The Fibonacci sequence.

This looks a lot better. You saw nothing.

COUNTS TO

It's a Fibonacci sequence.

It is the Fibonacci sequence.

The leaves are hard to keep track of.

It's a great way to teach recursion.

Your grandfather's Fibonacci sequence.

So here I'm (Applause)

And when I'm going to figure these out, I don't have to do a huge amount of work.

The ratios of consecutive numbers also approach phi.

Or the Brussels sprouts on this stalk are a beautiful, delicious 3 and 5.

13.

The rest happens auto math ically.

The rest takes care of itself.

Or why not go back one more to 1, and so on?

You might have noticed that up here on the right, I made a very simple chart to try and explain how Fibonacci behaves to myself.

Just additional those 2 chart cells together and get the answer.

Another kind of statement is the return statement.

And you can keep going! try fibonacci of 3! it'll work! because we know fibonacci of 0,1,or 2 works cuz fibonacci of 3 will boil down to fibonacci of 2 plus fibonacci of 1. we know that fibonacci of 2 is 1, fibonacci of 1 is 1 1 1 is 2 so it'll just keep working. now we could try it out

Fibonacci numbers show up occasionally in nature.

The fibonacci sequence makes numerous appearances throughout the natural world.

This is the most practical application in the world.

This is certainly true.

Look familiar? 8 and 13 are both numbers in the 'Fibonacci series.'

Let's just go check.

This one was 8 and 13.

So there are Fibonacci numbers in pine cones.

Maybe plants just like these numbers, though.

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.

Let me try to prove that to you.

Now I can fill in the little pine coney things.

This pinecone that has 7 spirals one way and 11 the other, might be showing Lucas numbers.

21 13 just a little to small and 34 21 is even close than too big and so on.

It's 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.

The Fibonacci numbers only make sense when they're in order.

So just as we applied memoization to Fibonacci, we're going to apply memoization to our attempts to parse a grammar.

If you count the number of spiral going left, and the number of spirals going right, they'll be adjacent Fibonacci numbers.

We initialize our chart to be the empty mapping, and I'm going to define a procedure named memofibo.

The three sets of spirals in any pineapple are pretty much always adjacent Fibonacci 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

And now that we've seen all the theory behind it, we're going to code it up in Python together.

Either way is a perfectly good series, and it's got another similarity with the Fibonacci series.

And since Fibonacci numbers and Lucas numbers are related, maybe that explains it.

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.