Pentagonal hexecontahedron

Pentagonal hexecontahedron
(Click here for rotating model)
Type	Catalan solid
Face type	irregular pentagon
Faces	60
Edges	150
Vertices	92
Vertices by type	12 {5} 20+60 {3}
Face configuration	V3.3.3.3.5
Symmetry group	I, [5,3]+, 532
Dihedral angle	153° 10' 24"
Properties	convex, face-transitive chiral
Snub dodecahedron (dual polyhedron)	Net

In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. It is also well-known to be the Catalan solid with the most vertices. Among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices.

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 29, Pentagonal hexecontahedron)
• 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 hexecontahedron )

External links

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