build a body that can do all of this "from the ground up"? Looking at the richness that Reality must have to embody this amount of coherence certainly puts the phrase "irreducible complexity" and the idea that "God is a watchmaker" in a completely different perspective. Although, it might also be said that there are simple principles that underlie all things. Knowing how the Flux Thruster Atom Pulsar works tells us what the first paragraph in the quotation from the website is describing, and gives us the means to resolve number 2 on the list I've made above. Now what about these other two features? In Vortex Based Mathematics, a Binary Triplet is when the numbers of Family Number Group 1 are of a particular polarity, while those of Family Number Group 2 are of the other. On The Mathematical Fingerprint of God, this causes an oscilation of positive and negative when you travel along The Doubling Circuit (in addition to the polarities of both the left and right sides made by the 9 acting as a mirror). What exactly would this mean in a computer system? I leave it to the reader's imagination. Maybe it means that the Arithmetical Logic Unit (i.e.: the part that does the math inside the processor) is made up of 2 trinary logic gates connected in an flip-flop. Whatever it might be, we do know one thing however: It has to be able to "calculate while inverting". I take this to mean that it is still performing calculations even in a state where a normal computer would be considered "off", just as our bodies still function while we are sleeping. A regular computer uses "binary code", which in the simplest sense means that everything is represented by only 2 digits (or powers of 2). The processor itself is made up of a material called a "semi-conductor" (i.e.: it can be made to conduct electricity or not). This allows the transistor pathways within it to act as little switches to represent our binary code. The flow (or current) of electricity makes it a 1 ("on"), and the lack of current makes it a 0 ("off"). There is more that can be said about the idea of computing in multiple-dimensions as well. While we don't usually acknowledge this feature of the Cartesian Coordinate System, it actually contains 6-dimensions, not 3 (i.e.: we have 3 positive axes and 3 negative ones because the coordinate system itself is made out of 3 number-lines that converge on 0). The coordinate system that makes up the skin of the toroid does something similar. Clockwise and counterclockwise along both of the rings that make up our Y-axis ("the vertical") and X-axis ("the horizontal") accounts for two of these number-lines. The Z-axis accounts for the other one by either moving away from center (forward-positive), or towards center (backward-negative). They all converge on 9. What you do on one hemisphere is reflected in the other at the same time, in the same way that The Doubling Circuit is doing all functions of mathematics at once (i.e.: adding-multiplying through doubling in one direction, and subtracting-dividing through halving in the other direction). Again, how you would make use of this in a computer? I suppose it could be used to do an amazingly large amount of math very quickly as you are doing multiple complex operations simultaneously. And based on the name, it is probably related to the structure of a "binary tree". This type of branching action is probably the key to understanding the function of the brain (as dendrites branch), emulating it, and repairing brain damage (e.g.: reconnecting the nerves of stroke

victims, persons with Alzheimer's, or the mentally handicapped) through Biophysical Harmonics. Humanity has the knowledge to resolve every need of every being in a constructive manner if we work together to do so. Technology (like that provided by The Rodin Solution) can indeed help us there, but it must be handled with the utmost of care. It is a tool that should only be directed towards the enrichment of everyone's lives. Channeling our powers towards killing each other or our environment is to kill ourselves. Let us instead make a smooth transition into a "Heaven on Earth" by being Self-Responsible, by practicing Virtue, and by contributing whatever each of us can to the betterment of ALL. "Man's nature is essentially good. The evil in man springs from fear of the safety and security of his body, and from greed for the satisfaction of bodily desires. Remove these and man will naturally respond to the good in him, for all men seek the peace, happiness and security which only a balanced system of human relations will give to him." - Walter & Lao Russell, A New Concept of the Universe (pg. 123) Resources: Trinary Logic *Simple Introduction *Advanced Introduction Examples of its Use +The Setun Computer made in 1956 by Sergei Sobolev and Nikolay Brusentsov at the Moscow State University. The aim was "to create a small, inexpensive computer, simple in use and service ...". +The ATAMIRI multilingual translation system that is based off of the trivalent logic of the Aymara Language. The work of Katya Walter on Non-Linear Computation seems very close (if not exactly the same in some respects) as to what this type of computer would be doing in my opinion. Maybe the left and right halves of The Doubling Circuit, Family Number Groups 1 & 2, and both hemispheres of the torus are "period-3 windows" that induce "co-chaos" in the circuit and allow the information to selforganize. I also find it plausible that perhaps a similar function takes place with the 1/f "noise" in neurons.

Addendum – Sine Waves It has been briefly mentioned before that The Doubling Circuit produces a series of pure sine waves when projected into 2-dimensions. I will use many different works throughout this paper to attempt to demonstrate the significance of this.

Is it possible that this is really a - / + Phi spiral moving in two directions simultaneously? Dan Winter & The Physics of Phi: Each Phi spiral is actually a series of pure sine waves. It is a well-known principle in physics that any complex wave shape can be created from the sum of simpler pure sine waves with different frequencies and amplitudes. This principle is called the Fourier principle. The Phi spiral is constructed from a series of harmonics with wavelengths that comply with the Golden Mean version of the Fibonacci sequence: 1/ Ф

1

Ф

Ф²

Ф³

0.61803

1.00000

1.61803

2.61803

4.23606

Ф= 1.618033988749894848204586834365638117720309180 When pure sine waves with wavelengths of 1/ Ф, 1, Ф, Ф ², Ф ³ etc. are added together, they will form a perfect Phi spiral. - From pg. 83 of Souls of Distortion Awakening by Jan Wicherink (The book can be downloaded for free here.) The above quotation is actually in reference to the work of Dan Winter, whom also points out the fascinating property of Phi being able to both add and multiply simultaneously. For example: Ф0 = 1 Ф1 = 1.6180339..., Ф2 = 2.6180339..., Ф3 = 4.2360672..., etc. Ф0 + Ф1 = Ф2, Ф2 + Ф1 = Ф3, etc.

Notice that when we add the powers together we are actually doing a recursion process like the Fibonacci Sequence (e.g. 1 + 1 = 2, 2 + 1 = 3, 3 + 2 = 5, etc.). The ratios of the numbers in the Fibonacci Sequence also approximate Phi. Vortex Based Mathematics is said to "do all functions of math simultaneously". This is easy to see in The Doubling Circuit. Going forwards you are doubling (or multiplying), and going backwards on the same path you are halving (or dividing). And what are multiplying and dividing but more complex forms of adding and subtracting? Therefore, in one motion you do all the basic functions of math at once. If Phi (1.618...) can also add and multiply at the same time by going in one direction, does it subtract and divide in the other (maybe through its reciprocal, 0.618...)? Have we given Phi a whole number representation almost like we did with division by 7? The golden section is part of an endless series of numbers in which any number multiplied by 1.618 gives the next higher number, and any number multiplied by 0.618 gives the next lower number:

Like phi itself, this series of numbers has many interesting properties: • each number is equal to the sum of the two preceding numbers; • the square of any number is equal to the product of any two numbers at equal distances to the left and right (e.g. 1.618² = 0.618 x 4.236); • the reciprocal of any number to the left of 1.000 is equal to the number the same distance to the right of 1.000 (e.g. 1/0.382 = 2.618). (To obtain perfectly exact results, the numbers would have to be extended to an infinite number of decimals.)

