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By Barmak J.A., Minian E.G.

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9. 31 32 Figure 11: Some surfaces Problems 1. Take some strips and join the opposite ends of each strip together as follows: with no twists; with one twist (half-turn); this is called a M¨ obius strip; with two twists; with three twists. 2. Imagine that you are a two-dimensional being who lives in one of these four surfaces. To what extent can you tell exactly which one it is? 3. Now cut each of the above along the midline of the original strip. Describe what you get. Can you explain why? 4. What is the Euler number of a disk?

Here are the types of some of the patterns shown in section 31: 47 Figure 16: •; Figure 17: ∗2222; Figure 18: ∗∗; Figure 19: ∗◦. Figure 20: ∗632. Figure 21: 632. Apart from the spots and circles, these can be read directly from the pictures. The important thing to remember is that if two things are equivalent by a symmetry, then you only record one of them. A dodecahedron is very like a sphere. The orbifold corresponding to its symmetry group is a spherical triangle having angles π/5, π/3, π/2; so its symmetry group is ∗532.

A wave pattern is the next example. This pattern repeats horizontally, with no reflections or rotations. The wave pattern can be rolled up into a cylinder. It can be constructed by rolling up a strip of paper around a cylinder, and cutting a single wave, through several layers, with a sharp knife. When it is unrolled, the bottom part will be like the waves. When a pattern repeats both horizontally and vertically, but without reflections or rotations, the quotient orbifold is a torus. You can think of it by first rolling up the pattern in one direction, matching up equivalent points, to get a long cylinder.

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2-Dimension from the Topological Viewpoint by Barmak J.A., Minian E.G.

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