Course:Harris, Fall 08: Diary Week 16

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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