Truncated Icosahedral Symmetry

A truncated icosahedron is a polyhedron with 32 total faces, 12 pentagons and 20 hexagons. It has 60 vertices, each with 3 edges meeting for a total of 90 edges. A TI has three defining dimensions: the two dihedral angles, 138 degrees 11 min (hexagon-hexagon) and 142 degrees 37 min (hexagon-pentagon), and the length of an edge. All other dimensions can be derived from these three using trigonometry.

One advantage to using a truncated icosahedron is its high symmetry. As shown in Figure 1, the TI has 2-, 3-, and 5-fold symmetry. This high symmetry greatly simplifies both the design and machining of the prototype.

Figure 1: TI symmetry.