The Golden Ratio: Is It Myth or Math?

Summary

This script delves into the allure of the golden ratio, a mathematical constant believed by some to underlie beauty in nature, art, and architecture. It explores the ratio's irrational nature, historical significance from Euclid to Fibonacci, and its prevalence in unexpected places. The video challenges the myth of the golden ratio's ubiquity, highlighting both its genuine appearances in natural phenomena like plant spirals and the human tendency to perceive patterns where they may not exist, concluding that while math can offer insights into the world's order, life's beauty often transcends such rigid constructs.

Takeaways

• 🔍 The Golden Ratio, often symbolized by the Greek letter phi (Φ), is a mathematical constant that appears in various aspects of art, architecture, and nature.
• 📏 The Golden Ratio is an irrational number, with a value of approximately 1.618033988749895, and is represented by a non-repeating, non-terminating decimal.
• 📐 Euclid, the ancient Greek mathematician, was one of the first to study the Golden Ratio, recognizing it as a special way to divide a line segment.
• 🌱 The Fibonacci sequence, introduced by Leonardo of Pisa (Fibonacci), is closely related to the Golden Ratio, with the ratio of consecutive Fibonacci numbers approaching phi as the sequence progresses.
• 🌻 The Golden Ratio is observed in the arrangement of leaves and other plant structures, which often exhibit spiral patterns related to Fibonacci numbers.
• 🏛 Historical structures like the Great Pyramid of Giza and the Parthenon are often claimed to incorporate the Golden Ratio, although these claims are sometimes disputed.
• 🎨 Some artists and architects, such as Salvador Dalí and Le Corbusier, have intentionally used the Golden Ratio in their work, but its presence is not a requirement for beauty in art.
• 🐚 The nautilus shell is frequently associated with the Golden Ratio, but actual measurements often show variations from the ideal golden spiral.
• 🤔 The human tendency to seek patterns can lead to the perception of the Golden Ratio in places where it may not actually exist, highlighting the importance of distinguishing myth from reality.
• 🌿 The Golden Ratio's occurrence in nature is not due to a conscious mathematical calculation by plants but is a result of evolutionary processes that optimize growth and resource capture.
• 💡 The fascination with the Golden Ratio reflects a human attraction to patterns and the search for underlying order in the universe, even amidst the complexity and messiness of life.

Q & A

What is the Golden Ratio and why is it considered significant?

The Golden Ratio, often denoted by the Greek letter phi (φ), is an irrational number approximately equal to 1.6180339887... It is considered significant because it appears in various aspects of art, architecture, and nature, and is believed by some to represent an aesthetically pleasing balance.

How is the Golden Ratio related to the Fibonacci sequence?

The Golden Ratio is closely related to the Fibonacci sequence because as you progress through the sequence, the ratio of successive Fibonacci numbers converges towards the Golden Ratio, indicating a natural mathematical progression that often appears in organic growth patterns.

What is a Golden Rectangle and why is it considered aesthetically pleasing?

A Golden Rectangle is a rectangle in which the ratio of the length to the width is the same as the Golden Ratio. It is considered aesthetically pleasing because this proportion is believed to be inherently harmonious and balanced, which is why it has been used in various forms of art and architecture.

Can you explain the concept of a Golden Triangle and its properties?

A Golden Triangle is a triangle with sides in proportion to the Golden Ratio. Specifically, if you divide a side of a Golden Triangle according to the Golden Ratio, the resulting smaller triangle is also a Golden Triangle with angles of 72, 72, and 36 degrees. This self-similarity and the unique angle properties make it distinct.

What is a Golden Spiral and how is it formed?

A Golden Spiral is a logarithmic spiral that grows outward by a factor of the Golden Ratio for every quarter turn it makes. It is formed by drawing quarter-circles based on the Golden Rectangle, with each quarter-circle smaller by a factor of the Golden Ratio, and then connecting these arcs with a smooth curve.

How did Fibonacci contribute to the understanding of the Golden Ratio?

Fibonacci, through his sequence, indirectly contributed to the understanding of the Golden Ratio. The sequence, which starts with 0 and 1 and each subsequent number being the sum of the two preceding ones, approximates the Golden Ratio as it progresses, revealing a natural occurrence of the ratio in a simple numerical pattern.

