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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/
相關課程
- 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.
- 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 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.
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