- Email: [email protected]

Welcome!

The natural world is alive with beautiful and amazing shapes.

These forms are all connected by a single number. Mathematics’ most mysterious number.

Ancient people held this number in awe and reverence, and gave it names like “The Divine Proportion” and “The Golden Ratio.”

It has inspired some of the greatest art and architecture of all time

Inspired some of the greatest art and architecture of all time

It is considered to be “the most beautiful number”

It’s found in many places in nature.

What is this number? Why is it so amazing? Why does it show up often in nature? Recent research has answered this question and revealed a deep connection between mathematics and science.

Euclid (c. 300 BCE) Known as the “Father of Geometry”, he wrote a math book called Elements that is still used today –– 2300 years later!

The Divine Proportion Cut a segment into “mean” and “extreme” ratios.

x longerx part shorter 1 part

1

=

whole x + thing 1 longer x part

x 1

x+1 x

x = x

2

=

x

x+1

If you square this number, it’s the same as adding 1.

1? 3?

1

2

? =

3

2

? =

1+1 3+1

2? 1.5 ?

2

2

? =

2

? 1.5 + 1 = 2.5 =

1.5 2.25

2+1

x

Hurray for the Quadratic Equation!

2

x+1

2

x – x–1=0 –b ± √ b – 4ac 2

x=

2a

a=1

x=

=

b = –1

–(–1) ± √ (–1) – 4(1)(–1)

c = –1

2

2(1)

=

1±√5 2

x 1±√5 2

2

=

x+1

= 1.618033988749895....

Sooooooooooooooooooooo beautiful! 1.618033988749895 = 2.618033988749895

2

x x

2

x+1 x x

=

= –1

1 1+ x –1

x–1

1 x

x

1 1.618033988749895

=

= 0.618033988749895

This number is so important, it gets its own letter: the Greek letter phi, written like this: Φ

How can you use a number like 1.618... in art? And why would you want to?

4.4

6.8

8.6

12.20

20.12

In ancient times, people thought of numbers as shapes ... ... and Φ makes some of the most amazing shapes.

Φ does number tricks: 1 = 0.618033988749895 Φ

=Φ–1

Φ = 1.618033988749895 Φ

2

= 2.618033988749895

=Φ+1

Φ does geometry tricks too....

Golden Rectangle Φ (~1.618)

1

Square

smaller Golden Rectangle

Golden Rectangle

This is golden, too!

Golden Rectangle Φ 1

1

Φ ?– 1 0.618 1 Φ

1

Golden Rectangle

Ratio of =

1 1 Φ

long short =

Φ 1

short side x Φ = long side

This small rectangle is also golden! 1 Φ

1

r=e

–kt

72°-36°-72° Triangle

ratio of short/long

36°

small: large: (x–1) / 1 = 1/x

1? x

x–1 = 1/x 1?

72°

36° 72°36°

72° 1

x=Φ x–1 ?

Golden Triangle

36° Φ

72° 36° 72°36°

also Golden!

1

72°

Φ

2

Φ 1 Φ

3

Phidias (ca. 500 BCE), the greatest scultper of Classical Greece, used this ratio extensively.

It is reported than in many of Phidias’ statues: Height of Figure Height of Belly Button

= Φ = 1.618...

It was thought that this was the most beautiful position for one’s belly button.

It is also said that his work contained many golden rectangles. length width

= Φ

Statue of Zeus, one of the 7 wonders of the ancient world, built by Phidias.

Phidias’ work once filled the Parthenon, which seems to contain many golden rectangles.

Φ is named for Phideas! Φιδίας

Throughout the ages, many artists have used the golden ratio, most famously Italian Renaissance man Leonardo DaVinci (ca. 1500)

Many famous artists have intentionally used the golden ratio in their art.

DaVinci

Dali

Michaelangelo

Seurat

Raphael

Mondrian

But... are these shapes really more beautiful?

Many psychological studies have been done over the past 200 years... and there is no conclusive evidence that people prefer this ratio.

There are many claims that the ratio can be found all over the body...