- From Patterns in Nature by David Pratt It is interesting the way in which Dan Winter applies this feature of Phi within physics. To continue the quote on the previous page: When these Phi spirals circle around the torus they meet and interfere. As a result of this harmonic interference, two new additional waves will be created. What is important to notice is that both new waves will have wavelengths that are again in the Fibonacci series. This allows for the harmonic interference to be nondestructive since the interference will simply result in more harmonics in the Fibonacci series. While destructive interference is the norm in wave interference, the only exception in nature is when the waves interfere with Golden Mean ratio wavelengths! In other words, the Phi spiral can re-enter itself around the torus shape without destroying itself. So the Phi spiral is the universe’s only possible way to nest and become self-organizing. This is exactly how and exactly why stable matter can be formed from electromagnetic energy as a ‘form’ of pure wave interference. Electromagnetic energy in a straight line is what we usually call light. When this same light chases its own tail around the surface of the torus shape we now call it

matter. In other words, the atom is pure electromagnetic energy in a form that we no longer perceive as light but as matter, or to put it in Daniel Winter’s own words; ‘So now we have this dualism that waves in a line are energy and waves in a circle are mass, and because we don’t know how the wave got into a circle from the line and out, we conceive mass as separate from energy. E=MC^2 simply said that yes loop the speed of light back around on itself, and you made mass of energy’. The Golden Mean spirals of the torus shape eventually spiral into a perfect zero ‘still point’ in the nucleus of the vortex that coincides with the nucleus of the atom. So these sine waves implode inwards into increasingly smaller wavelengths. The implosion of the Golden sine waves into smaller and smaller wavelengths not only increases the frequency of the waves, but also increases the speed of the waves to become super-luminal waves (travelling faster than light). According to Daniel Winter, this is what gravity really is, the cascade of Golden Mean electromagnetic sine waves that gain an ever-increasing velocity, thereby exceeding the speed of light. Einstein had always assumed that electromagnetism and gravity were related and Daniel Winter shows us just how this connection is firmly established. When doughnuts are nested to form the electron shells of the atom, the only requirement to continue this form of non-destructive interference is that these doughnuts align according to the Platonic Solid geometries. When these nested doughnuts inside the atom are arranged according to the Platonic solids symmetries, all waves will rush into the center of the atom, creating repetitive, recursive, or fractal patterns that not only shape the electron shells but the nucleus as well. Eventually the fractal patterns disappear into a ‘zero point’ in the nucleus of the atom. The implosion and condensing of the electromagnetic waves into shorter and shorter wavelengths is what gravity really is. In a way, the torus is a miniature black hole that attracts the light into itself, creating gravity. The harmonious wave cycles that follow the Golden Mean ratio, the Phi cycles, may be the origin of the word ‘physical’ (Phi cycle). Physics would simply be (well, not too simply) the study of Phi cycles! - From pg. 84 of Souls of Distortion Awakening by Jan Wicherink There is much that can be said about this from the perspective of many different philosophies and sciences, but we will only put it in the context of Vortex Based Mathematics for now. *In Dan Winter's cosmology, it is sometimes stated that there is a "dimple" or "chalice" on the top of each nested doughnut that allows the "perfect fractal compression of charge" (i.e.: a point on the torus that allows the series of Phi waves of Electricity to converge and produce Gravitation). While approached in a slightly different manner, this point is quite similar to the Northern Hemisphere of the torus associated with the inward motion of the blackhole. It might also be related to the negative Emanations (of the female tetrahedron in The Unit) as well because these are also said to produce Gravity (by creating negative Nested Vortices that link up to make Negative Backdraft Counterspace). *In one mode, Light travels as a wave (like The Doubling Circuit sine wave on the first page). In the other mode, Light folds in on itself to form matter (like when The Doubling Circuit makes the Bounded Infinity inside The Mathematical Fingerprint of God or wraps around the skin of the

torus). Remember that what we term "Light", being an Electric wave, is not wholly linear (like a "ray"). It actually spirals around the Magnetism (the 3 and 6) caused by The Emanations (of number 9). This has been demonstrated physically by the photophoresis experiments of Felix Ehrenhaft (PDF Link). Evidence substantiating this spiral movement was produced by Prof. Felix Ehrenhaft at the Physics Department of Vienna University in 1949 through a process known as photophoresis. Reported in the Acta Physica Austriaca (Vol. 4, 1950 and Vol. 5, 1951), the behavior of barely perceptible particles of matter and gas particles enclosed in glass tubes were observed when illuminated by concentrated light-rays of various frequencies. Observations of this phenomenon were made under conditions varying from high pressure to high vacuum (30 atm to 1 x 10-6 mm Hg [Hg = mercury]) and it was concluded that since the spiral movement of the observed particles was caused by light-rays, the particles had to be propelled along the same spiral path as the light itself (fig. 1.6). It was also determined that light magnetises matter and noted that while some particles spiralled away from the light source, others such as chlorophyll, gyrated towards it. Measurements also determined that the observed particles orbited up to 650 times per second while rotating at 4,000 cps about their own axes, an effect only possible because the calculated energies involved, apparently endowed with antigravitational properties, were 70 times more powerful than gravity. - From pg. 24 of Living Energies by Callum Coats It quickly becomes evident why a properly made and energiezed Flux Thruster Atom Pulsar can transmute matter (through toroidal pinch) and make a reactionless drive. It harnesses the same energy that stars and galaxies use to transmute in their centers, or fly around in their orbits. *And finally, although this may seeem a little "esoteric", the fact that "the Phi spiral can re-enter itself around the torus shape without destroying itself" is the same reason as to why it is said that The Binary Combinational Explosion Tree "enables Rodin to map this process through the event horizon of a torus and into the vortex-well singularity where it inverts. No mathematics, other than Rodin's can calculate while inverting, since all existing branches of mathematics self-destruct before emerging on the other side of the toroid." As it is known in the Blackhole Information Paradox, going into the singularity will make you One with everything (because it "destroys information") unless you have a means to get back out on the other side. For example, if you want to use a blackhole to teleport to other locations (because all locations are entangled in a singularity just like how every point is infinity in a fractal), then you need Coherence.

Richard Merrick & Harmonic Interference Theory: Richard Merrick is the author of the book Inteference: A Grand Scientific Musical Theory (which can be downloaded for free here), and the creator of what he calls Harmonic Interference Theory. While originally developed to describe the physiological basis of music perception, it was extended into a means of describing many aspects of Reality based on the principles of musical harmony. I have assembled some quotations and diagrams from his website and one of his papers, which I will then follow with my interpretation: ...this universal pattern can be easily found using a “Blackman spectral analysis” of two musical tones diverging at a constant rate from unison upward to an octave (see figure). Reproduced here with a built-in function in Adobe Audition®, the analysis reveals the spacing and size of resonant gaps that form naturally according to small whole number harmonic ratios, just as Pythagoras had discovered over 2,500 years ago. Each gap corresponds to a simple musical proportion, such as the 3:2 ratio of a perfect fifth, 4:3 perfect fourth and the highly resonant 5:3 major sixth - the widest gap of all.

