and your teacher's talking about ...

Well, who knows what your teacher's talking about.

Probably a good time to start doodling.

And you're feeling spirally today, so yeah.

Oh, and because of overcrowding in your school,

your math class is taking place in

greenhouse number three.

Plants.

Anyway.

You've decided there are three basic

types of spirals.

There's the kind where, as you spiral out,

you keep the same distance.

Or you could start big but make it tighter

and tighter as you go around, in which case

the spiral ends.

Or you could start tight but make the spiral

bigger as you go out.

The first kind is good if you really want

to fill up a page with lines.

Or if you want to draw curled up snakes.

You can start with a wonky shape to spiral around

but you've noticed that, as you spiral out,

it gets rounder and rounder.

Probably something to do with how the ratio

between two different numbers approaches one

as you repeatedly add the same number to both.

But you can bring the wonk back by

exaggerating the bumps and it gets all

optical illusiony.

Anyway, you're not sure what the second

kind of spiral is good for, but I guess it's

a good way to draw snuggled up slug cats,

which are a species you've invented just

to keep this kind of spiral from feeling useless.

This third spiral, however, is good for

all sorts of things.

You could draw a snail or a nautilus shell.

And elephant with a curled up trunk,

the horns of a sheep, a fern frond, a cochlea

in an inner ear diagram, an ear itself.

Those other spirals can't help but be jealous

of this clearly superior kind of spiral.

But I draw more slug cats.

Here's one way to draw a really perfect spiral.

Start with one square and draw another

next to it that is the same height.

Make the next square fit next to both together,

that is each side is length two.

The next square has length three.

The entire outside shape will always

be a rectangle.

Keep spiraling around, adding bigger

and bigger squares.

This one has side length one, two, three,

four, five, six, seven, eight, nine,

10, 11, 12, 13.

And now 21.

Once you do that you can add a curve going

through each square, arcing from one corner

to the opposite corner.

Resist the urge to zip quickly across

the diagonal, if you want a nice smooth spiral.

Have you ever looked at the spirally pattern

on a pine cone and thought, "Hey, sure are

"spirals on this pine cone?"

I don't know why there's pine cones

in your greenhouse.

Maybe the greenhouse is in a forest.

Anyway, there's spirals and there's not

just one either.

There's one, two, three, four, five, six,

seven, eight going this way.

Or you could look at the spirals going

the other way and there's one, two, three,

four, five, six, seven, eight, nine, 10, 11, 12, 13.

Look familiar?

Eight and 13 are both numbers in the

Fibonacci series.

That's the one where you start by adding

one and one to get two, then one and two

to get three, two and three to get five.

Three plus five is eight, five plus eight

is 13, and so on.

Some people think that instead of starting

with one plus one you should start with

zero and one.

Zero plus one is one, one plus one is two,

one plus two and three, and it continues

on the same way as starting with one and one.

Or, I guess you could start with one plus zero

and that would work too.

Or why not go back one more to negative one

and so on?

Anyway, if you're into the Fibonacci series,

you probably have a bunch memorized.

I mean, you've got to know one, one, two,

three, five.

Finish off the single digits with eight

and, ooh with 13, how spooky.

And once you're memorizing double digits,

you might as well know 21, 34, 55, 89 so that

whenever someone turns a Fibonacci number

you can say, "Happy Fib Birthday."

And then, isn't it interesting that 144, 233, 377?

But 610 breaks that pattern, so you'd better

know that one too.

And oh my goodness, 987 is a neat number

and, well, you see how these things get

out of hand.

Anyway, 'tis the season for decorative

scented pine cones and if you're putting

glitter glue spirals on your pine cones

during math class, you might notice that

the number of spirals are five and eight

or three and five or three and five again.

Five and eight.

This one was eight and thirteen and one

Fibonacci pine cone is one thing, but all of them?

What is up with that?

This pine cone has this wumpy weird part.

Maybe that messes it up.

Let's count the top.

Five and eight.

Now let's check out the bottom.

Eight and 13.

If you wanted to draw a mathematically

realistic pine cone, you might start

by drawing five spirals one way and eight

going the other.

I'm going to mark out starting and ending

points for my spirals first as a guide

and then draw the arms.

Eight one way and five the other.

Now I can fill in the little pine coney things.

So there's Fibonacci numbers in pine cones

but are there Fibonacci numbers in other things

that start with pine?

Let's count the spirals on this thing.

One, two, three, four, five, six, seven, eight.

And one, two, three, four, five, six, seven,

eight, nine, 10, 11, 12, 13.

The leaves are hard to keep track of,

but they're in spirals too.

Of Fibonacci numbers.

What if we looked at these really tight spirals

going almost straight up?

One, two, three, four, five, six, seven,

eight, nine, 10, 11, 12, 13, 14, 15, 16, 17,

18, 19, 20, 21.

A Fibonacci number.

Can we find a third spiral on this pine cone?

Sure, go down like this.

And one, two, three, four, five, six, seven,

eight, nine, 10, 11, 12, 13 (muttering) 19, 20, 21.

But that's only a couple examples.

How about this thing I found

on the side of the road?

I don't know what it is.

It probably starts with pine, though.

Five and eight.

Let's see how far the conspiracy goes.

What else has spirals in it?

This artichoke has five and eight.

So does this artichoke looking flower thing.

And this cactus fruit does too.

Here's an orange cauliflower with five and eight

and a green one with five and eight.

I mean, five and eight.

Oh, it's actually five and eight.

Maybe plants just like these numbers though.

Doesn't mean it has anything to do

with Fibonacci, does it?

So let's go for some higher numbers.

We're going to need some flowers.

I think this is a flower.

It's got 13 and 21.

These daisies are hard to count, but they have

21 and 34.

Now let's bring in the big guns.

One, two, three, four, five, six, seven,

eight, nine, 10, 11, 12, 13, 14, 15, 16, 17,

18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,

31, 32, 33, 34.

And one, two, three, four, five, six, seven,

eight, nine, 10, 11, (muttering) 17, 24,

(muttering) 42, 53, 54, 55.

I promise, this is a random flower and I didn't

pick it out specially to trick you into thinking

there's Fibonacci numbers in things, but you should

really count for yourself next time you see

something spirally.

There's even Fibonacci numbers in how

the leaves are arranged on this stalk,

or this one, or the Brussels sprouts on this stalk

are a beautiful delicious three and five.

Fibonacci is even in the arrangement of

the petals on this rose, and sunflowers have

shown Fibonacci numbers as high as 144.

It seems pretty cosmic and wondrous, but the cool

thing about the Fibonacci series and spiral

is not that it's this big complicated

mystical magical super math thing beyond

the comprehension of our puny human minds

that shows up mysteriously everywhere.

We'll find that these numbers aren't weird at all.

In fact, it would be weird if they weren't there.

The cool thing about it is that these

incredibly intricate patterns can result

from utterly simple beginnings.