By Conway, Doyle, Gilman, Thurston.
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An n-way crossing point is one where precisely n mirrors meet. At an n-way crossing point, adjacent mirrors meet at an angle of π/n. ) 37 mirror Figure 13: A heart is obtained by folding a sheet of paper in half, and cutting out half a heart. The half-heart is the orbifold for the pattern. A heart can also be recreated from a half-heart by holding it up to a mirror. 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.
1 The prefix The prefix tells the number of directions from which you can view the pattern without noticing any difference. The possibilities for the prefix are: hexa-; tetra-; tri-; di-; mono-. For example, if you are looking at a standard brick wall, it will look essentially the same whether you stand on your feet or on your head. This will be true even if the courses of bricks in the wall do not run parallel to the ground, as they invariably do. Thus you can recognize right away that the brick-wall pattern is di-something-or-other In fact, it is dirhombic.
We will begin by discussing Conway’s names for the orbifolds. A reproduction of Conway’s manuscript appears at the end of the section. You should refer to the reproduction as you try to understand the basis for the names. 1 Conway’s names Each of Conway’s 17 names consists of two parts, a prefix and a descriptor. 1 The prefix The prefix tells the number of directions from which you can view the pattern without noticing any difference. The possibilities for the prefix are: hexa-; tetra-; tri-; di-; mono-.