To be clear, this is not a random, variable or contrived pattern, but the one universal pattern of interference produced by all harmonic standing waves as they vibrate through any medium. This spectral pattern is not limited to sound only, but exists everywhere harmonics form, including electromagnetic fields, laser light, ... natural vibrations in the Earth, the spacing and sizes of planets in our solar system and the coherent cellular structures of life. We can represent it mathematically using a statistical curve called a first-derivative Gaussian distribution (shown in red in figure below).

You will probably never learn in school how important this harmonic curve really is, but it is present everywhere in Nature. It approximates the change in the number of spots on the Sun, describes the change in diameter of blood vessels in living organisms and estimates the thickness of tree bark as it reduces upward in a tree, to name but a few. As a representation of the velocity change in a Gaussian “normal distribution” (or “Bell Curve”), this one function is the foundation of probability science and the very cornerstone of modern statistics. But while most scientists accept and use this distribution and its strange equation without question, some of us might still wonder what physical process is at work underneath it to cause harmonics to always self-organize in this way. How can we understand what this Gaussian equation is trying to tell us about Nature? When we stop to consider the curve from a philosophical perspective, we can begin to understand the basic principles underneath it, driving the harmonic patterning process. A more intuitive and organic description of the resonance pattern can be expressed as the square of the linear harmonic series as it is curved (one might say carved) by the Fibonacci series. (see diagram below)

As we see in the figure, the interference curve is a natural byproduct of harmonics as they resonate and damp one another. It results from the square of the first twelve frequencies of the harmonic series, as {1, 4, 9, 16, ..., 144}, divided by the first twelve Fibonacci frequencies {1, 1, 2, 3, ..., 144}. This is summarized by the Harmonic Interference function in the figure, expressing the balance between spatial resonance and temporal damping in a simpler, more beautiful symmetric form. With this one equation, we can see how Nature balances itself between the finite and the infinite, harmonizing a closed resonating circle or harmonic wave with an open Fibonacci spiral. It represents nothing less than the geometric harmony of πsquared divided by the golden ratio Φ, otherwise known as the “squaring of the circle.” ...

So when we now take this harmonic interference pattern and geometrically square it again, folding it back upon itself as if reflecting inside a circular container, we arrive at perhaps the most important geometry in the universe and the one guiding pattern at work in the evolution of life – the symmetrical Reflective Interference Model (see diagram below).

So, in summary: *Two tones diverging at a constant rate over an octave (and their associated overtones) interfere to produce a series of resonant gaps. These gaps are the familiar small, whole number musical intervals. They apply to all forms of energy and not just sound. *These gaps can be represented with a mathematical curve ("first-derivative Gaussian distribution", basically a "Bell Curve" that takes into account motion). This curve is "the cornerstone of statistics" and is ubiquitous throughout Nature. *We can make points on this curve by the powers of 2 of the first twelve frequencies of the Harmonic Series divided by the first twelve Fibonacci Numbers. Reflecting the wave upon itself (such as in the diagram to the left directly above) produces the "Linear Reflective Interference" model. We now have two curves, one that represents perfect resonance and one that represents perfect damping. Note the - / + Phi ratios as you near wave amplitude1. Projecting these curves into polar coordinates (such as in the diagram to the right directly above labeled "Polar Reflective Interference") produces this cardioid shape (which looks almost like a polar-view of a toroid with the two curves being the north and south hemispheres). These curves can naturally approximate many organic forms:

1 Because the resonant gaps will shift over if we begin with a different frequency, what would happen if our fundamental was something else, A = 216 Hz for example?

You can make more complicated forms by using a "recursive Hilbert Space" (i.e.: compounding the wave), and can sculpt its shape through "Fourier Transforms" (i.e.: adding or subtracting sine and cosine waves). In 3 steps we've shown how all energy follows musical patterns, how even the subtlest of things approached through statistical analysis are not actually "random", and came to a mathematical tool that can give us the morphology of many seemingly unrelated things of vastly different sizes. There is a lot more to this theory, but we will pause here and continue on.

Nassim Haramein & Scale Unification: Another researcher by the name of Nassim Haramein has independently come to the conclusion that all physical things are made up of what he calls "White Wholes / Black Wholes" that are akin to divisions out of the infinite potential of the "Quantum Vacuum Energy Density" (in the exact same way The Primal Point of Unity is really a blackhole-whitehole pair that is seen as a "coalescence of Aether"). Working in tandem with a group of several others, he has produced a scaling law for "all organized matter" within the known Universe. (The paper describing this can be found here). By plotting the radius of various things throughout Nature by their frequency on a logarithmic scale, we produce a nearly straight line:

(Image from The Resonance Project. Color inverted for clarity.)

On the upper-left of the left-most graph in the diagram above, we have the Planck distance of (10-33 cm). To the lower-right we have the size of the Universe (i.e.: as far out as is "measurable"). There are also several other data points of things whose radius and frequency lie between these two extremes (as shown in the table directly below it). By dividing adjacent points by one another (such as in the graph to the right in the above diagram), we produce a value very close to Phi! Not only do we find Phi inherent in the dimensions of all things, but they even interrelate in increments of Phi. It should also be pointed out that this includes the biological as well. The paper states: "It is of interest that the microtubles of eukaryotic cells, which have a typical length of 2 X 10-8 cm and an

estimated vibrational frequency of 109 to 1014 Hz lie quite close to the line specified by the scaling law and intermediate between the stellar and atomic scales." A similar scaling law is also used to formulate a different atomic model where protons act like miniblackholes. (The paper that describes this specifically can also be found here).

Here is a diagram from the paper showing that when you make a logarithmic scale of objects plotting their radius versus their mass (rather than their frequency), we get another scaling law that shows the deviation between the mass of this "Schwartzchild proton" and the measured value for the "standard model" proton. There is more to this theory as well, but we again pause here. Before we continue on however, I would like to add one more thing. In the case of Vortex Based Mathematics, every single one of these data points on each of these graphs may also be thought of as a toroid or collection of tori.

