8 Symmetries of a Regular Thirty-one Zone Star

8   Symmetries of a Regular Thirty-one Zone Star

1The dodecahedron and the icosahedron are duals of each other - the vertices of one match the face midpoints of the other and vice versa.

2 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.

3 The fifteen $C$ lines go through the edge midpoints of either the icosahedron or the dodecahedron.

4 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.

5 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.

6 All pairs of lines lie in one of-these six kinds of sections.

7 See Sections for a discussion of different angles and the polygons they form.