# Course:Harris, Fall 08: Diary Week 16

From EscherMath

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.)

- So that puts its dimension in between 1 and 2. Specifically, log(3)/log(2) = 1.6 (approx.)

- 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.

- I collected the papers on Shape of Space.
- Groups finished up last explorations.

Wed:

- no class

Fri:

- Final exam, noon