Hartmut Müller & Global Scaling Theory: We aren't going to go into much depth as to what this particular theory consists of, although at its root, it is quite similar to the scaling laws given in the above theory. To quote an introductory article about it: The scientific division of labour according to the example of large-scale industries also had its positive consequences ("Nothing so bad that it wouldn't be useful" - an old Russian saying goes). The physical compatibility of completely different mathematical models made it necessary to bring precision of physical measurements to unprecedented heights. Over decades a priceless colossal data base was accumulated. It contains the spectral lines of atoms and molecules, the masses of the elementary particles and atomic nuclei, atomic radii, dimensions, distances, masses and periods of revolution of the planets, moons and asteroids, the physical characteristics of stars and galaxies, and much more. The need for measurements of the highest precision promoted the development of mathematical statistics which in turn made it possible to include precise morphological and sociological data as well as data from evolutionary biology. Ranging from elementary particles to galactic clusters this scientific data base extends over at least 55 orders of magnitude. Yet, despite its tremendous cosmological significance this data base has never been the object of an integrated (holistic) scientific investigation until 1982. The treasure lying at their feet was not seen by the labour-divided, mega-industrial scientific community. First indication of the existence of this scientific gold mine came from biology. As the result of 12 years of research Cislenko published his work "Structure of Fauna And Flora With Regard to Body Size of Organisms" (published by LomonosovUniversity Moscow, 1980). His work documents what is probably the most important biological discovery in the 20th century. Cislenko was able to prove that segments of increased species representation were repeated on the logarithmic line of body sizes in equal intervals (approx. 0.5 units of the decadic logarithm). The phenomenon is not explicable from a biological point of view. Why should mature individuals of amphibians, reptiles, fish, bird and mammals of different species find it similarly advantageous to have a body size in the range of 8 - 12 cm, 33 - 55 cm or 1,5 - 2,4 m? Cislenko assumed that competition in the plant and animal kingdoms occurs not only for food, water or other resources, but also for the best body sizes. Each species tries to occupy the advantageous intervals on the logarithmic scale where mutual pressure of competition also gave rise to crash zones. Cislenko, however,was not able to explain why both the crash zones and the overpopulated intervals on the logarithmic line are always of the same length and occur in equal distance from each other, nor could he figure out why only certain sizes would be advantageous for the survival of a species and what these advantages actually are. Cislenko's work caused the German scientist Dr. Hartmut Müller to search for other scale-invariant distributions in physics, as the phenomenon of scaling has been well known to high-energy physics. In 1982 he was able to prove that there exist statistically identical frequency distributions with logarithmic-periodically recurrent maximums for the masses of atoms and atomic radii as well as for the rest masses and for life spans of elementary particles. Müller found similar frequency distributions along the logarithmic line of the sizes, orbits, masses, and revolution periods of the planets, moons and asteroids. Being a mathematician and physicist he did not fail to recognise the cause for this phenomenon in

the existence of a standing pressure wave in the logarithmic space of the scales/measures. - From A Universe of Scale by Sepp Hasslberger Within the theory Hartmut Müller also demonstrates an interesting correlation between time and Phi by using continuous fractions (the mathematics of which has been excluded here for sake of brevity). To paraphrase: In Nature, The Golden Mean is built "automatically" almost everywhere. It is the most basic recursion (growth) process. In Global Scaling all natural structures are granular and are finally composed of gravitons. Therefore the creation of structure is eqivalent to the clustering of such gravitons. Or in other words, in Nature all mass elements can be represented by a whole natural number. No decimal numbers are required. In Nature no decimal numbers exist. Also The Golden Mean is not a decimal number as often written in text books. Moreover Nature is "only" capable of constructive or destructive growth processes which happens always at least in parts recursively. ... Nature "calculates" – or better, approximates – The Golden Mean step by step according to this very simple rule. This consumes time. And with every time step the value of Ф gets more and more accurate. This conclusion is very important and should not be underestimated. One direct conclusion is, that time somehow must elapse in quanta rather than being absolutely continuous. This important conclusion is also supported by Global Scaling, as we will see later in this document. - From pg. 16 (with slight corrections) of The Global Scaling Theory: A Short Summary by André Waser What an intriguing quotation, and how especially relevant it is to the topic at hand. The idea of time being related to Phi leads us to another one...

Stephen J. Puetz & Unified Cycle Theory: Stephen J. Puetz is author of a book entitled The Unified Cycle Theory: How Cycles Dominate the Structure of the Universe and Influence Life on Earth. Although I haven't read it, the general premise is given on his webpage: The Unified Cycle Theory separates physical cycles into four distinct categories. These cycles heavily influence nature and mankind in many different ways. The various categories follow. •

Milankovitch Cycles - Gravitational forces modulate these cycles related to Earth's eccentricity, obliquity, and precession.

•

Solar Cycles - Magnetic cycles internal to the Sun regulate frequencies of 27 days, 11 years (Schwabe cycle), 88 years (Gleissberg cycle), and 208 years (Suess cycle).

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Geomagnetic Cycles - The solar cycles act as the primary modulators of magnetic cycles on Earth. However, a semi-annual cycle, the eclipse cycle, and the lunar cycle also play minor roles in geomagnetic oscillations.

•

EUWS Cycles - A set of mysterious cycles, named the Extra-Universal Wave Series (EUWS), link themselves precisely by a factor of three. The EUWS cycles dominate all areas of the universe. These cycles reflect their presence throughout all timescales on Earth.

Events correlated to the EUWS cycles include global temperature changes, the carbon cycle, geological timescales, evolution of species, extinctions, the rise and fall of civilizations, and stock market fluctuations. All major long-term cycles in human behavior link themselves to these mysterious cycles. Physical cycles affect human behavior through emotions. As collective emotions shift, human social patterns oscillate between states of confidence, cooperation, and trust and times of fear, individualism, and mistrust. So what exactly is this "Extra-Universal Wave Series"? From a review: For the “unification” whereof he speaks, Puetz simply takes the number 22.176 billion years and divides it by 3, getting 7.392 billion years, and then divides the result again by 3 to get 2.464 billion years, and so on. At first, I was quite skeptical, thinking that all this was mere numerology. That was until Puetz used the calculation to show that all of the geologic epochs fell right into this cyclic behavior. My favorite, because I have worked with Quaternary geology for decades, is the 41,728-year EUWS cycle. The data for the eastern equatorial Pacific sea temperature for the last million years shows remarkable agreement with that cycle (Fig. 1). The precision surpasses that of the 41,000-year cycle observed long ago by Milankovitch (1941) as well as the more recent correction to 42,000 years, attributed to cyclic variations in the axial tilt of the earth. Of course, in nearly all cycle theories, various subcycles can exhibit constructive and destructive interference. This results in extra peaks or diminished peaks that at first appear to be random variations within the major intercyclic periods. As in all of science, real data generally give skeptics plenty of chance for doubt. Puetz handles part of this intercyclic noise with something called the Turning Point Distribution Principle: “This principle describes how the EUWS cycles reverse direction at tops and