What is the significance of the Golden Angle in the context of the Golden Ratio?

The Golden Angle, approximately 137.5 degrees, is the angle that results when dividing a circle according to the Golden Ratio. This angle is significant because it allows for the formation of spiral patterns found in nature, such as in the arrangement of leaves on a stem or the spirals of a pineapple, without overlapping.

How do plants use the Golden Ratio in their growth patterns?

Plants use the Golden Ratio in their growth patterns to optimize light exposure for their leaves or to efficiently pack seeds into their fruits. The spiral arrangement of leaves or seeds often follows the Golden Angle, which ensures that each element is spaced to maximize exposure to sunlight or space without overlap.

Is the Golden Ratio found in the human body, and if so, where?

Some theories suggest that the Golden Ratio can be found in the proportions of the human body, such as the ratio of a person's height to the distance from their navel to their feet. However, these claims are not universally accepted, as beauty standards and body proportions vary widely across different cultures and individuals.

What are some misconceptions about the Golden Ratio in nature?

Misconceptions about the Golden Ratio in nature include the belief that it is the fundamental building block of all natural structures. While it does appear in certain growth patterns, such as the spirals of some plants, it is not universally present in all aspects of nature. Some claims, like the ratio of DNA helix turns to its width, have been debunked as incorrect.

How has the Golden Ratio been used in art and architecture?

The Golden Ratio has been used in art and architecture to create compositions that are considered aesthetically pleasing and balanced. Artists like Salvador Dalí and architects like Le Corbusier have intentionally incorporated the Golden Ratio into their works. However, it is important to note that not all beautiful art or architecture relies on the Golden Ratio.

Outlines

00:00

🔍 The Golden Ratio: Nature's Mathematical Mystery

This paragraph delves into the concept of the golden ratio, a mathematical property believed by some to be intrinsically linked to beauty and nature. It discusses the golden rectangle and its aesthetic appeal, the irrational nature of the golden ratio symbolized by phi, and its historical recognition by ancient Greeks like Euclid. The paragraph also explores the golden ratio's connection to sacred geometry and its alleged presence in various aspects of life, from art to the human body, questioning whether it truly underlies everything or if it's an overrated myth.

05:03

🌀 Fibonacci, Phi, and the Mythological Status of the Golden Ratio

The second paragraph focuses on the Fibonacci sequence and its relationship with the golden ratio. It explains how the ratio of consecutive Fibonacci numbers approximates phi as the sequence progresses. The narrative highlights the historical figure Leonardo of Pisa, known as Fibonacci, who introduced Hindu-Arabic numerals to Europe. The paragraph also touches on the mythologizing of the golden ratio, suggesting that its prevalence in nature and human constructs might indicate a deeper universal secret, despite the lack of concrete evidence linking it to specific natural phenomena.

10:04

🏛️ Debunking the Golden Ratio in Architecture and the Human Body

This paragraph critically examines the claims of the golden ratio's presence in architectural marvels and the human body. It challenges the idea that the golden ratio defines beauty standards or is a key element in the design of historical structures. The paragraph points out the fallacy in attributing the golden ratio to DNA's structure and the inaccuracies in measuring the ratio in natural forms like nautilus shells. It emphasizes the human brain's pattern-seeking tendency, which may lead to misinterpretations of the golden ratio's prevalence.

15:04

🌿 The Golden Ratio in Nature: A Genuine Phenomenon

The fourth paragraph presents a more authentic connection between the golden ratio and nature, particularly in the growth patterns of plants. It discusses the occurrence of Fibonacci numbers in the arrangement of leaves, seeds, and petals, which can be explained by the golden angle and the plant's evolutionary strategy to maximize sunlight exposure and space utilization. The paragraph also explains the biological and chemical processes that lead to these patterns, emphasizing that while the golden ratio may appear in nature, it is not a deliberate mathematical construct by plants.

20:05

🎨 Beauty, Perception, and the Golden Ratio in Art and Life

The final paragraph reflects on the subjective nature of beauty and the role of the golden ratio in art and design. It acknowledges that while some artists and architects have intentionally used the golden ratio, beauty is not solely defined by mathematical principles. The paragraph concludes by celebrating the messiness of life and the world, suggesting that the search for order within this chaos is a part of human curiosity and appreciation for beauty in its many forms.

