Pentagonal hexecontahedron
In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images 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.
Construction
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.Geometry
The faces are irregular pentagons with two long edges and three short edges. Let be the real zero of the polynomial, where is the golden ratio.Then the ratio of the edge lengths is given by:
The faces have four equal obtuse angles and one acute angle. The obtuse angles equal, and the acute one equals. The dihedral angle equals.
Note that the face centers of the snub dodecahedron cannot serve directly as vertices of the pentagonal hexecontahedron: the four triangle centers lie in one plane but the pentagon center does not; it needs to be radially pushed out to make it coplanar with the triangle centers. Consequently, the vertices of the pentagonal hexecontahedron do not all lie on the same sphere and by definition it is not a zonohedron.
To find the volume and surface area of a pentagonal hexecontahedron, denote the longer side of one of the pentagonal faces as, and set a constant t.
Then the surface area is:
And the volume is:
Variations
variations can be constructed with pentagonal faces with 3 edge lengths.This variation shown can be constructed by adding pyramids to 12 pentagonal faces and 20 triangular faces of a snub dodecahedron such that the new triangular faces are coparallel to other triangles and can be merged into the pentagon faces.
Snub dodecahedron with augmented pyramids and merged faces | Example variation | Net |
Orthogonal projections
The pentagonal hexecontahedron has three symmetry positions, two on vertices, and one mid-edge.| Projective symmetry | + | ||
| Image | |||
| Dual image |