Pentagonal icositetrahedron

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Pentagonal icositetrahedron
Pentagonal icositetrahedron, anticlockwise twistPentagonal icositetrahedron
Click ccw or cw for spinning versions.
Type Catalan
Conway notation gC
Coxeter diagram CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Face polygon DU12 facets.png
irregular pentagon
Faces 24
Edges 60
Vertices 38 = 6 + 8 + 24
Face configuration V3.
Dihedral angle 136° 18' 33'
Symmetry group O, ½BC3, [4,3]+, 432
Dual polyhedron snub cube
Properties convex, face-transitive, chiral
Pentagonal icositetrahedron

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.


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.


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]

Orthogonal projections[edit]

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

Orthogonal projections
[3] [4]+ [2]
Image Dual snub cube A2.png Dual snub cube B2.png Dual snub cube e1.png
Snub cube A2.png Snub cube B2.png Snub cube e1.png

Related polyhedra and tilings[edit]

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.


External links[edit]