By Artstein-Avidan S.
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Extra resources for A Bernstein-Chernoff deviation inequality, and geometric properties of random families of operators
1 First proof Think of 2π(V − E + F ) as putting +2π at each vertex, −2π on each edge, and +2π on each face. 64 We will try to cancel out the terms as much as possible, by grouping within polygons. For each edge, there is −2π to allocate. An edge has a polygon on each side: put −π on one side, and −π on the other. For each vertex, there is +2π to allocate: we will do it according to the angles of polygons at that vertex. If the angle of a polygon at the vertex is a, allocate a of the 2π to that polygon.
38 Figure 14: A string of paper dolls Figure 15: This wave pattern repeats horizontally, with no reflections or rotations. The quotient orbifold is a cylinder. We obtain a mirror string by starting somewhere on a mirror and walking along the mirror to the next crossing point, turning as far right as we can so as to walk along another mirror, walking to the next crossing point on it, and so on. ) Suppose that you walk along a mirror string until you first reach a point exactly like the one you started from.
It gives an object with interesting topological and geometrical properties, called an orbifold. 36 The first instance of this is an object with bilateral symmetry, such as a (stylized) heart. Children learn to cut out a heart by folding a sheet of paper in half, and cutting out half of the pattern. When you identify equivalent points, you get half a heart. A second instance is the paper doll pattern. Here, there are two different fold lines. You make paper dolls by folding a strip of paper zig-zag, and then cutting out half a person.