Exercise 3 - Math in The Modern World
September 2, 2022 | Author: Anonymous | Category: N/A
Short Description
Download Exercise 3 - Math in The Modern World...
Description
Unisite Subdivision, Del Pilar, City of San Fernando, 2000 Pampanga, Philippines
MATHEMATICS MATHEMATIC S IN THE MODERN WORLD
FIBONACCI NUMBERS and
THE GOLDEN RATIO Submitted by: Andeza, Micah P. David, Kristel Rosfer P. Gomez, Ma. Carmela D. Mercado, Jasmine Erica S. Yusi, Via Rizza V. Submitted to: Carina Sun Dizon (Professor in MMW)
1|Page
Table of Contents Activity 1: Fibonacci numbers and Golden ratio in surroundings su rroundings ……………………………………………………………………………….. Activity 2: Boxes and Cans ………………………………………………………………………………………….. Activity 3: Golden Rectangle ……………………………………………………………………………………….. Activity 4: Fibs …………………………………………………………………………………… Activity 5: Lucas Numbers ………………………………………………………………………………………. Activity 6: SINGKO Sequence ………………………………………………………………………………………..
2|Page
ACTIVITY 1 FIBONACCI NUMBERS AND GOLDEN RATIO IN SURROUNDINGS Give 5 examples that follow the Fibonacci number and of the Golden ratio you can find in surroundings. Include photographs and as well as estimates of the base and height for each examples. Explain how each example follows the Fibonacci number and of the Golden ratio.
3|Page
1. Cauliflower
The cauliflower cauliflower is a vegetable which exhibits exhibits the golden golden ratio. Why? Why? This is because because its florets (is one of the small, flower-shaped pieces flower-shaped pieces of a vegetable) are arranged into spiral forms which represent the golden spiral.
2. Sunflower
The sunflower's head is also subject to Fibonaccian Fibonaccian process wherein the seeds are produced into center, and then it migrates outside to fill all spare space. These seed heads will form into a golden spiral and upon counting these spirals; it totally tends to match a Fibonacci number.
4|Page
3. Kamantigue
The Kamantigue Kamantigue also known known as Impatiens Impatiens Balsamina, Balsamina, is a flower which also follows follows the Fibonacci number. The petals of this flower flowe r are usually 5 but there are certain times, based on my observation that this flower has 7 petals wherein both numbers are seen under Fibonacci numbers.
4. Cochlea
At the outer part of the ear, also represents the golden ratio. This is because as seen on the figure it forms a golden spiral, known as the logarithmic logarith mic spiral wherein it abounds in nature.
5|Page
5. Rose
Rose also depicts the Fibonacci number and the golden ratio rati o pattern. This is because its sepals and petals are usually counted as 5 and it has many spirally arranged stamens. Each petal is also placed at 0.618034 per turnout of 360° circle for its exposure into sunlight and other factor.
6. Cactus
Cactus and other succulents also form geometric spiral similar to sunflowers, pinecones and others as seen on the picture shown. In addition, the arrangement of plant leaves along the stem is phyllotaxis and mathematically, the spiral phyllotaxis follows Fibonacci sequence.
6|Page
7. Bottle Cleaner
Likewise from the other examples, the bottle cleaner which is usually seen at home (especially if you have a younger sibling which is eventually a baby) also represents that golden ratio and the golden spiral is everywhere. How? This is because its spiky figures tend to show the golden spiral as shown on the figure.
8. Egg
Egg also exhibits golden ratio which is obviously seen on its shape or it’s curved which forms into a golden spiral from the outside curve to inside.
7|Page
9. Pepsi Logo
Phi is usually applied in the product logos and one example of these is the Pepsi Logo. The Pepsi logo was redesign in the pattern of the Phi so that it will be more appealing for the consumers.
10.
BPI Debit Card
If you will compute the ratio of the BPI card from its length, 8.5 cm and its width, 5.3 cm the answer will be 1.6 wherein it is almost or is roughly the golden ratio, 1.618.
8|Page
ACTIVITY 2 BOXES AND CANS Collect 5 boxes and 5 cans of different labels. Measure the height and width of the boxes. Measure the height and diameter of the cans. Compute the ratio of each and write the measurement in a table. In each include photographs. Use cm or mm scale.
9|Page
BOXES NAME
PICTURE
HEIGHT
WIDTH
12.5cm
14.3cm
RATI
=
.
1989 Taylor Swift CD Album
=
.
=1.14
Miniso Keyboard Box
14.2cm
38.6cm
=
.
=
.
=2.71
22.8cm
29cm
=
= Fila Shoe Box
10 | P a g e
.
=1.27
9cm
5.5cm
=
Chuckie
=
.
=1.63
Swiss Miss
11cm
28cm
=
=
=2.24
CANS 11 | P a g e
NAME
PICTURE
HEIGHT
DIAMETER
10.2cm
7.5cm
RATIO
=
.
= Alaska Evaporated Milk
.
=1.36
3.8cm
7.2cm
=
Reno Brand Liver Spread (85 grams)
.
=
.
