# 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[1] is a Catalan solid which is the dual of the snub cube. In crystallography it is also called a gyroid.[2][3]

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 kised 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:[4]

{\displaystyle {\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.

Projectivesymmetry Image Dualimage [3] [4]+ [2]

### 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

Spherical pentagonal icositetrahedron

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.

## References

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, MR 0730208 (The thirteen semiregular convex polyhedra and their duals, Page 28, Pentagonal icositetrahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)