I got the idea for this design some weeks ago and have been itching to create it. It's not my idea (I saw it here, but this version is my own. As some of you know I'm remodeling a viking camp into a math learning center, and I'm having a blast making art, designing rooms and much more.
I had a free afternoon on Tuesday and so I wrote some code (with Processing) to create the design. Advantage of coding it instead of using Illustrator is that (a) it's perfect and (b) I can make an image 8000 pixels wide for example so I can print it big. (It will be 1 meter x 1.618 meters, printed on acrylic, and mounted on a door in the banquet room).
It's a golden rectangle, divided along the diagonal. It is then filled with the largest squares possible, then the next largest, then the next, and so on. There's some amazing properties:
(1) In each half, there is 1 large square, 1 next large, 2 of the next, then 3, 5, 8, 13, 21, ... Fibonacci numbers. Yes indeed. Yummy.
(2) If we say the base is 1 and the height is phi ( (1 + √5) / 2, the golden ratio, ≈1.618), the side length of the largest square is 1/phi, the next is 1/phi^2, the next 1/phi^3, etc.
(3) We can then see in the picture that the vertical stack of squares implies 1/phi + 1/phi^2 + 1/phi^3 + ... = phi.
(4) The corners of the squares on each half align perfectly. By looking how the large square on one side matches with the second-largest square on the other side, we see that 1/phi + 1/phi^2 = 1, the width.
and finally...
(5) #1 - 4 above are fun to prove! I found #1 the trickiest as it is not just straightforward algebra manipulation, but oh so satisfying. Anyone want to take a stab at proofs? Or are there other things you see in there?
Awesome! If I ever own a house again, I want to do a dragon curve in the kitchen tile.