Pentagonal icositetrahedron

Pentagonal icositetrahedron
Click ccw or cw for spinning versions.
Type	Catalan
Face polygon	irregular pentagon
Faces	24
Edges	60
Vertices	38 = 6 + 8 + 24
Face configuration	V3.3.3.3.4
Dihedral angle	136° 20'
Symmetry group	O, [4,3]+, 432
Dual polyhedron	snub cube
Properties	convex, face-transitive, chiral
Net

In geometry, a pentagonal icositetrahedron is a Catalan solid which is the dual of the snub cube. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other.

If it has unit edge length, its surface area is $\scriptstyle{3}\sqrt{\tfrac{22(5t-1)}{4t-3}} \scriptstyle{\approx 19.29994}$ and its volume is $\sqrt{\tfrac{11(t-4)}{2(20t-37)}} \scriptstyle{\approx 7.4474}$. Here t is the tribonacci constant (see snub cube).

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.

V3.3.3.3.3 (332) and (532)	V3.3.3.3.4 (432)	V3.3.3.3.5 (532)
V3.3.3.3.6 (632)	V3.3.3.3.7 (732)

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, ISBN 978-0-521-54325-5, MR730208  (The thirteen semiregular convex polyhedra and their duals, Page 28, Pentagonal icositetrahedron)
• The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal icosikaitetrahedron)

External links

• Eric W. Weisstein, Pentagonal icositetrahedron (Catalan solid) at MathWorld.
• Pentagonal Icositetrahedron – Interactive Polyhedron Model