Pentagonal hexecontahedron

In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are s (or "enantiomorphs") of each other. It has 92 vertices that span 60 pentagonal faces. It is the Catalan solid with the most vertices. Among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices. Using the Icosahedral symmetry of the Weyl orbits of order 60 gives the following Cartesian coordinates with is the golden ratio: Twelve vertices of a regular icosahedron with unit circumradius centered at the origin with the coordinates Twenty vertices of regular dodecahedron of unit circumradius centered at the origin scaled by a factor from the exact solution to the equation , which gives the coordinates and Sixty vertices of a unit circumradius chiral snub dodecahedron scaled by . There are five sets of twelve vertices, all with even permutations (i.e. with a parity signature=1). A group of two sets of twelve have 0 or 2 minus signs (i.e. 1 or 3 plus signs): and another group of three sets of 12 have 0 or 2 plus signs (i.e. 1 or 3 minus signs): Negating all vertices in both groups gives the mirror of the chiral snub dodecahedron, yet results in the same pentagonal hexecontahedron convex hull. The pentagonal hexecontahedron can be constructed from a snub dodecahedron without taking the dual. Pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal hexecontahedron. An alternate construction method uses quaternions and the icosahedral symmetry of the Weyl group orbits of order 60. This is shown in the figure on the right. Specifically, with quaternions from the binary Icosahedral group , where is the conjugate of and and , then just as the Coxeter group is the symmetry group of the 600-cell and the 120-cell of order 14400, we have of order 120.