Keywords

💡Golden Ratio

The Golden Ratio, often denoted by the Greek letter phi (Φ), is an irrational number approximately equal to 1.6180339887... It is characterized by the property that the ratio of the sum of two quantities to the larger one is equal to the ratio of the larger one to the smaller. In the video, the Golden Ratio is presented as a mathematical property believed by some to be a fundamental aspect of beauty and nature, with the script exploring its prevalence in art, architecture, and the natural world.

💡Sacred Geometry

Sacred Geometry refers to the idea that there are certain geometric shapes and proportions that have spiritual or symbolic significance and are found in religious traditions and sacred structures. The video script mentions sacred geometry in the context of the Golden Ratio, suggesting that this mathematical concept is embedded within nature and has been revered throughout history in various cultures.

💡Irrational Number

An irrational number is a real number that cannot be expressed as a simple fraction – that is, its decimal representation is infinite and non-repeating. The Golden Ratio is an example of an irrational number, as discussed in the script, which differentiates it from rational numbers like 5 or 0.5 that can be expressed as a ratio of two integers.

💡Fibonacci Sequence

The Fibonacci Sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. This sequence is intimately connected to the Golden Ratio, as the ratio of consecutive Fibonacci numbers approximates the Golden Ratio as the sequence progresses. The script uses the Fibonacci Sequence to illustrate the prevalence of the Golden Ratio in nature and mathematics.

💡Golden Rectangle

A Golden Rectangle is a rectangle in which the ratio of the length to the width is the same as the Golden Ratio. The video script describes a Golden Rectangle as aesthetically pleasing and suggests that this shape has been used in art and architecture to achieve balance and beauty.

💡Golden Triangle

A Golden Triangle is a triangle with sides in proportion to the Golden Ratio. The script mentions that the angles of a Golden Triangle are 72, 72, and 36 degrees, and that smaller Golden Triangles can be created by dividing one of its sides according to the Golden Ratio, illustrating the self-similarity inherent in this geometric figure.

💡Golden Spiral

The Golden Spiral is a logarithmic spiral that grows outward by a factor of the Golden Ratio for every quarter-turn it makes. The script describes how a Golden Spiral can be drawn by connecting the corners of successive Golden Rectangles, and it is often associated with the growth patterns seen in nature, such as in sunflowers and pinecones.

💡Golden Angle

The Golden Angle is the angle corresponding to a turn of approximately 137.5 degrees, which is derived from the reciprocal of the Golden Ratio. The video script explains that the Golden Angle is significant in the arrangement of leaves and other plant structures, as it allows for an efficient packing of elements without overlap.

💡Fibonacci Numbers

Fibonacci Numbers are the individual numbers in the Fibonacci Sequence. The script highlights that certain plant structures, such as the spirals on a pineapple or a pinecone, often exhibit a number of spirals that are Fibonacci Numbers, suggesting an underlying mathematical pattern in natural growth.

💡Logarithmic Spiral

A Logarithmic Spiral is a spiral curve that expands or contracts by a constant ratio as it rotates. The script clarifies that while the Nautilus shell is often incorrectly associated with the Golden Ratio, it is actually an example of a logarithmic spiral, which grows at a constant proportion rather than the Golden Ratio.

💡Evolution

Evolution in the context of the video refers to the biological process by which plants have developed and adapted over time to optimize their growth and survival. The script suggests that the prevalence of the Golden Ratio and Fibonacci Numbers in plant structures is a result of evolutionary processes that have found these patterns to be beneficial for capturing sunlight and resources.

Highlights

The golden ratio, often denoted as phi, is found in various aspects of nature, art, and architecture, suggesting a universal principle of aesthetics and proportion.

The golden rectangle, with sides in the golden ratio, is considered the most aesthetically pleasing shape by many due to its balanced proportions.

Phi is an irrational number, meaning it cannot be expressed as a simple ratio of two integers, similar to the square root of two.

Euclid, the ancient Greek mathematician, was one of the first to study the properties of the golden ratio in his book 'Elements'.

The golden ratio appears in the Fibonacci sequence, where the ratio of consecutive numbers approximates phi as the sequence progresses.

Johannes Kepler connected the Fibonacci sequence with the golden ratio, further cementing the idea that phi is a fundamental aspect of the universe.