... and that the most beautiful faces conform closely to the golden ratio.

(as if we don’t have enough reasons to feel bad about how we look!)

Is it true? Is this ratio part of the way our bodies are assembled?

Distance A A

B

Distance B = 1.6 Golden Ratio! WOW!

Distance A Distance B A B

= 1.6 Golden Ratio again! WOW!

Distance A Distance B A

=2

B (never mind, ignore that...)

Wait a minute...

22 points

22 points # lengths? # ratios (≠1)?

22 points 22 x 21÷2 =

231 lengths 231 x 230 =

53,130 ratios

I looked at 50,000 ratios between 0 and 20, and some of them were close to 1.6! Amazing?

Not very amazing.

There are many other claims about the golden ratio that are • coincidences • wishful thinking • bad approximations

There are many other claims that are • coincidences • wishful thinking • bad approximations

If you’re looking for it, you can find it! You can also find any other number you want.

Be wary! It is not enough to find a few measurements that are “close” to the ratio. Ask: Is there a reason? The Golden Ratio DOES appear in some places in nature. The reasons are amazing. The reasons are MATHEMATICAL.

Leonardo Pisano (ca. 1170 – 1250) Leonardo of Pisa Son of Bonacci Fibonacci

Leaning Tower of Pisa (ca. 1174 – present)

Fibonacci’s Travels (ca. 1180–1200)

Liber Abaci “The Book of Calculating”

1

2

3

4

5 6 7

8

9 0

(Hindu-Arabic Numerals) Also in the book: “The Rabbit Problem” The solution method produces numbers that are extremely important in nature.

January February March April May

June

December?

Pairs of:

Babies

Jan

1

Adults

Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec

1 1 2 3 5 8 13 21 34 55

?

1 1 2 3 5 8 13 21 34 55 89

total 144

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ... The sequence is called the Fibonacci sequence.

It was so named by Edouard Lucas. (1842–1871, Paris)

Bee Families

Fertilized Egg = Female

Unfertilized Egg = Male

Bee Family Tree =M Q

Q

=F Q

Q

Draw the next two steps!

Bee Family Tree 8 5 3 2 1 1

Sneezewort

Many plants often have Fibonacci numbers in them.

21 petals each

34 petals on this sunflower

34 petals on this Gerbera flower

Trilliums have 3 petals, Buttercups and pansies have 5 petals Delphiniums have 8 petals, Marigolds & Black-eyes Susans & Ragwort have 13 petals, or 21 petals or 34 petals.

Plant Spirals

Artichokes Pineapples

Pinecones

How many spirals...

21 and 34

Count the spirals...

55 and 89

More Fibonacci Numbers

Yes, Fibonacci Numbers!

WHY? Coincidence? Because they’re “Nature’s special numbers”? Nature is always efficient... what is efficient about using Fibonacci numbers?

A connection between Fibonacci numbers and The Golden Ratio

A ratio is a multiplicative relationship between two numbers.

Fractions are ratios. 22 7

355 113

144 89 55 34 21 13 8 5 3 2 1 1

= = = = = = = = = = =

1.6179 1.6181... 1.6176... 1.6190... 1.6153... 1.625 1.6 1.6666... 1.5 2 1

Ratios of Fibonacci Numbers

1

1.5

~1.618

2

1.618033988749895....

The ratios of Fibonacci Numbers converge on THE GOLDEN RATIO.

Plant “use” the Golden Ratio to distribute their leaves, or petals, or branches, or seeds.

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°

Many plants produce leaves or petals or seeds so that the angle between a new part and the previous part is always the same. This makes the spirals we see.

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°

A spiral with angle = 1/8 of revolution 1/8 x 360° = 45° a lot of empty space

Leaves block each other. a lot of crowding

Seeds crowd each other.

Instead of a 45° angle between seeds, plants use Φ x 360° = 582.5° .... 582.5° – 360° = 222.5° 222.5° is called the Golden Angle. It is 1.618 full turns.

