In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron.

It has two distinct forms, which are mirror images (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.

The faces have four equal obtuse angles and one acute angle (between the two long edges).

The obtuse angles equal

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 shorter side of one of the pentagonal faces as

Using these, one can calculate the measure of sphericity for this shape:

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.

[2] An alternate construction method uses quaternions and the icosahedral symmetry of the Weyl group orbits

Specifically, with quaternions from the binary Icosahedral group

is the symmetry group of the 600-cell and the 120-cell of order 14400, we have

gives the 60 twisted chiral snub dodecahedron coordinates, where

is one permutation from the first set of 12 in those listed above.

is obtained by taking the solution to

Using the Icosahedral symmetry in the orbits of the Weyl group

of order 60[4] gives the following Cartesian coordinates with

Negating all vertices in both groups gives the mirror of the chiral snub dodecahedron, yet results in the same pentagonal hexecontahedron convex hull.

Isohedral 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.

The pentagonal hexecontahedron has three symmetry positions, two on vertices, and one mid-edge.

This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n).

(The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.