The golden ratio is claimed to be present in various natural phenomena, such as the spirals of a nautilus shell, although some of these claims have been debunked.

Plants like pineapples, sunflowers, and pinecones exhibit a special kind of spiral that relates to the Fibonacci sequence and the golden ratio, suggesting an evolutionary advantage.

The golden angle, approximately 137.5 degrees, is the angle at which new growth in plants occurs to maximize sunlight exposure without overlapping.

The golden ratio's prevalence in art and architecture is sometimes overstated, with many examples being coincidental rather than intentional.

The human body's proportions are sometimes claimed to follow the golden ratio, but this is a subjective measure and varies widely across individuals.

The golden ratio's mathematical properties, such as its relationship to itself when squared or inverted, contribute to its intrigue and mystique.

Leonardo Fibonacci's introduction of Hindu-Arabic numerals to Europe through his book 'Liber Abaci' indirectly led to the popularization of the golden ratio.

The golden spiral, formed by drawing a curve through successive golden rectangles, is a shape often associated with the golden ratio in nature and art.

The golden ratio's presence in nature is more than mere coincidence, as it offers practical advantages in growth patterns and resource optimization for plants.

The perception of beauty and the role of the golden ratio in art is subjective, with many artists using the ratio without necessarily adhering to it strictly.

The golden ratio's significance in mathematics, nature, and human perception reflects our fascination with patterns and the search for underlying order in the universe.

Transcripts

a number so perfect perfect we find it

everywhere everywhere sacred

sacred geometry a mathematical property

hardwired into nature

secrets the golden ratio the golden

ratio the golden ratio what's the answer

what's the answer what's the answer

history

sacred geometry sacred geometric

geometry the golden

ratio the golden ratio

wait wait wait wait hold on i mean

really

is there actually one special number

that underlies

everything from sunflowers to seashells

everything from pineapples and pine

cones

to the pyramids and the parthenon i mean

a number

that can link beauty in art music and

the human body

one number that links nature's order to

the rules of mathematics

well some people think so but like uncle

carl says extraordinary claims require

extraordinary evidence so let's take a

closer look

at what the golden ratio is really about

i mean after all the universe is a

strange place

full of surprises

[Music]

hey smart people joe here which of these

rectangles is the

most perfect give them a look which one

just feels most

balanced the most beautiful did you pick

this one

that's a golden rectangle and many

people believe that this

shape is the most aesthetically pleasing

quadrilateral that there is

this one not so much this one ew

gross yeah get that away the golden

rectangle

the ratio of its long side to its short

side

is exactly this this is the golden ratio

abbreviated as phi or fee depending on

how you prefer to pronounce your greek

these numbers after the decimal point

they go on forever without repeating

like the better known pie phi is an

irrational number

it's irrational because it can't be

written as the ratio of

two integers five that's rational

because we can write it as the integer

five over the integer one the number

zero point

five also rational we can write it as 3

over 4.

even 0.3333 infinitely repeating

that's rational because it can be

written simply as 1 over 3.

