and then a triangle.

So, I've converted this square into a nice rectangle,

the sides are one, two,three, four, five,

by 1, 2, 3, 4,

5, 6, 7,8, 9, 10,

11, 12, 13, a five by 13 rectangle.

So, the number of small boxes here,

is five times 13 is 65.

So, we've gone from 64 small boxes,

to 65 small boxes.

If I could do that with gold,

that would have created one small box of gold.

So, I would be extremely wealthy man.

So, obviously, I can't do that.

I have bamboozled you.

That's why it's called the Fibonacci bamboozlement.

So, what's special about the size of the square and the rectangle?

The square has side eight,

the rectangle has sides five and 13.

Five, eight and 13 are Fibonacci numbers.

There's a relationship between these numbers called Cassini's Identity,

which together with the golden ratio,

can explain how this puzzle can bamboozle you.

We'll derive Cassini's identity in this course,

and we'll also discuss

other interesting mathematical relationships that

connect the Fibonacci numbers and the golden ratio.

I'll also show you how to construct what, I think,

are two of the most beautiful pictures in mathematics.

One that leads to a spiral,

called the golden spiral,

and one that leads to another spiral called the Fibonacci spiral.

In fact, I used the Fibonacci spiral with

the first six Fibonacci numbers as the icon for this course.

Look at this beautiful picture.

How did I get interested in this subject?

Well, at my University,

I teach a course on Mathematical Biology.

In this course, I discuss Fibonacci's famous rabbit problem,

which is one of the first applications of Mathematics to Biology.

I also talk about how the Fibonacci numbers can pop up unexpectedly in nature.

One of my favorite examples has to do with the arrangement of florets,

the small little flowers in the head of a sunflower.

The Fibonacci numbers appear as the number of spirals seen in the head.

Their appearance here, is related to the relationship to the golden ratio.

This special number, square root of five plus one divided by two.

This is an irrational number that turns out to be

very difficult to approximate by a rational number.

We'll talk about that in some detail in this course.

So, how do I hope that you take this course?

I hope that you'll watch all the videos,

where I explain the mathematics about the Fibonacci numbers and the golden ratio.

I hope you'll try to solve some of the problems

that I'll post in the discussion sections.

You really need to solve problems to learn mathematics.

After you work on the problems,

you can then try and take the multiple choice quizzes that I've written,

that will test your knowledge on the course material and the problems.

Of course, you're free to just watch the videos and enjoy the course.

I hope you will enjoy the course.

But if you really want to learn the mathematics,

then I urge you to try and solve some of the problems yourself.

So, I hope you enjoy the course,

and I look forward to interacting with many of you in

the discussion forums. See you later.