Course:Harris, Fall 08: Diary Week 16

Mon:

• I mentioned likely exam topics:
  • Compare/contrast the three geometries.
  • Show (with details) why there are five and only five regular tessellations of the sphere.
  • definitions:
    • regular polygon
    • tessellation
    • regular tessellation
    • fractal object (don't worry about dimension)
  • symmetry groups:
    • Identify a symmetry group of a design (using notes for Frieze or Wallpaper).
    • Draw a design with a specific symmetry group.
    • Find the symmetries--or similarity transformations--in a design.
    • Identify the product of two elements of a symmetry group.
• I tried to indicate the intention behind the construction of the Sierpinski triangle.
  • Drawing a large-scale version of it with six stages showing makes it a little clearer that if you separate out the top third and blow it up by a linear factor of two, you get the same thing again.
  • That is what makes it a fractional dimension:
    • For a 1-dimensional object, take 1/2 of it, blow it up by a linear factor of 2, and you get the same thing again.
      • Note: $2^{1}=2$.
    • For a 2-dimensional object, take 1/4 of it, blow it up by a linear factor of 2, and you get the same thing again.
      • Note: $2^{2}=4$.
    • For a 3-dimensional object, take 1/8 of it, blow it up by a linear factor of 2, and you get the same thing again.
      • Note: ${2}^{3}=8$.
    • For the Sierpinski triangle, take 1/3 of it, blow it up by a linear factor of 2, and you get the same thing again.
      • So that puts its dimension in between 1 and 2. Specifically, log(3)/log(2) = 1.6 (approx.)
        • i.e., $2^{1.6}$ = 3 (approx.)
• I collected the papers on Shape of Space.
• Groups finished up last explorations.

Wed:

• no class

Fri:

• Final exam, noon