Day 16, MAT115

• Please read Pascal's Triangle, Idea #13, p. 52.
• To get an early start on our next topic, visit http://www.mathsisfun.com/platonic_solids.html to learn about the platonic solids.

Make sure that you send me the final image, and not just a link (that won't do anything!). Right click on the image, and save it to your computer -- then past the image into your email. That's the easiest way, I think.

Let's see how we're doing so far....

• "The Sexiest Rectangle" - or is it just cute?
  • Important property: if you cut out the largest square possible, you still have a golden rectangle! There's a "world within a world" in the golden rectangle....

  • The golden mean (or golden ratio, golden section) -- A "precious jewel" (Johannes Kepler)
    • derived by ratios of Fibonacci numbers, or
    • using the rectangle directly. The Greeks defined a golden rectangle by calling it the rectangle which, when the largest square possible is removed, retains its shape (which means that the ratio of side lengths doesn't change).
    • The golden rectangle has side lengths in the ratio

      Note that the value is only approximately 1.618: is an irrational number, like , and its decimal representation goes on forever and never repeats:

      

• Now: what kind of rectangle will you get if you repeat the Fibonacci spiral process with another kind of rectangle to start (rather than a square)?

  Turns out, that if you do it over and over and over again, you'll get a rectangle that gets more and more golden.

  Let's try: we'll use a piece of grid paper (looking short and squat), and a 3x1 rectangle (start it running the tall-way at the 12th row down, 8th cell over).

  Now, let's figure out what the recurrence relation is for the dimensions of our rectangles, and then figure out if we're getting golden....

• The A0 Paper Fractal

  "A paper" is something you've no doubt encountered before: it's the long sheets we occasionly use (usually "A4" paper), which is far more common in Europe.

  "A paper" is constructed so that, if folded in half (do we say the long way or the short way?), you get a sheet which has exactly the same shape -- that is, the ratio of its side lengths is the same as the original sheet.

  

  Let's see what the dimensions of A paper must be so that this requirement is fulfilled....

  Call the long dimension of the A1 paper above , and the short dimension 1.

  • Then what are the dimensions of the A2 paper, according to the design of A paper?
  • What is the value of ?

• Why not an A0 paper spiral? Let's try that.... This is the spiral, created using non-squares each time:
  

  In both cases there's "the world within the world". Both are fractals.

• This triangle is much older than Blaise Pascal, who lived in the 17th century (it dates at least to China, long before -- 13th century AD -- so at about the same time as Fibonacci was doing his thing in Italy....)

  One thing this allows us to do is to understand how this culture wrote its numbers (Chinese Bamboo Counting Rods). How did the Chinese of the 13th century write 15? 28? 30?

• The other Pascal's Triangle:

• How is it constructed? Easy!
  

• What's it good for?
  • Pascal was interested in gambling -- that's why he studied it so hard.
  • How many ways can you put 7 people into 2 cars to go to the party?
  • Triangular numbers
  • Tetrahedral numbers

• Where's Fibonacci?