Welcome to my course, on the Fibonacci numbers and the golden ratio. I'm standing here in the hallway of the Mathematics department at my university, the Hong Kong University of Science and Technology. As you can see, we have some beautiful art here, all mathematically related, but I think my favorite painting is this one right here, of Friar Luca Pacioli, doing mathematics. He's here demonstrating a theorem of Euclid, by doing a sketch of this 12 sided platonic solid, beautiful crystal, half filled with water. You can see the line of water there. The reason I love this painting, is because the Friar was indeed a renaissance man, and he published a book called, On the Divine Proportion, which is about the golden ratio. Illustrations in that book were done by Leonardo da Vinci, the most famous Renaissance man. There's even some speculation that this painting was also painted by the famous Leonardo. My course, is a course that's not usually offered in the university, but I have the opportunity to present it on the web, just for fun. What I want to do, is to act as the curator of all of the Mathematics on Fibonacci numbers and the golden ratio. If you look on the web, you can find lots of information, and there's lots of information in the university library also. So, let's go now. I want to show you a fascinating puzzle, that was a favorite of Lewis Carroll, the author of Alice in Wonderland. In front of me, I have a puzzle called the Fibonacci bamboozlement. The puzzle starts as a square, with sides one, two, three, four, five, six, seven, eight, by one, two, three, four, five, six, seven, eight. A square, eight by eight, so it has 64 of these little boxes. I can take this square, and rearrange the pieces. So, I put the trapezoid on the bottom. Put a triangle on the top, and then another trapezoid, 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.