Doodling in math: Spirals, Fibonacci, and being a plant [2 of 3] Part 1: http://youtu.be/ahXIMUkSXX0Part 3: http://youtu.be/14-NdQwKz9wMore on Angle-a-trons: http://www.youtube.com/watch?v=o6W6P8JZW0oNote: Beautiful spirally non-Fibonacci pinecones are very rare! If you find one, keep it.
Doodling in math: Spirals, Fibonacci, and being a plant [2 of 3]
- Say you're me and you're in math class
- and you're doodling a flowery petaly things,
- if you want something with lots of overlapping petals
- you're probably following a loose sort of rule
- that goes something like this:
- Add new petals where there are gaps between old petals.
- You can try doing this precisely starting with
- some number of petals, say 5,
- then add another layer in between.
- But in the next layer you have to add 10,
- and the next has 20.
- The inconvenient part of this is that
- you have to finish a layer before everything is even.
- Ideally you'd have a rule that just let's you add petals
- until you get bored.
- Now imagine you're a plant and
- you want to grow in a way that spreads out your leaves
- to catch the most possible sunlight.
- Unfortunately, and I hope I'm not presuming too much
- in thinking that, as a plant, you're not very smart.
- You don't know how to add numbers to create a series,
- you don't know geometry and proportions,
- and can't draw spirals or rectangles, or slug cats,
- but maybe you could follow one simple rule.
- Botanists have noticed that plants
- seem to be fairly consistent, when it comes to
- the angle between one leaf and the next.
- So let's see what you could do with that.
- So you grow your first leaf
- and if you didn't change angle at all,
- then the next leaf you grow would directly above it.
- So that's no good because
- it blocks out all the light, or something.
- You can go 180 degrees,
- to have the next leaf directly opposite, which seems ideal.
- Only once you go 180 again
- the third leaf is right over the first.
- In fact, any fraction of a circle,
- with a whole number as a base, is going to have
- complete overlap, after that number of turns.
- And unlike when you're doodling, as a plant
- you're not smart enough to see
- you've gone all the way around and
- should now switch to adding things in between.
- If you try and postpone the overlap by making the fraction really small
- you just get a ton of overlapping in the beginning
- and waste all this space, which is completely disastrous.
- Or maybe other fractions are good,
- the kind that positions leaves in a starlike pattern.
- It'll be a while before it overlaps,
- and the leafs will be more evenly spaced in the mean time.
- But what if there're a fraction that never completely overlapped?
- For any rational fraction, eventually the star will close,
- but what if you use an irrational number?
- The kind of number that can't be expressed as whole numbered ratio.
- What if you used the most irrational number?
- If you think it's weird to say that one irrational number
- is more irrational than another...
- Well you might wanna become a number theorist.
- If you're a number theorist, you might tell us that
- phi is the most irrational number, or you might say
- that's like saying of all the integers, 1 is the integeriest,
- or you might disagree completely.
- But anyway, phi ...
- is more than one, but less than two;
- more than 3/2, less than 5/3;
- greater than 8/5, but 13/8 is too big.
- 21/13 just a little to small and
- 34/21 is even close than too big and so on.
- Each pair of adjacent Fibonacci numbers creates a ratio
- that gets closer and closer to phi as numbers increase.
- Those are the same numbers on the sides of these squares.
- Now stop being a number theorist and
- start being a plant again.
- You put your first leaf somewhere
- and the second leaf at an angle that is 1 phi-th of a circle,
- which, depending on whether you going one way or the other,
- could be about 225,5 degrees or about 137,5. Great!
- Your second leaf is pretty far from the first,
- gets lots of space and sun.
- And now let's add the next one of the phi-th of a circle away,
- and again, and again... You can see how new leaves tend to
- pop up in the spaces left between old leaves,
- but it never quite fills things evenly,
- so there's always room for one more leaf,
- without having to do a whole new layer.
- It's very practical, and as a plant you probably like this.
- It would also be a good way to give lots of room
- to see pods and petals and stuff.
- As a plant that follows this scheme,
- you'd be at an advantage.
- Where do the spirals come in?
- Let's doodle a pinecone using the same method.
- By the way, you can make your own phi angle-a-tron
- by dog-earing a corner of your notebook.
- If you folded is so the edges are on a line,
- you have 45 degrees plus 90, which is 135,
- pretty close to 137,5.
- If you're careful you can slip in a couple more degrees.
- Detach your angle-a-tron and you're good to go.
- Add each new pineconey thing a phi-angle around,
- and make them a little farther out each time,
- which you can keep track of
- by marking the distance on your angle-a-tron.
- Check it out, the spirals form by themselves.
- And if you count the number of arms, look, it's 5 and 8.
- If you're wondering why spirals would form
- and why always with (phi)bonacci numbers,
- you could morph back into a number theorist
- or geometer, or something.
- Here's just a little bit of intuition:
- One simple way to doodle a flowers is
- to start with a certain number of petals, say 5.
- And when you go back around, add the next layer
- close to the first but bigger.
- Each layer adds 5 new petals and the 5 arms spiral out.
- Looks pretty spiraly to me.
- Now go back to phi. You put out 3 petals
- before you go back around.
- And if I make them really wide
- the next lap adds 3 petals that
- overlap a bunch with the first, and so on...
- If I started with skinnier petals though
- the second time you go around
- it doesn't quite overlap so much and
- it takes 8 petals before it goes around twice
- and they overlap enough for you to see the spirals.
- So this time I get 8 and 13.
- I mean none of the spirals
- are actually physically there, on any of these plants.
- It's just the plant bits are close enough
- that you can see the pattern.
- So all a plant needs to do
- to get awesome Fibonacci numbers of spirals
- is add new bits at a 137.5 degree angles.
- The rest takes care of itself.
- That the Fibonacci series is in so many things
- really says less about those things
- and more about mathematics.
- I mean, that's what mathematics is all about:
- simple rules, complex consequences.
- A process so easy that even a plant can do it,
- can turn it into these amazing structures all around us.
- Just like a few simple postulates can give us
- an incredibly powerful geometry.
- I mean that's all assuming that a plant can do it!
- But measuring the angles between plant bits
- you can see that they obviously do do it.
- I mean it's not like they have angle-a-trons.
- But plants have been around a long time,
- and have had a lot of practice,
- so that probably explains everything.
- And thus we always get spirals
- of 5 and 8 on this flower, 5 and 8 on this artichoke,
- 5 and 8 on this pinecone.
- Even this cauliflower has 1,2,3,4,5,6,7... um...
- Anyway we always get 1,2,3,4,5,6,7... huh..
- And 1,2,3,4!...
- It's easy to dismiss these as mutant anomalies.
- But just because they're different and unusual
- doesn't mean we should ignore them.
- 4,7,11... What could does these numbers mean?
- Maybe things aren't as simple as I thought...
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