bottoms. Variations from theoretical turning points distribute themselves in a nonrandom manner. …If a EUWS cycle misses its theoretical turning point, alternatives include theoretical turning points of its sub-cycles. This tendency produces non-random, stair-step distributions” (p. 5). Being a neomechanist, I can appreciate this fully. The ideal cyclic motion of a simple pendulum, for instance, is either forward or back. To get subcycles, there would have to be additional causes (other bodies impacting the pendulum). Puetz goes on to calculate 13,909-year, 4,636-year, 1,545-year, 515-year, 172-year, 57-year, 19-year, 6.36-year, and 2.12-year cycles. There even are 258.11-day, 86.04-day, and 26.68-day cycles, all of which he supports with data drawn from the literature, including the stock market. Given his focus, there probably is some cherry picking involved, but too many of the peaks and troughs are precisely as predicted by the calculation. I haven’t checked all the charts for accuracy, but the references for the raw data are meticulously included, with many readily available on the Internet. You can judge for yourself. Reconstructed temperature variations for Greenland appear to be remarkably in agreement with the 1,545-year cycle (p. 281). He shows the correlation between sunspot and climate cycles with the rise and fall of various civilizations. And most pertinent to our present situation, he shows that the 57-year EUWS cycle coincides with the highly controversial Kondratieff Wave cycle in economics. The 57-year cycle postdicts four market crashes in a row: 1720, 1778, 1835, and 1892. He calculates that the possibility of this happening by chance is 1 in 10 million (1/57 X 1/57 X 1/57 X 1/57). The 1929 market crash came one 19-year panic cycle early. Most surprising: Both the 57-year and 172-year cycles predicted a major depression to begin in 2007. Sound familiar? He also points out that “Since 1934, the Dow Jones Industrial Average correlated with the 2.12-year EUWS frequency for 34 cycles.” If you think there is profit potential in all this, you can get Puetz’s financial newsletter (http://www.uctnews.com), which at the moment, is more bearish than anything else I have seen. The book goes into some interesting esoteric detail: Who knew that the price of rice in China pretty much followed the 172-year cycle ever since 976 AD? Who knew that the rise and fall of civilizations followed a 515-year EUWS cycle? One needs to appreciate a lot of data in chart form and bear with quite a bit of necessary repetition. Some of the conclusions will be a stretch for sociologists, historians, and others more knowledgeable about the details, but the shear extent of this first attempt is outstanding. It has only been mentioned briefly before that Vortex Based Mathematics shows that time occurs in frames because the Phasing of The Emanations happens in thirds. Maybe this is related to the division of the "Extra-Universal Wave Series" by 3 to make the other cycles? If the positive Emanations (of the male tetrahedron in The Unit) produce momenuntum and spin, will their patterns show up in the cyclical motions of all things? Just as we derive our units of space from the seemingly static dimensions of objects around us, we derive our units of time from their dynamic rhythms. And whether we keep time through the periodic oscillations of the Earth or an atom of Caesium, we can see that all material bodies are both made up of a Bounded Infinity, and moving along a Spin Continuum.

Wave Theories: Related to the idea of physical forms really being collections of processes are the various models that propose matter is entirely composed of "spherical standing waves": *Milo Wolff's Space Resonance Theory (and by extention Daniel Fitzpatrick Jr.'s A-Laws) *Gabriel LaFreniere's Matter Waves *Yuri Ivanov's Rhythmodynamics Theory *Caroline Thompson's Phi Wave Aether Theory etc. It is important to keep in mind that the waves on an oscilliscope (which includes sine waves) describe a periodic motion. It can be applied to axial spin/rotations, beating/pulsations, orbits/spiralling, cycles consisiting of two or more states, and other like phenomena, not just "waves" in the traditional sense.

Addendum – Changing Axes There are only six variations possible in the assignment of axes. We will cover all of them here, and show the changes that take place when moving between them. But first, a simple recap:

Now for the grids:

Axes: Y – 1-8 / X – 4-5 / Z – 2-7 Domain Schematic:

1-6-5-2-9-7-4-3-8-8-3-4-7-9-2-5-6-1 8-3-4-7-9-2-5-6-1-1-6-5-2-9-7-4-3-8

Axes: Y – 2-7 / X – 1-8 / Z – 4-5 Domain Schematic:

2-3-1-4-9-5-8-6-7-7-6-8-5-9-4-1-3-2 7-6-8-5-9-4-1-3-2-2-3-1-4-9-5-8-6-7

Axes: Y – 4-5 / X – 2-7 / Z – 1-8 Domain Schematic:

4-6-2-8-9-1-7-3-5-5-3-7-1-9-8-2-6-4 5-3-7-1-9-8-2-6-4-4-6-2-8-9-1-7-3-5

Axes: Y – 1-8 / X – 2-7 / Z – 4-5 Domain Schematic:

2-3-1-4-9-5-8-6-7-7-6-8-5-9-4-1-3-2 7-6-8-5-9-4-1-3-2-2-3-1-4-9-5-8-6-7

Axes: Y – 2-7 / X – 4-5 / Z – 1-8 Domain Schematic:

4-6-2-8-9-1-7-3-5-5-3-7-1-9-8-2-6-4 5-3-7-1-9-8-2-6-4-4-6-2-8-9-1-7-3-5

Axes: Y – 4-5 / X – 1-8 / Z – 2-7 Domain Schematic:

1-6-5-2-9-7-4-3-8-8-3-4-7-9-2-5-6-1 8-3-4-7-9-2-5-6-1-1-6-5-2-9-7-4-3-8

It is very important to keep in mind how every feature is inextricably connected. When altering one thing, we simultaneously affect the others. This is especially evident as we switch axes. The first thing you might have noticed is that the reciprocals have changed:

The Nexus Keys change as well because the repeated numbers within the reciprocals are both the middle and ends of each of them. By paying attention to the numbers within the Domain Schematics when tracing out their patterns with your finger, you can see that they are the Nexus Keys associated with each Polar Number Pair. This has been described once before, but we will reiterate their features here for sake of clarity (using Polar Number Pair 1-8 as an example): *Each sequence has 18 digits (to account for both the positive and negative 1-9). *There are two sequences, each being the mirror of the other. One is forwards (beginning on a positive 1, 2, or 4 – i.e.: all the numbers on the right side of The Mathematical Fingerprint of God), and the other is backwards (the Antithesis, beginning on a negative 8, 7, or 5 – i.e.: all the numbers on the left side of The Mathematical Fingerprint of God).

Technically, there are many Nexus Keys on the toroid skin, just as there are many Doubling Circuits and Gap Spaces, but I have only highlighted two of them here.

*Taking half of one of the sequences and moving towards the 9 in the middle from each end gives you all the Polar Number Pairs:

If the sequences that make up the Nexus Keys are intact, then all of your multiplication tables are full rings (because it is literally made up of many positive and negative X-axes or Y-axes in a row, as depicted in the diagrams below). This is handy for quickly checking the accuracy of your grid and toroid.

Y-axes

X-axes

As for the set of grids shown earlier, each Domain Schematic corresponds to the Polar Number Pair of the Y-Axis for those in the first row, and with the Polar Number Pair of the X-Axis for those in the second row. Although it might be a little hard to see within the diagrams, by comparing the two rows of grids, one notices that the stagger of the Nested Vortices also shifts over to accomadate different pairs of axes (e.g.: in order for Polar Number Pair 4-5 to be the X-Axis and 2-7 to be the Y-Axis at the same time, the layout of the Nested Vortices must be like the orange grid to the right):

There are a few other little patterns to pick out of the grids, like the oscillation of the Polar Number Pairs around the 9 in adjacent Nested Vortices, and the mirroring of the reciprocals. Try to see if you can find any more.