=1.895
11.8m
10cm
=
=.
Fiesta Fruit Cocktail
=1.18
5cm
9.6cm
=
12 | P a g e
.
Golden Country Corned Beef
=
=1.92
11cm
6.8cm
=
Jolly Mushrooms
13 | P a g e
=. =1.618
A. Are the ratios close to the golden ratio? Which one are close? Based on the computed ratios of the given boxes and cans seen in the table above, only one of the examples listed reaches to the Golden Ratio (which is 1.618) and few are close to the Golden Ratio while the rest are too far from it. Among the example of boxes, only the box of Chuckie is close to the golden ratio. With the height of 9cm and width of 5.5cm, the comput computed ed ratio of the box of Chuckie is 1.626 which is almost close to the Golden Ratio. For the cans, based on the computed ratios of the different cans (or canned goods) seen in the table above, out of the 5 examples, 4 of them are a bit far to the Golden Ratio while 1 of them the m reach or is exactly the same as the Golden Ratio Rat io which is 1.618. Evidently seen in the table, one of the examples which is the can of the Jolly Mushroom reach or exactly the same same with the Golden Golden Ratio. With the height height of 11cm and diameter of 6.8cm its computed ratio is 1.618 which is very similar with the Golden ratio (1.618).
B. Why do you think many consumer items follow the golden ratio? We think consumer items follow the golden ratio because packaging design is very important in consumer products. One goal is to create aesthetically pleasing products. Also, the proportion of the packaging is also a factor of the package appearance since there should be “visual communication” between the product and consumers. Based on the proportion of the Golden Ratio, we think that most consumers have that impulse to purchase items which package proportions are close to the golden ratio that is why many consumer items follow the Golden ratio.
14 | P a g e
ACTIVITY 3 GOLDEN RECTANGLE Cut out the Golden rectangle of different dimensions and show that a considerate number of cutouts give out the Golden ratio.
15 | P a g e
8.4 cm
13.6 cm
2 cm
1.2 cm 0.4 cm
0.8 cm
3.2 cm
5.2 cm
38 mm
62 mm
2 cm
10 mm
24 mm 14 mm
16 | P a g e
7.4 cm
1 cm 0.8 cm
4.6 cm
0.6 cm
12 cm
1.8 cm
2.8 cm
34 cm
55 cm
-5 cm
8 cm
13 cm
17 | P a g e
21 cm
11.74 cm
7.26 cm 1.7 cm
1.08 cm
2.78 cm
18 | P a g e
4.48 cm
ACTIVITY 4 FIBS A fib is a poem where the number of syllabus per line follows the Fibonacci sequence. Invented by Greg Pincus, it caught on so well that the New York Times actually wrote an article about it, which naturally spawned many more fibs.
19 | P a g e
Example 1: She Is The girl Who has fought Her love but scattered Can somebody mend it like these, These painful painful things never been been happened to her before? before?
-
Maria
Example 2: I’m reminiscing
The times we we spent together, together, friend How did we end up like this? From friends to strangers – quick quick Come back and tell me what did I do wrong? I’m sick of hearing that it is all my fault
-
Jazzy
Example 3: Please Don’t hide
Yourself behind Those fake fake smiles, you you know I always got your back, my friend Even if I am away a million miles, I’m here Beside you, to comfort you; let’s talk about everything about life until the night,
- Example 4: 20 | P a g e
Rica
I’m
Tired. Tired of Listening To the sound of my Tears, tired of wasting wasting my days just just Pretending Pretendin g to be so fine and okay, too tired of Constantly battling with my fear that one day I’ll flunk and fail, but after every day, I simply kneel and pray, that someday this fear would merely go away, and so as pain and will eventually lead me to a brighter way.
-
Mics
Example 5: Stay strong. Powerful Words that changed my life. They make me me sane, strong strong and hopeful hopeful In times of challenges I always repeat those words. Always remember the reason and the sacrifices why you held on for so long.
