Final:

1. Compare and contrast Euclidean, spherical and hyperbolic geometry. What can you say about the following topics in each of the geometries: geodesics, polygons (which ones exist, which ones don’t), sum of the angles in a triangle, regular and semi-regular tessellations (how many?), isometries, area, Escher’s use of each.

2. Explain everything you know about hyperbolic geometry. (Discuss: geodesics, polygons, sum of the angles in a triangle, regular and semi-regular tessellations (how many?), isometries, area, Escher’s art based on hyperbolic geometry)

3. Explain everything you know about flatland and the 4th dimension. (Discuss: the inhabitants of Flatland, the class structure of flatland, how they recognize one another. Describe A. Square’s trip to Line Land. Describe A. Square’s meeting with the sphere. What would happen if we met a being from the 4th dimension?

4. Compare and contrast the axioms for Euclidean, spherical and hyperbolic geometry.






Expect to be asked to do the following:
1. Compute the area of a polygon in hyperbolic space. (I will give you the formula)
2. Create an ideal hyperbolic tessellation.
3. Find a wallpaper pattern for a given tessellation.

Look at all your homework and worksheets. The final is cumulative, so all of that is “fair game”. I will not put the theory of Symmetry groups on the Final (so this means no questions about: list the elements, find subgroups etc. I MAY ask you to identify the symmetry group of a rosette, border or wallpaper pattern though!)