Pentagonal icositetrahedron

[2][3] It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

The pentagonal icositetrahedron can be constructed from a snub cube without taking the dual.

The pyramid heights are adjusted to make them coplanar with the other 24 triangular faces of the snub cube.

(See snub cube for a geometric explanation of the tribonacci constant.)

Then Cartesian coordinates for the 38 vertices of a pentagonal icositetrahedron centered at the origin, are as follows: The convex hulls for these vertices[4] scaled by

result in a unit circumradius octahedron centered at the origin, a unit cube centered at the origin scaled to

, and an irregular chiral snub cube scaled to

, as visualized in the figure below: The pentagonal faces have four angles of

If its dual snub cube has unit edge length, its surface area and volume are:[5] The pentagonal icositetrahedron has three symmetry positions, two centered on vertices, and one on midedge.

Isohedral variations with the same chiral octahedral symmetry can be constructed with pentagonal faces having 3 edge lengths.

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.

The pentagonal icositetrahedron is second in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n.

The pentagonal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.