Math 497 Assignment #4

Do several of these problems. There are rich sources for these idea on the Web, for example at these links.

Geometry and Golden Ratio

  1. If a regular pentagon has vertices 0, 1, 2, 3, 4, then the segments connecting 0, 2, 4, 1, 3, 0 form a 5-pointed star (a pentagram). Explain why the ratio of the pentagram side to the pentagon side is the golden ratio. (You can assume the pentagon side = 1 unit if you wish. Look for nested isosceles triangles that share an angle; they will be similar.

  2. Construct an approximate golden spiral using quarter-circles in the square of spiraling golden rectangles. Be sure you understand where the golden ratio comes in to the golden rectangle. Also, consider building up rectangles with successive Fibonacci sides.

Interpretation - Dudeney's Cows - see web link

  1. Look into some situations modeled by Fibonacci numbers besides rabbits. See the Web.

Lucas Numbers

  1. We saw that there were two sequences with the Fibonacci rule consisting of powers of tau and sigma. If you add these two you get the Lucas numbers. Check the Lucas formula.

Fibonacci Puzzles

  1. Find the web page of these puzzles, try some and report back.

Open ended

  1. Follow up one of the online sources and write up a page of some interesting aspect of the Fibonacci numbers and/or the Golden Ratio. Or do the same with the 4-dimensional shapes or spirals.

History or biography

  1. Look up some biographical about Fibonacci or about the history of the golden ratio or other current topic and write a half-page about what you found.