How seeds are arranged in nature

0°

~222.5° Seed 1

Angle = 222.5°

Angle = Φ revolutions

0°

Seed 2

Angle = 222.5° x 2

Angle = Φ x 2 revolutions

0°

Seed 3

Angle = 222.5° x 3

Angle = Φ x 3 revolutions

0°

Seed 4

Angle = 222.5° x 4

Angle = Φ x 4 revolutions

0°

Seed 5

Angle = 222.5° x 5

Angle = Φ x 5 revolutions

0°

Seed 6

Angle = 222.5° x 6

Angle = Φ x 6 revolutions

(distance from center to seed n = √n)

0°

Seed n

Angle = 222.5° x n

Angle = Φ x n revolutions

Find spirals, find patterns...

Φ 0°

Seed 8: angle = 8 x Φ rotations

12.944 rotations (same as 0.944 rotations) Φ 0°

0.944 x 360° = 340° ... which is about 20° short of of 360°

~20°

~20° ~20° ~20°

~20°

~20° ~20°

Φ ~20°

~20° 0° ~20°

~20° ~20° ~20° ~20°

~20° ~20° ~20°

~20°

Φ 0°

Every eighth seed:

8 x Φ rotations

13 rotations ~ 8x 8

= ~13 rotations

Nearly a whole number of rotations Every thirteenth seed:

13 x Φ rotations

21 rotations ~ 13 x 13

= ~21 rotations

Every 13th seed 13 x Φ rotations

= 21.034... rotations Φ 0°

= 12.4° past the previous seed in its spiral

Φ 0°

Seed 7 isΦrotated 7 xsome π revolutions = 21.9912 turns Why and not other number like π? = about 357° = 3° from start... VERY CLOSE!

π = 3.14159265358979...

~ π =

22 7

(3.142857....) 7 arms

Compare: Seed 8 in the Φ spiral is 20° off... much worse for making spirals.

Seed 113 is rotated 113 x π revolutions = 354.9999699 turns = about 359.989° = 0.011° from start!

113 arms 355 113 (3.14159292...) ~ π =

Pi has GOOD approximations. Good approximations make strong spiral arms. Strong spiral arms cause crowding and empty space.

22 7

355 113

Good approximation!

GREAT approximation!

Good rational approximations = Bad seed distributions

The number with the worst possible approximations is...

Φ The Golden Ratio is the Most Irrational Number... The Golden Ratio gives the best seed placement!

Note: This can be analyzed with Hurwitz’s Theorem which gives a measure of how good or bad approximations are, and using continued fractions to generate convergents.

Every real number, rational or irrational, can be written as a continued fraction as follows: 1

N = a+

1

b+ c +

1 1 d+ e + ...

where a, b, c, d, e, ... are positive integers.

Here’s the beginning of π as a continued fraction. 1

~ π = 3+

1

7+ 15 + The greater the integers, the better the rational approximations.

1 1 1+ 292 + ...

For fun...

355 339 16 1 ~ π = 3 + + (3.14159292035...) 113 112 113 1 7 + 1 16 16 16 15 + 1

Φ =

1 1+ Φ 1 Φ 1

Φ = 1+ 1+

Φ1

1+ 1+ These are the lowest we can go... Φ has the worst approximations!

Φ 1 Φ 1 1+ 1 Φ 1+ 1 + ... Φ

How does this work in plants? Mathematical Hypothesis: Some plants twist uniformly as they grow. The twist produces equal angles between leaves/ branches/seeds. The plant grows to a point which: • maximizes light on leaves or • minimizes stress (crowding) on seeds.

BIG IDEA

The resulting angle is automatically the Golden Ratio, and the Golden Ratio produces Fibonacci Numbers.

The mechanism may be • active... • evolutionary... • or both.

feedback influences growth shaped over many generations

The mechanisms differ in different plants, but the mathematical result is the same.

CRAZY IDEA: Nature seeks the most chaos. From this chaos, comes amazing order.

FIN

Copyright © 2019 PROPERTIBAZAR.COM. All rights reserved.