but what about the diagonal of a square

whose sides are

one unit long the pythagorean theorem

tells us that the diagonal has a length

of the square root of two

which is a number but one that can't be

written as a ratio of two nice and tidy

integers

it's irrational likewise phi also can't

be written as a simple integer ratio

and an ancient greek named euclid was

one of the first to notice that

this euclid guy was big into geometry in

fact most of the geometry that we learn

in school is named after

him it can be a pretty big deal to get a

whole section of math class named after

you

so around 300 bc euclid wrote a book

called elements

a collection of most of what was known

about math at the time

and until the 20th century it was the

best-selling book

ever other than the bible you could

notice that there was one

special way to divide a line where the

ratio of the whole

to the longer segment was the same as

the ratio of the longer segment to the

shorter one

and that ratio is phi well euclid called

it the extreme

and mean ratio which sounds like what

happens when i make a bad

tweet the names phi and golden ratio

they didn't show up until almost the

20th century

anyway the greeks and mathematicians of

that time

they didn't think of numbers like we do

as these strings of digits from zero to

nine

to them phi was this ratio

just like to them pi wasn't 3.14159 etc

pi was just the ratio of a circle's

circumference to its diameter

this is literally the golden ratio and

you can do

some weird stuff with it the ratio of

the long sides of this triangle to its

short side is

you guessed it phi this is a golden

triangle also called a

sublime triangle and the angles of that

triangle are 72

72 and 36 degrees now if i

divide one of the long sides according

to the golden ratio

and make a smaller triangle there it's

another golden triangle

same angles and all and that other

triangle we just created

the length of these sides to the base is

one

over phi we call this squatty shape the

golden gnomon

and if i take one golden triangle and

stick two golden gnomons on the side

i get a regular pentagon yeah we're just

getting started

let's overlap two golden omans and add a

smaller golden triangle on the side

you make a pentagram and going back to

our golden rectangle

if we put another golden rectangle here

another inside that

and another and so on and so on and draw

a curve

through all these shapes we get a shape

called the golden

spiral if that looks familiar it's

probably because you've seen an image

like this before on the internet and we

will be getting back to that

very soon no there's even more

strangeness if you multiply phi

times itself that's the same as one plus

phi

take one over phi and that's the same as

phi minus one

this is a weird number okay fine so what

phi is weird there's infinite numbers so

some of them

are gonna be a little strange what makes

phi

special is that it shows up in a bunch

of really unexpected places that are

pretty far off from geometry class

or at least people claim to find phi in

a lot of unexpected places

and this is the really interesting thing

about phi because as cool

of a number as it is on its own it's

achieved this almost

mythological status this is like the

elon musk of numbers

and many people say that because we find

it in

so many places it can't just be a

coincidence

it must be a sign of some deeper secret

about the universe

well where does the real story of phi

end and the myth begin well if there's

one person responsible for the

mythological status of phi

it's this guy leonardo of pisa aka

fibonacci around the year 1200 fibonacci

was responsible for bringing

hindu arabic numerals into common use

across europe

these are the numerals that we use today

zero through nine and merchants quickly

realized that doing arithmetic with

these was

way easier than roman numerals which is

what everyone in europe was using at the

time

so to teach people how to use these new

numbers which had actually already been

in use in asia for like a thousand years

fibonacci wrote a math textbook liber

abakai which just means the book of

calculation

this book was full of math problems to

teach people how to add

and exchange currencies and divide and

multiply with these new numbers

and tucked inside of chapter 12 was this

weird problem about

rabbits doing what rabbits are known to

do that would end up making

fibonacci famous imagine you have a pair

of rabbits in a field

one male and one female no rabbits die

or get eaten

starting from the second month she's

alive every female reproduces

each month making a new pair of rabbits

one male one female

so how many rabbits will they produce

after one year

you can pause and take a minute to work

it out if you want to but it ends up

looking like this

you might notice something special about

the number of pairs of rabbits

each month the number of pairs is equal

to the sum of the previous

two months and after 12 months you'd

have 144

pairs this is the famous fibonacci

sequence

you can carry it on forever just add the

previous two numbers to get the next

and so on until the end of the universe

or until you get bored

the reason that we're talking about the

fibonacci sequence in a video about the

golden ratio

is because as you go on in the fibonacci

sequence

the ratio between numbers gets closer

and closer to phi

in fact any sequence of numbers that

follows the fibonacci rule

