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]
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Doodling in Math Class: Snakes + Graphs
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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)
Binary Hand Dance
Re: Visual Multiplication and 48/2(9+3)
The Gauss Christmath Special
Rhapsody on the Proof of Pi = 4
A Song About A Circle Constant
Doodling in Math: Spirals, Fibonacci, and Being a Plant [Part 3 of 3]
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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=0--3rcO7dMYC&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/
- 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 phi-th 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 angle-a-tron.
- It's easy to get a 90 degree angle-a-tron,
- and if I take a third of a third of that,
- that's a ninth of 90 which is another 10 degrees.
- 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 angle-a-tron.
- 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 phi-th 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 pop-up 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 self-perpetuating cycle.
- All that these flower bits are doing
- is growing where there is most room for them.
- The rest happens auto-math-ically.
- 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.