Oscillation of Polar Number Pairs

Mirrored Reciprocals

Addendum – Energy Dynamics This section is just some miscellaneous observations and ideas about energy. Much of this needs to be "fleshed out" still, so this paper will most definitely be extended at a later date. Polarity: Polarity is present all throughout the mathematics and associated physics. Halving (Subtracting-Dividing)

Doubling (Adding-Multiplying)

Left-Side of The Mathematical Fingerprint of God (Negative)

Right-Side of The Mathematical Fingerprint of God (Positive)

Backwards Motion of Binary Triplet (6)

Forwards Motion of Binary Triplet (3)

Negative Numbers (1-9)

Positive Numbers (1-9)

Reverse Doubling Circuit (on toroid)

Forward Doubling Circuit (on toroid)

CW Rings (of X or Y axes)

CCW Rings (of X or Y axes)

Antithesis Nexus Key

Nexus Key

Southern Hemisphere (CW Rotation)

Northern Hemisphere (CCW Rotation)

Whitehole (Decompression-Explosion)

Blackhole (Compression-Implosion)

Positive Nested Vortice (Outward)

Negative Nested Vortice (Inward)

Female Tetrahedron of The Unit

Male Tetrahedron of The Unit

Negative Emanation (Gravity)

Positive Emanation (Spin & Momentum)

Cooling

Heating Etc.

Some of these might have to be switched around a little bit, but hopefully this table gives some idea as to the importance of polarity and how all the characteristics of the toroid balance out. Exploring The Unit:

While we haven't spoken much about it, The Seed of Life pattern to left is actually a polar-view of the torus (you can find more about The Seed of Life here). It takes the form of a hexagon. The hexagram to the right is a flat projection of The Unit viewed from the same axis. It too makes a hexagon. Because of this feature, we may nest hexagrams inside of one other, each decreasing or increasing in size logarithmically as we move inward or outward. Notice that with each successive iteration it seems to rotate 90 degrees.

Alternatively, one might also think of it as twisting CW or CCW in 30 degree increments (e.g.: 30, 60, 90, 120, etc.). 12 turns of 30 degrees each makes one full rotation (360 degrees). In 3-dimensions the Cosmic Eggbeater motion of The Unit is a little more complex.

The orange and green rings depict the rotation of the Male and Female Tetrahedra of The Unit along The First 9-Axis in red. At the same time, they spin together in the way depicted by the yellow ring to make The Second 9-Axis in blue. It is this blue line that is the axis of spin for the two vortexes that make up The Hourglass of the torus. Since it is spinning in two different planes at once, like the crank and wisks of a mechanical eggbeater, this is how it gets its name. It also probably acts somewhat like a differential gear system. I feel that it cannot be emphasized enough that understanding the dynamics of The Unit is fundamental to getting a complete understanding of the toroid because: *The centroids of each tetrahedra combine to form the Aeth Coalescence. *The angles of the tetrahedra determine the Phasing of The Emanations, and thus the polarization and shape of the tiles that make up the skin of the torus. (Here is an animation showing their relation to the Binary Triplet). *The rate of rotation of each tetrahedra and their combined motion determines the frequency of the torus, and the two axes of spin described above. We also know that the ratcheting of The Unit is the cause of the harmonic star patterns in 2dimensions (like those shown in the Magic Squares). This idea gives an interesting avenue for further exploration because only particular star patterns can be considered harmonious, and there are certain rules to derive them (1, 2). Another interesting thing to note is that the tetrahedron has 7 axes of spin symmetry that all seem to converge on the centroid:

(Image from Math Expression. There is a beautiful video there demonstrating these rotations.)

Exploring the Star Patterns: (Please see table and description on the following pages).

2 divisions of 3 divisions of 4 divisions of 5 divisions of 6 divisions of 8 divisions of 9 divisions of 10 divisions 12 divisions 15 divisions 18 divisions 180-degrees 120-degrees 90-degrees 72-degrees 60-degrees 45-degrees 40-degrees of 36-degrees of 30-degrees of 24-degrees of 20-degrees 3-sides / 60-degrees (Triangle)

4-sides / 90-degrees (Square)

5-sides / 108-degrees (Pentagon)

6-sides / 120-degrees (Hexagon)

8-sides / 135-degrees (Octagon)

9-sides / 140-degrees (Nonagon)

5-points (Pentagram)

6-points (Unicursal Hexagram)

8-points (Octagram)

[5 / 2]

[Irregular]

[8 / 3]

2 Triangles

12-sides / 180-degrees

15-sides / 156-degrees

9-points (Nonagram)

12-points

15-points

[9 / 4]

[12 / 5]

[15 / 7]

3 Triangles

10-sides / 144-degrees (Decagon)

2 Pentagrams

4 Triangles

18-sides / 160-degrees

3 Pentagrams 2 Nonagons or 5 Triangles or 6 Triangles (not shown) (not shown)

Key To Table: *Because 360 is a "highly composite number" it has many factors (i.e.: can be divided evenly into many parts or groups). The grey box shows how many increments each circle has been divided into. *By linking up each of these divisions we can form polygons. The light green box shows how many sides each of these polygons will have, and the number of degrees in each of their internal angles. *Some of these also naturally form star polygons. The light blue box shows how many points each of them will have. *The light purple box shows alternate star patterns based on combinations of shapes. Notice that if we take the table as a whole, we account for all of the star patterns associated with the Magic Squares. Possible Connections: The orange box contains what is called a "Schläfli symbol", which looks like a ratio. This tells us different things about that star polygon, and is a handy way to link together all of the features described above. To use the pentagram as an example:

The Schläfli symbol for a pentagram is [5 / 2]. The number on the left tells you how many points it has, and the number on the right tells you how many you skip to draw the line segments that make up the star itself. Lines within a star polygon always cross themselves. The two numbers in the Schläfli symbol follow particular rules: *They only make a regular star polygon if they are coprime (i.e.: their greatest common factor is 1). *If the second number is a 1, then they will make a regular polygon. For example, [5 / 1] would make a pentagon instead of a pentagram. There are other rules for "irregular star polygons" (i.e.: those made up of the same shape compounded – for example, a hexagram is made of two triangles) and for "degenerate star polygons" (i.e.: those made up of only line segments radiating away from the center).

(Diagram from Wikipedia) In Aerodynamics (pg. 16), it is stated that "stars are logarithmic spirals". This is easy to see. We make "turns" in drawing out our stars, and every spiral has a certain number of "turns". If the star is unicursal (i.e.: begins and ends at the same point), then it describes a spiral whose begining and end are one and the same. How archetypal! "Alpha, Omega."

A spiral and a star of 5 turns. Keeping this in mind, it is interesting to note the similarity of the rules that stars follow to the ones that flowers follow in making their seed spirals. To quote this webpage: In many cases, the head of a flower is made up of small seeds which are produced at the centre, and then migrate towards the outside to fill eventually all the space (as for the sunflower but on a much smaller level). Each new seed appears at a certain angle in relation to the preceeding one. For example, if the angle is 90 degrees, that

is 1/4 of a turn, the result after several generations is that represented by figure 1.