adding the two previous to get the next

trends to phi

like this set the lucas numbers follow

the pattern and carry it on and the

difference between the terms

all trends to phi yeah i know

it's weird but fibonacci never made that

connection himself

a guy named johannes kepler did a few

hundred years later

the same kepler who figured out the math

that explains how planets move

pretty smart guy it's after that when

the fibonacci sequence and phi got

together that the myth

really took off and people started to

claim these numbers

were more than just numbers

despite phi's seemingly mystical

mathematical origins

the truly mind-boggling aspect of phi

was its role as a fundamental building

block in nature

plants animals and even human beings all

possessed dimensional properties that

adhered with eerie exactitude

to the ratio of phi to one

phi's ubiquity in nature clearly exceeds

coincidence

that is how one of the greatest writers

in all of history

put it and that's really the question

isn't it

does phi the golden ratio the divine

proportion whatever grand name you want

to give it

really show up everywhere in nature or

is it our

pattern sensing brains making us think

that we see it everywhere

like when you notice a license plate

from another state and suddenly you

start seeing out-of-state license plates

more than you used to

or think you used to well let's look at

some places people

claim to see phi the human body it's

claimed that the ideal ratio of a

person's height to the distance from

their navel to their feet

is phi i probably don't need to tell you

that beauty standards in different

cultures vary

a lot and people come in way too many

shapes and sizes for that to be a rule

the great pyramid of giza the parthenon

notre dame cathedral in paris the taj

mahal

a handful of ancient buildings that

people claim

were built with golden ratio dimensions

the thing is for an object that's fairly

big like a building or complex like a

body

there are so many ways to measure it and

so many measurements you can take

that some are bound to be somewhere

around a golden ratio apart from each

other

i mean this video is a 16 by 9 aspect

ratio

that's pretty close to 1.6 but it's not

phi 16 by 10 would be even closer

actually

lots of places that people claim to see

phi in nature

are just plain wrong like the ratio of

one turn of a dna helix to its width

google that and you'll see results that

say it's 34 angstroms high per turn

and 21 angstroms wide both fibonacci

numbers ooh

intriguing unfortunately that's wrong

these are dna's actual measurements

here's the key

thing in any example that you find phi

isn't

approximately 1.6 give or take it's

exactly this if people go measuring

things

looking for the golden ratio they often

measure them in ways that ensure that

they find the golden ratio

our brains love patterns and once we

learn

a pattern like the usual arrangement of

a mouth

and a nose and two eyes to make a face

we see that pattern everywhere and that

brings us

to this a nautilus shell a nautilus is a

cool little mollusky thing that swims

around with a spiral shell

and a face full of spaghetti it's become

basically the official

mascot of the golden ratio in nature the

claim is that if you trace the spiral of

this shell

each ring is a golden ratio away from

the next smallest ring

etc but people have gone out and

actually measured

loads and loads of nautilus shells and

they aren't golden spirals the ratios

vary quite a bit

like snail shells or sheep horns it's an

example of what's called a

logarithmic spiral i mean it's really

cool each turn of the spiral grows by

the same proportion

because the nautilus grows at the same

rate but that proportion

isn't phi and that's too bad because

logarithmic spirals are cool but no one

pays attention to them because of this

obsession with phi

my point is close is not enough

if phi is a fundamental building block

in nature

we should be able to show that it's more

than a coincidence

that there's some reason behind it being

there

which brings us to these not every claim

about

seeing phi in nature is fake

fi does show up in nature in a really

interesting way and if you've ever

looked closely at a pineapple

or a pinecone or a sunflower or an

artichoke

i don't know who closely studies

artichokes but maybe you have

all of these plant parts show a special

kind of spiral let me show it to you

here's a pineapple and if you notice on

a pineapple there's these spirals going

one direction like this and then we can

see a spiral

going the other direction the other way

around the pineapple

let me trace this out and make it a

little easier for you to see so a little

arts and crafts time here and it's okay

to be smart this is going to be fun

[Music]

there's 13 spirals in that direction

well now let's count these spirals in

the other direction

so we've got eight spirals going in one

direction

and 13 spirals going in the other

direction and if those numbers

sound familiar that's because they're

both

fibonacci numbers and you remember from

before that within the fibonacci

sequence

we find the golden ratio okay that's

just one pineapple

and maybe that's a big pineapple

conspiracy coincidence

so let's count something else i don't

know if you've ever noticed this but

pine cones have these adorable little

spirals

too let's see if there's any five magic

going on in there eight spirals

going that direction it's pretty spooky

[Music]

