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

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Doodling in Math Class: Spirals, Fibonacci, and Being a Plant [2 of 3] : Part 1: http://youtu.be/ahXIMUkSXX0 Part 3: http://youtu.be/14-NdQwKz9w More on Angle-a-trons: http://www.youtube.com/watch?v=o6W6P8JZW0o Note: Beautiful spirally non-Fibonacci pinecones are very rare! If you find one, keep it.

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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.
Somehow.
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...