deciphering nature’s code - mike naylor · deciphering nature’s code ... why is it so amazing?...
TRANSCRIPT
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Deciphering Nature’s CodeThe Secret Mathematics of the Natural World
– by Mike Naylor –
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Welcome!
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The natural world is alive with beautiful and amazing shapes.
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These forms are all connected by a single number.Mathematics’ most mysterious number.
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Ancient people held this number in awe and reverence, and gave it names like
“The Divine Proportion” and “The Golden Ratio.”
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It has inspired some of the greatest art and
architecture of all time
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Inspired some of the greatest art and architecture of all time
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It is considered to be“the most beautiful number”
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It’s found in many places in nature.
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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.
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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!
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The Divine ProportionCut a segment into “mean” and “extreme” ratios.
x 1
longer part
shorter part=
whole thing
longer part
x
1x + 1
x
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=x1
x + 1x
x x
x 2= x + 1
If you square this number, it’s the same as adding 1.
1? 1 2= 1 + 1? 2? 2 2
= 2 + 1?
3? 1.5 ? 1.52= 1.5 + 1?
2.25 = 2.53 2
= 3 + 1?
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x 2= x + 1
x 2 – x – 1 = 0
x = –b ± √ b – 4ac
2a
2
a = 1 b = –1 c = –1
x = –(–1) ± √ (–1) – 4(1)(–1)
2(1)
2
= 1 ± √ 5
2
Hurray for the Quadratic
Equation!
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x 2= x + 1
1 ± √ 5
2= 1.618033988749895....
1.618033988749895
2.618033988749895
2
=
Sooooooooooooooooooooo beautiful!
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x 2= x + 1
x x x
x = 1 +x1
x – 1 =x1
–1 –1
1.618033988749895
1= 0.618033988749895
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This number is so important, it gets its own letter:
the Greek letter phi,written like this: Φ
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How can you use a number like 1.618... in art?
And why would you want to?
In ancient times, people thought of numbers as shapes ...
... and Φ makes some of the most amazing shapes.
4.4 6.8 8.6 12.20 20.12
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= 1.618033988749895 Φ
Φ2 = Φ + 1= 2.618033988749895
Φ1
= 0.618033988749895 = Φ – 1
Φ does number tricks:
Φ does geometry tricks too....
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1
Golden Rectangle
Φ (~1.618)
Square
smallerGolden
Rectangle
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This isgolden, too!
Golden Rectangle
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1
1
Φ1
Φ – 1
1
Golden Rectangle
0.618
Φ
?
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Φ1
1
Ratio oflongshort
1
Φ1
Φ1
==
Thissmall
rectangleis alsogolden!
short side x Φ = long side
Golden Rectangle
Φ1
1
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r = e –kt
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72° 72°
36°
72°-36°-72°Triangle
1
36°36°
72°
x
1
1
x–1
(x–1) / 1 = 1/x
ratio of short/long
x–1 = 1/x
x = Φ?
?
?
small: large:
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72° 72°
36°
GoldenTriangle
1
Φ
36°36°
72°also
Golden!
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Φ
1
Φ2
Φ3
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Phidias (ca. 500 BCE), the greatest scultper of Classical Greece, used
this ratio extensively.
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Height of Figure
Height of Belly Button= Φ
It is reported than inmany of Phidias’ statues:
It was thought that this was themost beautiful position for
one’s belly button.
= 1.618...
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It is also said thathis work contained
many golden rectangles.
Statue of Zeus, one of the 7 wonders of the ancient
world, built by Phidias.
lengthwidth
= Φ
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Phidias’ work once filled the Parthenon,which seems to contain many golden rectangles.
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Φ is named for Phideas!
Φιδίας
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Throughout the ages,many artists have used thegolden ratio, most famously
Italian Renaissance manLeonardo DaVinci (ca. 1500)
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Many famous artists have intentionally used the golden ratio in their art.
DaVinci
Dali
Michaelangelo Raphael
Seurat Mondrian
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But... are these shapesreally more beautiful?
Many psychological studies have been done over the past 200 years...
and there is no conclusive evidence that people prefer this ratio.
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There are many claims that the ratio can be
found all over the body...
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... 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!)
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Is it true?
