Pentagonal icositetrahedron

Construction

The pentagonal icositetrahedron can be constructed from a snub cube without taking the dual. Square pyramids are added to the six square faces of the snub cube, and triangular pyramids are added to the eight triangular faces that do not share an edge with a square. The pyramid heights are adjusted to make them coplanar with the other 24 triangular faces of the snub cube. The result is the pentagonal icositetrahedron.

Cartesian coordinates

Denote the tribonacci constant by t\approx 1.839,286,755,21. (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 12 even permutations of (±1, ±(2t+1), ±t2) with an even number of minus signs
• the 12 odd permutations of (±1, ±(2t+1), ±t2) with an odd number of minus signs
• the 6 points (±t3, 0, 0), (0, ±t3, 0) and (0, 0, ±t3)
• the 8 points (±t2, ±t2, ±t2)

The convex hulls for these vertices{{cite journal

:[[File:Pentagonal Icositetrahedron.svg|400px|Combining an octahedron and snub cube to form the Pentagonal Icositetrahedron ]]

Geometry

The pentagonal faces have four angles of \arccos((1-t)/2)\approx 114.812,074,477,90^{\circ} and one angle of \arccos(2-t)\approx 80.751,702,088,39^{\circ}. The pentagon has three short edges of unit length each, and two long edges of length (t+1)/2\approx 1.419,643,377,607,08. The acute angle is between the two long edges. The dihedral angle equals \arccos(-1/(t^2-2))\approx 136.309,232,892,32^{\circ}.

If its dual snub cube has unit edge length, its surface area and volume are: :\begin{align} 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{align}

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.

Among the variations mentioned above, Scott Sherman reported another case in which the pentagonal faces are bilaterally symmetrical in 2014. He called it the Bilateral Pentagonal Icositetrahedron. In 2020, Tadaki Takahashi provided a geometric proof for this solid, naming it the Shogihedron because the shape of its faces resembles the pieces in the Japanese board game Shogi.

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.

References

• (Section 3-9)
• (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, https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)

References

1. Conway, Symmetries of things, p.284
2. "Promorphology of Crystals I".
3. "Crystal Form, Zones, & Habit".
4. "Pentagonal icositetrahedron".
5. http://makerhome.blogspot.com/2014_05_10_archive.html
6. https://hyogo-u.repo.nii.ac.jp/record/16830/files/6208.pdf {{Bare URL PDF. (January 2026)