Pentagonal icositetrahedron

Pentagonal icositetrahedron  (Click ccw or cw for rotating models.)
Type Catalan
Conway notation gC
Coxeter diagram     Face polygon irregular pentagon
Faces 24
Edges 60
Vertices 38 = 6 + 8 + 24
Face configuration V3.3.3.3.4
Dihedral angle 136° 18' 33'
Symmetry group O, ½BC3, [4,3]+, 432
Dual polyhedron snub cube
Properties convex, face-transitive, chiral Net

In geometry, a pentagonal icositetrahedron or pentagonal icosikaitetrahedron is a Catalan solid which is the dual of the snub cube. In crystallography it is also called a gyroid.

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

Construction

The pentagonal icositetrahedron can be constructed from a snub cube without taking the dual. The 6 square faces of the snub cube are raised to a height that the new triangles are coplanar with the triangles, and tetrahedra (not necessarily regular tetrahedra) are added to the 8 triangular faces that do not share an edge with a square to a height that the new triangles of the raised tetrahedra become coplanar to the triangles which do share an edge with a square. The result is the pentagonal icositetrahedron.

Geometry

Denote the tribonacci constant by t, approximately 1.8393. (See snub cube for a geometric explanation of the tribonacci constant.) Then the pentagonal faces have four angles of cos−1 (1 − t/2) ≈ 114.8° and one angle of cos−1 (2 − t) ≈ 80.75°. The pentagon has three short edges of unit length each, and two long edges of length t + 1/2 ≈ 1.42. The acute angle is between the two long edges.

If its dual snub cube has unit edge length, its surface area and volume are:

{\begin{aligned}A&=3{\sqrt {\frac {22(5t-1)}{4t-3}}}&&\approx 19.299\,94\\V&={\sqrt {\frac {11(t-4)}{2(20t-37)}}}&&\approx 7.4474\end{aligned}}

Orthogonal projections

The pentagonal icositetrahedron has three symmetry positions, two centered on vertices, and one on midedge.

Variations

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

This variation shown can be constructed by adding pyramids to 6 square faces and 8 triangular faces of a snub cube such that the new triangular faces with 3 coplanar triangles merged into identical pentagon faces. Snub cube with augmented pyramids and merged faces Pentagonal icositetrahedron Net

Related polyhedra and tilings

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.