# 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: [itex]2^1 = 2[/itex].
• 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: [itex]2^2 = 4[/itex].
• 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: [itex]{2}^3 = 8[/itex].
• 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., [itex]2^{1.6}[/itex] = 3 (approx.)
• I collected the papers on Shape of Space.
• Groups finished up last explorations.

Wed:

• no class

Fri:

• Final exam, noon