Notes on Vortex Based Mathematics PDF

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 Document Version – 2.0

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. Many of these patterns were pointed out to me by others for which I am very grateful. I humbly offer my opinions 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).

Photo Credits: The beautiful picture on the cover is by BJ White. (Careful, the link leads to the original file which is 6 MB large animation)! While many of the diagrams I have made or re-made from the images in Rodin's books and articles related to his work (unless otherwise noted), there are probably a few that do not have proper credits. Please let me know and I will update it or remove it.

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 apparent on both the triangular and hexagonal layouts, the Golden Rectangle is only noticible on the diamond layout if your tiles are made up of 2 Golden or 2 Equiangular Triangles.

Golden Triangle

Golden Triangle

60 Degree Equiangular Triangle This is a perfectly balanced tile.

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 in the middle as you approach The Primal Point of Unity. (See diagram below).

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.

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 that is either coming towards you (Northern Hemisphere) or moving away from you (Southern Hemisphere), 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. 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 tell which Polar Number Pair is your Z-Axis from that direction because the reciprocals will change.

1-8

2-7

4-5

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 one more thing in addition to the "pinching" described above: It makes Nested Vortices of magnetism along the surface (sometimes called Vector Spatial Interstices). 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" (as it is pointed out in the presentation Inexhaustable

Electricity Supply). To quote Randy Powell, they also "act like kickers for the electricity", allowing you to have two wires next to each other without insulation and without shorting out. This is called Synchronized Electricity and we will talk more about this later. *"Atom" – The geometry of the coil literally makes it a macroscopic model of an 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). They do not make "jumps", but change 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 "Parker Spiral" in the plasma that makes up the "Heliospheric Current Sheet" which fills the solar system. 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 (Least Mean Free Path), and should be fatter on the outside than on the inside. 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. The Unit 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). The Unit can also move through space to make very complex patterns with its motion, 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 movements 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. (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 this 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 both wires simultaneously but invert the voltage going into the second wire. (This is the activation of Family Number Group 1). *Then, on the next cycle, reverse this. (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 pulsating 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). *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 cycle all over again. You might have to play with the timing of it, but it is likely that it will achieve resonance even if you are off a 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 a Cosmic Eggbeater 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. Everything being alive, it is a living entity rather than "just a device" (e.g.: Plasmoid).

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 "sterographic 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 "sterographic 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 Ramsay and Rodin are reporting, and dramatically extended the communication range of a small CB radio from, say, its nominal 14 miles to about 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 deformations). 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.