January 16 Lecture (The Golden Section, or The Golden Ratio).

1. Mona Lisa again. Beautiful faces have proportions of the Golden Rectangle, according to a study.

2. A Golden Tree Not only faces are related to the Golden Ratio.

3. Here is link straight to a webMathematica script for playing with basic tree fractals. Additional information can be found in the MathInArt (webMathematica) page.

4. Penrose tiling, an important tiling of the plane using Golden Triangles.

January 23 Lecture (The Golden Section and Fibonacci Numbers).

1. Here is a Sunflower movie. You can control it by right mouse clicks or by control+click (Mac). How is this related to the Golden Section or Fibonacci numbers? It is related to the Golden Section in two non-obvious ways. Can you find them? One of these two ways is connected to the fact that 34/21=1.619 which is approximately the same as the Golden Section. Answers/explanations in the class.

2. This is the middle frame of the above movie. Where is the Golden Section? Something strange happens with the angles of 222.492 degrees: the two line segments that are the sides of these angle cross the points in the sunflower in almost exactly the same way. Here is a picture.

3. Just a sunflower movie, for the fun of it.

4. Use a web-search engine and type, say, Fibonacci and sunflower as keywords to find pictures of flowers displaying (in various ways) Fibonacci numbers. Here is one such page ; here is one more; and the last link here.

5. A script for making "sunflowers" can be accessed directly or through the webMathematica page for this course.