Vi Hart

Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3]

Doodling in Math Class: Spirals, Fibonacci, and Being a Plant [2 of 3]

Doodling in Math: Spirals, Fibonacci, and Being a Plant [Part 3 of 3]

Doodling in Math Class: Binary Trees

Doodling in Math Class: Stars

Doodling in Math Class: Snakes + Graphs

Doodling in Math Class: Infinity Elephants

Doodling in Math: Sick Number Games

Doodling in Math Class: Squiggle Inception

Doodling in Math Class: Triangle Party

Mobius Story: Wind and Mr. Ug

Math Improv: Fruit by the Foot

Wau: The Most Amazing, Ancient, and Singular Number

Are Shakespeare's Plays Encoded within Pi?

What is up with Noises? (The Science and Mathematics of Sound, Frequency, and Pitch)

Open Letter to Nickelodeon, Re: SpongeBob's Pineapple under the Sea

How To Snakes

Origami Proof of the Pythagorean Theorem

9.999... reasons that .999... = 1

Pi (還是)不好用 (英)

Fractal Fractions

自製量角器 (英)

Binary Hand Dance

Re: Visual Multiplication and 48/2(9+3)

The Gauss Christmath Special

Rhapsody on the Proof of Pi = 4

Doodle Music

畢格拉斯怎麼了? (英)

