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.
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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. |