thirteen spirals going the other

direction

those are fibonacci numbers too pinecone

pineapple maybe these should have been

called fine cones and pineapples but

i digress we can also find fibonacci

numbers in these

sunflowers this rose this

cauliflower this succulent thing

this is fun let's count the spirals on

this artichoke too

and five spirals going in that direction

eight spirals going in that direction

five and eight are fibonacci numbers as

well

and there's this the branch out

hold on i came prepared

for this thing and this the branch

of a monkey puzzle tree yes that is its

real name

this looks like some sort of medieval

weapon for the

like a plant night or something we can

count the spirals

on this thing too very carefully

i should be wearing safety goggles for

this

and eight spirals going in that very

dangerous direction

come here you 13

spirals going the other direction plants

can't do

math they can't count so why is there

this connection

well imagine that you're a plant you

eat light so the more sun you can catch

with your leaves the better

so as a stem or a branch grows

up or out where do you put your leaves

let's put one leaf right

here that looks fine and then let's go

say half a turn to where we'd be as far

as

possible from the first leaf that we

laid down we can do our half

turn rule again and

well wait now we're on top of the first

leaf and if we if we continue this well

we're not going to be catching maximum

rays man

let's start over and let's pick a let's

pick a new fraction of a turn

why not a third so we put our first leaf

down here

and we can turn a third of a turn right

here

we turn a third of a turn again for our

next leaf

right here looking pretty good so far a

third of a turn from

our third leaf now we're back over the

top of our first leaf again

so that's not gonna work either maybe

one over four

so we can start here a quarter turn a

quarter turn again

another quarter turn but there are

plants that actually grow like this

but we can quickly see as we continue

this pattern

we overlap our leaves again this can't

be the best

strategy out there for catching maximum

sun and it turns out that if we use any

fraction of a circle with a whole number

on the bottom our leaves will eventually

overlap

a rational number isn't going to work

but what if we used an irrational number

instead

and remember that irrational numbers

can't be expressed as

simple integer ratios and it turns out

that phi might be the most irrational

number that there is

so let's put our first leaf down

and then let's go a fraction of a circle

one over phi turns around and remember

that one over phi is equal to this

so if we express that as a fraction of

360 degrees

we're taking a turn of about 137.5

degrees each time which probably won't

surprise you

is called the golden angle that made

this little guide that's exactly that

angle

so let's fill in our leaves using our

new

phy guide all right so we go 137.5

degrees

from our first leaf and we lay another

one down

137 and a half degrees from there and

lay down our third leaf

as we see our new leaves fill the gaps

left from the leaves before we never

overlap so far let's keep going and see

what happens

[Music]

one two three four five six

seven eight the fibonacci number of

spirals they form

all on their own just from that golden

angle

turn rule why do these spirals form

you can actually do this yourself and

play around with different size leaves

or petals or whatever

and you'll find that with the golden

angle as your guide

you always put down a fibonacci number

of

things before you get to these layers

where things start to

almost overlap but not quite and the

spirals just sort of happen from there

and there's a fibonacci number of them

this works for more than leaves catching

sunlight too

it's also a useful pattern for catching

rain and funneling water down to your

roots

or for packing more seeds and flower

petals into a small space

looking at you here sunflowers these are

all things that can make you

a better plant quick side note

we didn't put a camera over there i'll

just stay over here

quick front note this explains a lot

about

why plants do this but the how is a lot

more complicated and scientists are

still working out a lot of the details

what we do know is that instead of

having some

internal leaf angle growth measure thing

or something

all these angles well they have to do

with newly growing plant parts

repelling other nearby plant parts

kind of like how the poles of magnets if

they're alike they repel

each other only this is thanks to growth

hormones and not

magnets but the point is there's actual

biology and chemistry

underneath all of this so anyway it's

not like there's a gene

in plants that's programmed to do math

or something but plants have had

gazillions of years of evolution

to find the best way to do all the cool

plant stuff that they want to do

and it's not like every plant even does

this plenty of plants follow

different rules and it works just fine

for them and that's all that really

matters in evolution

is that something works well enough not

that something reaches some

mathematical and irrational golden

perfection

but i mean you can't deny this is kind

of

beautiful and i think that's probably a

big part of why we're attracted to this

pattern more than other plant patterns

because you know ape brain like pretty

pattern

and that's the funny thing about beauty

it can take so many forms

we know that some artists like salvador

dali or the architect le corbusier

occasionally used the golden ratio

purposely in some of their work but i

mean

there's plenty of beautiful art that

makes no use of the ratio too

you remember when i asked you to pick

the most beautiful rectangle at the

beginning

lots of you probably picked one other

than the golden rectangle

math follows very particular rules and

things like beauty

and life they're a bit more messy

because

the world is a messy place and that's

part of the beauty isn't it

that's okay because sometimes in the

middle of that mess

if we look hard enough we can find some

order

after all stay curious

okay wow this is this is way creepier

than i thought it would be

can i be a person again i don't i don't

really like being a tropical fruit

anyone hello hello

you