-
21 | P a g e
Dems
ACTIVITY 5 THE GOLDEN RATIO AND HUMAN BODY Measure the following in cm or mm and record in a table - Distance from the ground to your navel - Distance from your navel to the top of your head - Distance from the ground to your knees - Length of your hand - Distance from your wrist to your elbow - Circumference of your waistline - Circumference of your hips
- Circumference of your bust line
22 | P a g e
Measurements of the body:
Distance from your ground to your navel Distance from your navel to the top of your head Distance from the ground to your knees Length of your hand Distance from your wrist to your elbow Circumference of your waistline Circumference of your hips Circumference of your bust line
a)
Distance from your ground to your navel
Distance from your navel to the top of your head
b)
Distance from your ground to your navel
c)
Distance from the ground to your knees
Distance from your wrist to your elbow
Length of your hand
d)
Circumference of your hips
Circumference of your bust line
-
-
-
-
Circumference of your waistline
23 | P a g e
-
Circumference of your waistline
e)
= 1.352
= 2.043
= 1.25
= 1.388
= 1.254
96 71 47 20 25 67 93 84
Measurements of the body:
a)
Distance from your ground to your navel
93
Distance from your navel to the top of your head
67
Distance from the ground to your knees
43
Length of your hand Distance from your wrist to your elbow
22 16
Circumference of your waistline Circumference Circumfer ence of your hips Circumference of your bust line
60
Distance from your ground to your navel
Distance from your navel to the top of your head
b)
Distance from your ground to your navel
c)
Distance from the ground to your knees
Distance from your wrist to your elbow
Length of your hand
d)
Circumference of your hips
Circumference of your bust line
-
-
-
-
Circumference of your waistline
24 | P a g e
-
Circumference of your waistline
e)
= 1.388
= 2.1 .16 627
= 0.7 .72 273
= 1. 1.33 3333 33
= 1. 1.28 2833 33
80 77
Measurements of the body: Distance from your ground to your navel Distance from your navel to the top of your head Distance from the ground to your knees Length of your hand Distance from your wrist to your elbow Circumference of your waistline Circumference of your hips Circumference Circumfer ence of your bust line
a)
Distance from your ground to your navel
Distance from your navel to the top of your head
b)
Distance from your ground to your navel Distance from the ground to your knees
your wrist to your elbow c) Distance from Length of your hand
d)
Circumference of your hips
Circumference of your bust line
-
-
-
Circumference of your waistline
25 | P a g e
-
Circumference of your waistline
e)
= 1.5
= 1.98
= 1.5
= 1.35
= 1.23
95 66 47 16 24 65 88 79
Measurements of the body:
a)
Distance from your ground to your navel Distance from your navel to the top of your head
108
Distance from the ground to your knees Length of your hand Distance from your wrist to your elbow Circumference of your waistline Circumference of your hips Circumference Circumfer ence of your bust line
47
Distance from your ground to your navel
Distance from your navel to the top of your head
b)
Distance from your ground to your navel
c)
Distance from the ground to your knees
Distance from your wrist to your elbow
Length of your hand
d)
Circumference of your hips
Circumference of your bust line
-
-
-
-
Circumference of your waistline
26 | P a g e
-
Circumference of your waistline
e)
68
18 25 90 92 90
= 1.5
= 2.17
= 1.39
= 1.02
= 1.13
Measurements of the body:
a)
Distance from your ground to your navel Distance from your navel to the top of your head
103
Distance from the ground to your knees Length of your hand Distance from your wrist to your elbow Circumference Circumfer ence of your waistline Circumference Circumfer ence of your hips Circumference Circumfer ence of your bust line
50
Distance from your ground to your navel
Distance from your navel to the top of your head
b)
Distance from your ground to your navel
c)
Length of your hand
Circumference of your hips
-
Circumference of your bust line
27 | P a g e
-
-
-
Circumference of your waistline
= 1.5 .56 61
= 2.0 .06 60
Circumference of your waistline
e)
-
Distance from the ground to your knees
Distance from your wrist to your elbow
d)
= 1.389
= 1.221
= 1.250
66
18 25 68 83 85
ACTIVITY 6 LUCAS NUMBERS
Lucas number are obtained just like Fibonacci numbereach term is obtained by adding two immediate previous terms- except that the first two terms are 2 and 1. Thus the Lucas sequence starts as 2, 1, 3, 4,….
28 | P a g e
I.
Give the first fifteen terms of the Lucas sequence
2 1 3 4 7 11 18 29 47 76 123 199 322 521 843 II.
Find the ratios of each term to the preceding term
2
47
1 = 2
1.621 29 = 1.6
1
76
2
4 3 7
= 0.5
47
= 1.6 1.617
123
= 3
76 = 1. 1.3 333
199 123
= 1.75
4 11 7 18 11 29 18
= 1.5 .57 71 = 1.6 .63 36
= 1.61 .618 = 1.61 .618
322
= 1.61 .618 199 521 = 1.61 .618 322 843 521
= 1.61 .618
= 1.6 .61 11 III.
Do the number approach any number?
Yes, the numbers of Lucas Sequence approach a certain number or ratio as it goes to the latter part of the sequence. As seen above, the first ratio of the 10 sequence does not merely approach a number but designates to approach a certain number. While seen at the 11th and 15th number of the sequence sequence it is much evident that they are now approaching a certain or definite number which is 1.618. Thus, the larger the sequence gets, the computed ratio will be approaching a definite number which is the Golden Ratio (1.618) even at the latter part of it.
29 | P a g e
ACTIVITY 7 SINGKO SEQUENCE Make your own “Fibonacci like number sequence” Name your sequence accordingly.
30 | P a g e
A. Give the first fifteen terms of your sequence
5 6 8 11 15 20 26 33 41 50 60 71 83 96 100 125
B. Find the ratios of each term to the preceding term 6/5 = 1.2 11/8 = 1.375 20/15 = 1.333 33/26 = 1.2692 50/41 = 1.2195 71/60 = 1.1833 96/83 = 1.1566 125/110 = 1.1364
C. Do the number approach any number? Given the ratios above it is seen that Singko Sequence approaches the numbers between 1.1 – 1.3. 1.3.
31 | P a g e
View more...
Comments
