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

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.