Of course, this is not the most efficient way of filling space. In fact, if the angle between the appearance of each seed is a portion of a turn which corresponds to a simple fraction, 1/3, 1/4, 3/4, 2/5, 3/7, etc (that is a simple rational number), one always obtains a series of straight lines. If one wants to avoid this rectilinear pattern, it is necessary to choose a portion of the circle which is an irrational number (or a nonsimple fraction). If this latter is well approximated by a simple fraction, one obtains a series of curved lines (spiral arms) which even then do not fill out the space perfectly (figure 2). In order to optimize the filling, it is necessary to choose the most irrational number there is, that is to say, the one the least well approximated by a fraction. This number is exactly the golden mean. The corresponding angle, the golden angle, is 137.5 degrees. (It is obtained by multiplying the non-whole part of the golden mean by 360 degrees and, since one obtains an angle greater than 180 degrees, by taking its complement). With this angle, one obtains the optimal filling, that is, the same spacing between all the seeds (figure 3). This angle has to be chosen very precisely: variations of 1/10 of a degree destroy completely the optimization. (In fig 2, the angle is 137.6 degrees!) When the angle is exactly the golden mean, and only this one, two families of spirals (one in each direction) are then visible: their numbers correspond to the numerator and denominator of one of the fractions which approximates the golden mean : 2/3, 3/5, 5/8, 8/13, 13/21, etc. These numbers are precisely those of the Fibonacci sequence (the bigger the numbers, the better the approximation) and the choice of the fraction depends on the time laps between the appearance of each of the seeds at the center of the flower. This is why the number of spirals in the centers of sunflowers, and in the centers of

flowers in general, correspond to a Fibonacci number. Moreover, generally the petals of flowers are formed at the extremity of one of the families of spiral. This then is also why the number of petals corresponds on average to a Fibonacci number. Are there rules like this for The Sunflower Hologram? It looks very much like the harmonic motions of a pendulum (see diagram below), and this most certainly is not coincidental as the torus skin is generated by periodic oscillations of energy. To be more specific, it is made by the frequency of The Unit (i.e.: the rate of rotation of the two tetrahedra along The First and Second 9-Axis). It is likely that this motion could be expressed as a ratio as well. And if we treat it as a musical interval, perhaps we can see which motions (and star patterns) are harmonic, thus producing stable tori.

Rose Curves produced by a pendulum. (Image from Wikipedia) Vortex Based Mathematics is Harmonics, just as Nature is Music. To quote another interesting article: In his book Harmonograph: A Visual Guide To The Mathematics of Music, author Anthony Ashton shows us how musical ratios interact to create geometric figures. Using a 19th century device called a harmonograph, Ashton set the frequencies, rotational direction and amplitudes of two vibratory arms connected to a pendulum ink pen to draw an assortment of shapes on paper. When the vibrating arms were concurrent (vibrating in the same direction), the pen drew a variety of nested circles

and cardioid or heart shapes. But when the vibrations were countercurrent (moving in opposite directions), the pen drew triangles (octave), squares (double octave), pentagons (perfect fifth), septagons (perfect fourth) and so on. When an open phase was used, spirals (unison), shells and eggs (near unison) were drawn. These geometric patterns of harmonic resonance are also found on vibrated plates of sand, powder in water and even plasma jets. More recently, they have been demonstrated to occur in spinning buckets of water, in the eyes of powerful hurricanes and beyond the Earth on other planets and in stellar clouds. Caused by the interplay between harmonic resonance and damping in all standing wave systems, the same simple geometrical shapes occur everywhere and in everything, whether animate or inanimate. Hopefully this document has stimulated some ideas that may be useful in connecting many aspects of VBM together, along with those of Science and Spirituality in general. Thank you for reading!

Addendum – Coil Considerations This section will cover things one should keep in mind when building variations of the coil. This is just my opinion, but hopefully it will be of help to others whom are experimenting with this aspect of the Flux Thruster Atom Pulsar. Star Pattern Variations: It was briefly touched upon in a previous section that the features of the traditional "Rodin Coil" embody One Quanta (i.e.: a 36 X 36 grid). To quickly recap: *The circle (or outer perimeter of the doughnut you are winding upon) is divided into 36 increments because it is 4 repetitions of The Harmonic Number Sequence (totaling 36 digits). This might also be thought of as one full ring of your X-Axis (made up of Polar Number Pair 4-5 in this case). The winding pattern is a polar-view of our toroid. *The wires themselves make two 12-pointed stars because they embody both the forward and reverse Doubling Circuits across the entire torus (12 forward + 12 reverse = 24 Doubling Circuits on a 36 X 36 grid. If you add 12 more to account for The Gap Space notice that you again have 36 digits total). See diagram below.

Northern Hemisphere of torus (CCW). The Harmonic Number Sequence / X-Axis is shown in pink, forward and reverse Doubling Circuits are in green and orange, and The Gap Space is in red. *The numbers you move through to make each turn of the wire are your Family Number Group 1 and 2, while the space you leave around them is Family Number Group 3.

Keeping all these things in mind, it seems that the coil scales like our grids: Number of divisions around doughnut

9

18

27

36

45

Number of turns per wire

3

6

9

12

15

etc.

The top row is also the number of digits that make up each side of our grid (e.g.: 9 X 9, 18 X 18, 27 X 27 etc.). Remember how the grids follow an Inverse Square Law? This row increases in increments of 9. The bottom row is also the number of forward or reverse Doubling Circuits for that grid (e.g.: in the case of 9 X 9, there are 6 Doubling Circuits, 3 forward and 3 reverse). This row increases in increments of 3. The highlighted numbers correspond to the "Rodin Coil" everyone is probably familiar with. The other numbers are possible variations of the star patterns that one forms with the two conductors of their coil. To give some examples:

9/3 This is the 9 X 9 grid coil. It is the simplest possible to make. Similar to the Triadic Coil of Gregor Arturo?

18 / 6

27 / 9

This is the 18 X 18 grid coil. It This is the 27 X 27 grid coil. Its might be a bit tricky to make winding patterns are like the because of its unicursal more familiar Ennead stars. hexagram windings.

The missing numbers from each of these designs are for The Gap Space.

Angle & Ratio:

The image to the left is the regular winding pattern. The image to the right is of The Doubling Circuits on the ABHA torus, and essentially what you are trying to accomplish with those wires. Because the angular relationship of the wire is what keeps the electricity from interfering with itself, the inner and outer diameter of our toroid has to be in a specific ratio. This allows our wire to arc in a particular way (thus creating the Longest Mean Free Path of Least Resistance), and allows one to control the waveform of the electricity itself.

The only place I have seen this mentioned is within a wonderful set of videos by Gregor Arturo, wherein he provides the following diagram:

It seems that for all the star pattern variations given above, these ratios would also change. Perhaps it would be easier to think of it in 3-dimensions rather than the way I have depicted it above...

(Image from Math is Fun)

Changing the size of the cross-sectional diameter of the torus, r, would be equivalent to changing the size of your Y-Axis ("vertical"); changing the diameter of the torus itself, R, would be a change in the size of your X-Axis ("horizontal"). Since altering these two radii changes the size of the grid that makes up the torus (i.e.: its surface area), it also alters the features it has (e.g.: the number of Doubling Circuits, Nested Vortices, etc.) according to the scaling law. It seems likely that in the case of the Flux Thruster Atom Pulsar, different wire geometries, dimensions of tori, and ways of powering it would yield different pressures in the center of it, thus allowing one to transmute elements, or perhaps even impart structure to the Aether to form elements "from scratch" when done correctly. With but a few variables we have enough variation to create many things. Augmenting the Effects of the Coil: In an article written for Nexus Magazine by Marko Rodin, it gives the following method for producing a Changing Aspect Ratio: "The conductor ideally should be wider at the torus equator and very thin, then tapering as it becomes thicker as it heads for the inside spindle of the inside magic circle. A good example of this is a wavy ribbon. This is called a changing aspect ratio. Thus the conductor width and density change with the inverse square law, while keeping the same amount of mass. Using a primitive technique, although still highly effective, this was accomplished by soldering together a multi-filament winding with the threads laying side by side on the outside and stacked on the inside."