A Song About A Circle Constant
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Doodling in Math: Spirals, Fibonacci, and Being a Plant [Part 3 of 3]: Part 1: http://youtu.be/ahXIMUkSXX0 Part 2: http://youtu.be/lOIP_Z_0Hs How to find the Lucas Angle: http://youtu.be/RRNQAaTVa_A References: Only good article I could find on the subject: http://www.sciencenews.org/view/generic/id/8479/title/Math_Trek__The_Mathematical_Lives_of_Plants Book of Numbers: http://books.google.com/books?id=03rcO7dMYC&pg=PA113&lpg=PA113&dq=conway+phyllotaxis&source=bl&ots=bTLzWkMtB&sig=XnbL9nRYQoWOCbvWdZPAlVa3Co0&hl=en&sa=X&ei=2afqTui9L6OUiAKapaC7BA&ved=0CCkQ6AEwAQ#v=onepage&q&f=false Douady and Couder paper with the magnetized droplets: http://www.math.ntnu.no/~jarlet/Douady96.pdf Pretty sane page on phyllotaxis: http://www.math.smith.edu/phyllo/
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 So say you're me and you're in math class
 and you're trying to ignore the teacher and
 doodle fibbonachi spirals while simultaneously trying to
 fend off the local greenery,
 only if you become interested in
 something the teacher said by accident.
 And so you draw too many squares to start with.
 So you cross them out, but you crossed out too many,
 and then the teacher gets back on track and
 the moment is over, so...
 Oh well, might as well try to do the spiral from here.
 So you make a 3 by 3 square,
 and here's a 4 by 4
 and then 7 and then 11...
 This works, as in you got a spiral of squares,
 so you write down the numbers.
 1, 3, 4, 7, 11, 18.
 is kind of like the Fibonacci series,
 because 1 + 3 is 4
 3 + 4 is 7 and so on
 Or maybe you start as 2 + 1
 or 1 + 2
 Either way is a perfectly good series,
 and it's got another similarity
 with the Fibonacci series.
 The ratios of consecutive numbers also approach phi.
 Ok, so a lot of plants have Fibonacci numbers of spirals,
 but to understand how they do it
 we can learn from the exceptions.
 This pinecone that has 7 spirals one way
 and 11 the other, might be showing Lucas numbers.
 And since Fibonacci numbers and Lucas numbers
 are related, maybe that explains it.
 One theory was that plants get Fibonacci numbers
 by always growing new parts
 a phith of a circle all around.
 What angle will give Lucas numbers?
 In this pinecone, each new pinecony thing
 is about a 100 degrees around from the last.
 We're going to need a Lucas angleatron.
 It's easy to get a 90 degree angleatron,
 and if I take a third of a third of that,
 that's a ninth of 90 which is another 10 degrees.
 There.
 Now you can use it to get spiral patterns
 like what you see on a Lucas number plants.
 It's an easy way to grow Lucas spirals
 if plants have an internal angleatron.
 Thing is, a hundred is pretty far from 137.5.
 If plants were somehow meassuring angles,
 you'd think the anomalous ones
 would show angles close to a phith of a circle,
 not jump all the way to 100.
 Maybe I believe different species
 use different angles,
 but two pinecones from the same tree,
 two spirals on the same cauliflower?
 And that's not the only exception.
 A lot of plants don't grow spirally at all.
 Like this thing with leaves growing opposite from each other.
 And some plants have alternating leaves,
 180 degrees from each other,
 which is far from both phi and Lucas angles.
 And you could say that these don't count,
 because they have fundamentally different growth pattern
 and they are different in class of plant or something.
 But wouldn't it be even better if there were
 one simple reason for all of these things?
 These variations are good clue that
 maybe these plants get this angle
 and Fibonacci number as a consequence of
 some other process and not just because
 it mathematically optimises sunlight exposure.
 If this sun is right over head
 which pretty much never is and
 if the plants are perfectly facing straight up which they aren't.
 So how do they do it?
 Well you could try observing them,
 that would be like science.
 If you zoom in on the tip of a plant, the growing part,
 there's this part called the meristem.
 That's where new plant bits form.
 The biggest plant bits were
 the first to form of the meristem,
 and the little ones around the center are newer.
 As the plant grows
 they get pushed away from the meristem
 but they all started there.
 The important part is that
 the science observer would see
 the plant bits pushing away
 not just from the meristem
 but from each other.
 A couple physicists want to try this thing
 where they drop drops of a magnetized liquid
 in a dish of oil.
 The drops repelled each other
 kind of like plant bits do and
 were attracted to the edge of the dish
 just like a plant bits move away from the center.
 The first couple drops would head
 in opposite direction from each other,
 but then the third was repelled by both,
 but pushed farther by the more recent and closer drop.
 It and each new drop would set off at a phi angle
 relative to the drop before
 and the drops ended up forming Fibonacci number spirals.
 So all the plant would need to do
 to get Fibonacci number spirals, is
 to figure out how to make the plant bits repel each other.
 We don't know all the details.
 But here is what we do know.
 There is the hormone that tells plant bits to grow.
 A plant bit might use up the hormone around it.
 But there is more further away,
 so it will grow in that direction.
 That makes plant bits move out from the meristem
 after they form.
 Meanwhile the meristem keeps forming new plant bits
 and they're gonna grow in places that aren't too crowded
 because that's where there's the most growth hormone.
 This leaves them to move further out
 into the space left by the other outward moving plant bits.
 And once everything get locked into a pattern
 it's hard to get out of it,
 because there's no way for this plant bit
 to wander off unless there were empty space
 with a trail of plant hormone to lead it out of its spot,
 but if there were
 all the nearer plant bits would use up the hormone
 in grow to fill out in space.
 Mathematicians and programmers
 was made their own simulations
 and found the same thing.
 The best way to fit new things in
 with the most space
 has some popup at that angle,
 not because plant knows about the angle,
 but because that's where
 the most hormone has build up.
 Once it gets started, it's the selfperpetuating cycle.
 All that these flower bits are doing
 is growing where there is most room for them.
 The rest happens automathically.
 It's not weird that all these plants
 show Fibonacci numbers,
 it would be weird if they didn't.
 It had to be this way.
 The best thing about that theory
 is that it explains why Lucas pinecones would happen.
 If something goes a bit differently in the very beginning
 the meristem will settle into a different but stable pattern
 of where there's the most room to add new plant bits.
 That is 100 degrees away.
 It even explains alternating leaf patterns.
 If the leaves are far enough apart,
 relative to how much growth hormone they like,
 that these leaves don't have
 any repelling force with each other,
 and all these leaves care about is
 being farthest away from the two above and below it,
 which makes 180 degrees optimal.
 And when leaves grow in pairs
 that are opposite each other
 the answer where there's most room
 for both of those leaves
 is at 90 degrees from the one below it.
 And if you look hard
 you can discover even more unusual patterns.
 The dots on the neck of this whatever it is
 come in spirals of 14 and 22
 which may be as like doubled a Lucas numbers,
 and this pinecone has 6 and 10 
 doubled Fibonacci numbers.
 So how is the pineapple like a pinecone,
 what do daisies and brussels
 perhaps have in common?
 Is not the numbers they show, it's how they grow.
 This pattern is not just useful, not just beautiful.
 It's inevitable.
 This is why science and mathematics are still much fun.
 You discover things that seem impossible to be true
 and then get to figure out
 why it's impossible for them not to be.
 To get this far in our understanding of these things
 it took the combined effort of mathematicians,
 physicists, botanists and biochemists,
 and we've certainly learned a lot,
 but there's much more to be discovered.
 May be you should keep doodling in math class?
 You can help figure it out.
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