Is this ratio part of the way our bodies are assembled?
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A
B
Distance A
Distance B
= 1.6
GoldenRatio!
WOW!
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A
B
Distance A
Distance B
= 1.6
GoldenRatioagain!
WOW!
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AB
Distance A
Distance B
= 2
(never mind,ignore that...)
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Wait a minute...
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22 points
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22 points
# lengths?# ratios(≠1)?
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22 points
231 lengths22 x 21÷2 =
53,130ratios
231 x 230 =
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I looked at 50,000ratios between 0 and 20, and some of them
were close to 1.6!
Amazing?
Not very amazing.
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There are many other claims about the golden ratio that are
• coincidences• wishful thinking
• bad approximations
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There are many other claims that are• coincidences
• wishful thinking• bad approximations
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If you’re looking for it, you can find it!You can also find any other number you want.
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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.
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Leaning Tower of Pisa(ca. 1174 – present)
Leonardo Pisano
Leonardo of Pisa
Son of Bonacci
Fibonacci
(ca. 1170 – 1250)
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Fibonacci’s Travels(ca. 1180–1200)
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Liber Abaci“The Book of Calculating”
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Also in the book: “The Rabbit Problem”
The solution method produces numbers that are extremely important in nature.
(Hindu-Arabic Numerals)
1 2 3 4 5 6 7 8 9 0
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January
February
March
April
May
June
December?
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Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Babies AdultsPairs of:
111123581321345589
1235813213455
?total 144
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Edouard Lucas.(1842–1871, Paris)
The sequence is calledthe Fibonacci sequence.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...
It was so named by
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Fertilized Egg = Female Unfertilized Egg = Male
Bee Families
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Bee Family Tree
Q
Q
= M
= F
Draw the nexttwo steps!
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Bee Family Tree
1
1
2
3
5
8
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Sneezewort
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Many plants often have Fibonacci numbers in them.
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21 petals each
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34 petals onthis sunflower
34 petals on this Gerbera flower
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Trilliums have 3 petals,Buttercups and pansies have 5 petals
Delphiniums have 8 petals,Marigolds & Black-eyes Susans & Ragwort have 13 petals,
or 21 petalsor 34 petals.
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Plant Spirals
Pineapples
Artichokes
Pinecones
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How many spirals...
21 and 34
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Count the spirals...55 and 89
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MoreFibonacciNumbers
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Yes,FibonacciNumbers!
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WHY?Coincidence?
Because they’re “Nature’s special numbers”?Nature is always efficient... what is efficient
about using Fibonacci numbers?
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A connection between Fibonacci numbers and
The Golden Ratio
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A ratio is a multiplicative relationship between two
numbers.
227
355113
Fractions are ratios.
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11235813213455
1 1.5 2
= 1
Ratios of Fibonacci Numbers89144
==========
21.51.6666...1.61.6251.6153...1.6190...1.6176...1.6181...1.6179
~1.618
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1.618033988749895....
The ratios of Fibonacci Numbersconverge on
THE GOLDEN RATIO.
Plant “use” the Golden Ratio to distribute their leaves, or petals, or
branches, or seeds.
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A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
Many plants produce leaves or petals or seedsso that the angle between a new part and the
previous part is always the same.
This makes the spirals we see.
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A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
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A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
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A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
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A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
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A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
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A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
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A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
![Page 91: Deciphering Nature’s Code - Mike Naylor · Deciphering Nature’s Code ... Why is it so amazing? ... “Father of Geometry”, he wrote a math book called Elements that is still](https://reader035.vdocuments.net/reader035/viewer/2022062909/5b04825d7f8b9a89208dc5f9/html5/thumbnails/91.jpg)
A spiral with angle = 1/8 of revolution 1/8 x 360° = 45°
a lot ofempty space
a lot ofcrowding
Leaves block each other.
Seeds crowdeach other.
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Instead of a 45° angle between seeds,plants use Φ x 360°
= 582.5° .... 582.5° – 360° = 222.5°
222.5° is called theGolden Angle.
It is 1.618 full turns.