Also, within the Gregor Arturo videos there are interesting ideas about different materials one might want to use (both as a conductor and as a base for winding their coil) to get different effects. A Reiteration of Expected Effects When Properly Constructed and Energized: By "properly constructed" we mean that the coil embodies the following features: *The area taken up by The Gap Space is equal to that of the two conductors, or Doubling Circuits. *The curve of the conductors is very gradual around the toroid and has the least acute incline as possible (i.e.: horizontal, not vertical). This creates the Longest Mean-Free Path of Least Resistance, the angular relationship that keeps the electricity from interfering with itself thus making it "super-conductive". *The conductors have a Changing Aspect Ratio, becoming skinny as you near the center of torus and flat along the outside of it, without changing mass (i.e.: change the shape, not the size). While the above two features are absolutely necessary, this allows it to work more efficiently, creating the Electrical Venturi phenomenon. And by "properly energized" we mean that it is powered with Sychronized Electricity, DC pulsed in time to the Binary Triplet (Number Activation Sequence). This creates Harmonic Shear, when the electricity moving around each conductor "shears" (or moves against) the other. This makes: *An (electrostatic?) wall around the wires that allows you to have them next to each other without insulation and without short-circuiting. *Nested Vortices in the magnetic field that keep the conductors cool, by acting as little heat sinks (i.e.: pumping the heat away from them). Since connecting each adjacent Nested Vortice is what produces the Spires, it seems that perhaps there will be a temperature gradient following this path (e.g.: generally cool on the outside, and hotter on the inside as you follow this equiangular spiral). All of the above is possible because of The Unit, a star tetrahdron in the center of the toroid that allows it to form. It is made up of a Male and Female tetrahedron (that produce Positive and Negative Emanations, respectively). The tetrahedra interpenetrate one another, spinning at different rates along The First 9-Axis. Simultaneously, they rotate together along another axis, The Second 9Axis, at 90-degrees to the First. This is the axis of spin for The Hourglass, the combination of the vortex on the Northern Hemisphere of the torus (associated with the blackhole), and the one on the Southern Hemisphere (associated with the whitehole). This double rotation along two axes is called a Cosmic Eggbeater. This is the "frequency" of the torus, the fundamental tone that produces the compression and decompression Ennead stars that generate its surface. In the case of the Flux Thruster Atom Pulsar, we are moving backwards (producing the toroid first, attempting to create a space for The Unit) instead of as Nature does, from the inside-out and sustaining it with LOVE. (Note here that I use the term "Love" in the sense of an archetypal Creative Principle that gives rise to reality itself, and not only in the sense of the human emotion). The Emanations are the Creator's Will (and thus your will on the deepest level because the same Intelligence both centers All things, surrounds All things, and is All things; every manifest thing is a type of torus). The Positive Emanations produce spin and momentum. This shows up in the Flux Thruster Atom Pulsar as the positive Nested Vortices that create the Torsion B-Field (i.e.: it causes the magnetic

field to rotate). The Negative Emanations produce compression. This shows up in the Flux Thruster Atom Pulsar as negative Nested Vortices that create the Negative Backdraft Counterspace (i.e.: a movement towards the center of the coil, a gravitational field). The centers of both tetrahedra fuse to form the Aeth Coalescence, a plasmoid, or pinch in the very center of the torus that produces fusion (Toroidial Pinch). Out of this point also comes a jet of energy (the Centroid Nozzle) that can be used for propulsion, thus making it a Reactionless Thruster. It is "reactionless" because we did not produce this force through some sort of mechanical input (just like the spin of the magnetic field), but only by moving energy in a particular geometry. All of it happens because of The Emanations.

VORTEX BASED MATHEMATICS REFERENCE KEY

The Sunflower Hologram [Polar-View, Pinched Center]

One Quanta (18 X 36 Grid – Diamond Carpet) Nested Vortices and Spires not labeled.

The Doubling Circuit (in red) *Forward (Doubling): 1 – 2 – 4 – 8 – 7 – 5 *Reverse (Halving): 1 – 5 – 7 – 8 – 4 – 2

The Gap Space (in blue): 3 – 3 – 9 – 6 – 6 – 9 Positive numbers are in white, negative are in black.

[Side-View, Cross-Section] Notice The Hourglass with The Primal Point of Unity (in yellow), The First 9-Axis (in orange) and The Second 9Axis (in purple).

The Polar Number Pairs: 1-8, 2-7, 4-5 These make up the X-Axis (in green), Y-Axis (in gold), and X-Axis (not shown).

The Nexus Key (in magneta and cyan) *Forward: 1-6-5-2-9-7-4-3-8-8-3-4-7-9-2-5-6-1 *Reverse (Antithesis): 8-3-4-7-9-2-5-6-1-1-6-5-2-9-7-4-3-8 They vary depending upon what numbers make up each axis. Note: While I have attempted to be as consistent as possible with the color-codings amongst the different views of the torus in relation to these number sequences, they serve more as a visual guide to how these features interact.

Family Number Group 1: 1 – 4 – 7 Family Number Group 2: 2 – 5 – 8 Family Number Group 3: 3 – 9 – 6

ABHA Torus w/ The Unit [Polar-View, Open Center]

[Isometric-View]

Blank Templates:

Top-Left: The Ennead Star (Nonagram) Bottom-Left: The Unit (Hexagram) Top-Middle: The Nexus Key (Enneagram) Bottom-Middle: Domain Schematic 1-8, 2-7, and 4-5 Right: The Mathematical Fingerprint of God

Tile to make larger grids. Reverse to change stagger of Nested Vortices.

To draw The Sunflower Hologram, array circles of equal diameters along a point on the circumfrence of one of those circles at regular intervals: 9 circles (9 X 9 Grid) = 40-degrees, 18 circles (18 X 18 Grid) = 20-degrees, 27 circles (27 X 27 Grid) = 13.333...-degrees, 36 circles (36 X 36 Grid) = 10-degrees, etc. Following the scaling law: 9 X 9 Grid = 81 Digits Total (9 instances of each number) / 36 Axis Rings (X and Y each have 18), 18 X 18 Grid = 324 Digits Total (36 instances of each number) / 72 Axis Rings (X and Y each have 36), 27 X 27 Grid = 729 Digits Total (81 instances of each number) / 27 X 27 Grid = 108 Axis Rings (X and Y each have 54), 36 X 36 Grid = 1296 Digits Total (144 instances of each number) / 36 X 36 Grid = 144 Axis Rings (X and Y each have 72), etc. The tiles only change in size, not in number, along the surface of the torus. Therefore all the axis rings contain the same number of tiles. There are as many tiles in one ring as there are tiles on one side of the grid. Remember, the features change as you shift the dimensions of the grid.