Math 497 Class 4 -- 1/31/02

Discussion of Higher Dimensional "Tetrahedra" and Hypercubes

• Defining and visualizing the n-simplex and n-hypercube

• How many vertices?

• Is this a symmetric 2-d picture of a simplex in 4-space? Or is the more pyramidal point of view clearer?

Return to Difference schemes and Newton

• Why it works. Notice that one can add and multiply difference schemes and get new ones. Working backward from simple diagrams is the Newton method.

Fibonacci Numbers

• Definition of Fibonacci numbers: 1 1 2 3 5 8 13 21 …
• Difference of Fibonacci sequence is a Fibonacci sequence
• Fibonacci Numbers in the Pascal Triangle
• Adding Fibonacci-like sequences

• The Golden Ratio as Fibonacci limit by dividing F(n+1) = F(n) + F(n-1) by F(n).
• Roots of x²-x-1 = 0 or x² = x+1 are numbers t and s which are the positive and negative roots:
• The sequences 1 x x² x³ x⁴ … are Fibonacci type sequences if x = t or x = s.
• Subtracting one from the other and dividing by t - s gives a remarkable formula for the Fibonaccis. Look ahead to pp. 202-3.

Golden Spiral and other spiraling

•  Golden spiral built from "Golden Rectangle", the only rectangle from which you can add (or cut) a square and get a rectangle of the same shape. See http://www.math.ubc.ca/~hoek/Teaching/Golden/Divina.html

• Modular arithmetic and spiraling around a 5-gon, 7-gon and 11-gon to find inverses, etc. (Contrast with 12-gon.)

o       Multiples of 2 mod 5:

o       Multiples of 3 mod 7

o       Multiples of 3 mod 12 (starting with 0)

•   Looking for spiral arms in the logarithmic spiral - more next time