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How seeds are arranged in nature
0°
~222.5°
Angle = 222.5°Seed 1Angle = Φ revolutions
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0°
Angle = 222.5° x 2Seed 2Angle = Φ x 2
revolutions
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0°
Seed 3 Angle = 222.5° x 3Angle = Φ x 3
revolutions
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0°
Seed 4 Angle = 222.5° x 4Angle = Φ x 4
revolutions
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0°
Seed 5 Angle = 222.5° x 5Angle = Φ x 5
revolutions
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0°
Seed 6 Angle = 222.5° x 6Angle = Φ x 6
revolutions
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0°
Seed n Angle = 222.5° x nAngle = Φ x n
revolutions
(distance from center to seed n = √n)
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Find spirals, find patterns...
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Φ
0°
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Φ
0°
Seed 8:angle = 8 x Φ
rotations
12.944 rotations
0.944 x 360° = 340°... which is about 20° short of of 360°
(same as 0.944 rotations)
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Φ
0°~20°
~20°
~20°~20°~20°
~20°~20°
~20°
~20°
~20°
~20°
~20°~20° ~20° ~20°
~20°
~20°
~20°
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Φ
0°
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8 x Φ rotations
8 x rotations 138
~ = ~13 rotations
Nearly a whole number of rotations
Every eighth seed:
13 x Φ rotations
13 x rotations 2113
~ = ~21 rotations
Every thirteenth seed:
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Φ
0°
Every 13th seed 13 x Φ rotations
= 21.034... rotations
= 12.4° past the previous seed in its spiral
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Φ
0°
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![Page 109: Deciphering Nature’s Code - Mike Naylor · Deciphering Nature’s Code ... Why is it so amazing? ... “Father of Geometry”, he wrote a math book called Elements that is still](https://reader035.vdocuments.net/reader035/viewer/2022062909/5b04825d7f8b9a89208dc5f9/html5/thumbnails/109.jpg)
![Page 110: Deciphering Nature’s Code - Mike Naylor · Deciphering Nature’s Code ... Why is it so amazing? ... “Father of Geometry”, he wrote a math book called Elements that is still](https://reader035.vdocuments.net/reader035/viewer/2022062909/5b04825d7f8b9a89208dc5f9/html5/thumbnails/110.jpg)
(3.142857....)
Why Φ and not some other number like π?
π =227
~
π = 3.14159265358979...
7 arms
Seed 7 is rotated 7 x π revolutions = 21.9912 turns= about 357° = 3° from start... VERY CLOSE!
Compare: Seed 8 in the Φ spiral is 20° off... much worse for making spirals.
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(3.14159292...)
π =355113
~
113 arms
Seed 113 is rotated 113 x π revolutions = 354.9999699 turns = about 359.989° =
0.011° from start!
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355113
Pi has GOOD approximations.Good approximations make strong spiral arms.
Strong spiral arms cause crowding and empty space.
227
Good rational approximations = Bad seed distributions
Goodapproximation!
GREATapproximation!
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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.
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N = a +1
b +1
c + 1
d +1e + ...
Every real number, rational or irrational, can be written as a continued fraction as follows:
where a, b, c, d, e, ... are positive integers.
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π = 3 +1
7 +1
15 + 1
1 +1
292 + ...
~
Here’s the beginning of π as a continued fraction.
The greater the integers, the better the rational
approximations.
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π = 31
7 +1
15 +11
16
112
16
113
16
16113
+339113
+355
(3.14159292035...)~
For fun...
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Φ = 1 + Φ1
1 + Φ1
Φ =
1 +1
11 +
1 +1
11 +
11 +
ΦΦΦΦΦ
1 + ...These are the lowest we can go...Φ has the worst approximations!
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Mathematical Hypothesis:
Some plants twist uniformly as they grow.
The twist produces equal angles between leaves/branches/seeds.
How does this work in plants?
The plant grows to a point which:• maximizes light on leaves or• minimizes stress (crowding) on seeds.
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The resulting angle is automatically the Golden Ratio, and the Golden Ratio produces Fibonacci Numbers.
BIG IDEA
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The mechanism may be
• active... feedback influences growth• evolutionary... shaped over many generations• or both.
The mechanisms differ in different plants, but the mathematical result is the same.
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Nature seeks the most chaos.From this chaos, comes amazing order.
CRAZY IDEA:
Thank you for coming! Enjoy the mathematical wonders of the world around you!
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FIN