# Pentagonal icositetrahedron

Pentagonal icositetrahedron

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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.

## 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 ${\displaystyle \cos ^{-1}\left({\frac {1-t}{2}}\right)\approx }$ 114.8° and one angle of ${\displaystyle \cos ^{-1}(2-t)\approx }$ 80.75°. The pentagon has three short edges of unit length each, and two long edges of length ${\displaystyle {\frac {t+1}{2}}\approx 1.42}$. The acute angle is between the two long edges.

If its dual snub cube has unit edge length, its surface area is ${\displaystyle \scriptstyle {3}{\sqrt {\tfrac {22(5t-1)}{4t-3}}}\scriptstyle {\approx 19.29994}}$ and its volume is ${\displaystyle {\sqrt {\tfrac {11(t-4)}{2(20t-37)}}}\scriptstyle {\approx 7.4474}}$.[4]

## Orthogonal projections

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

Projective Image Dual symmetry image [3] [4]+ [2]

## 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.