8 Symmetries of a Regular Thirty-one Zone Star
2The dodecahedron and the icosahedron are duals of each other - the vertices of one match the face midpoints of the other and vice versa.
3 The ten B lines of the thirty-one zone star go through the vertices of the dodecahedron or, equivalently, the face midpoints of the icosahedron while the six A lines go through the vertices of the icosahedron or, equivalently, the face midpoints of the dodecahedron.
4 The fifteen C lines go through the edge midpoints of either the icosahedron or the dodecahedron.
5 The middles of edges are commonly midway between vertices or face midpoints and C lines bisect all angles between A lines and three of the four kinds of angles formed between B lines.
6 In examining angles between lines, we are also examining equators. There are six different equators and slicing through them, we form the R, S, T, V , X, and Y sections.
7 All pairs of lines lie in one of-these six kinds of sections.
8 See Sections for a discussion of different angles and the polygons they form.
