# Pentagonal hexecontahedron

Pentagonal hexecontahedron

Type Catalan solid
Coxeter diagram
Conway notation gD
Face type V3.3.3.3.5

irregular pentagon
Faces 60
Edges 150
Vertices 92
Vertices by type 12 {5}
20+60 {3}
Symmetry group I, ½H3, [5,3]+, (532)
Rotation group I, [5,3]+, (532)
Dihedral angle 153° 10' 43"
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.

## Orthogonal projections

The pentagonal hexecontahedron has three symmetry positions, two on vertices, and one mid-edge.

Projective Image Dual symmetry image [3] [5]+ [2]

## Related polyhedra and tilings

Spherical pentagonal hexecontahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5

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.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+
Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3
332
[3,3]+
T
432
[4,3]+
O
532
[5,3]+
I
632
[6,3]+
P6
732
[7,3]+
832
[8,3]+...
∞32
[∞,3]+
Snub
figure

3.3.3.3.2

3.3.3.3.3

3.3.3.3.4

3.3.3.3.5

3.3.3.3.6

3.3.3.3.7

3.3.3.3.8

3.3.3.3.∞
Coxeter
Schläfli

sr{2,3}

sr{3,3}

sr{4,3}

sr{5,3}

sr{6,3}

sr{7,3}

sr{8,3}

sr{∞,3}
Snub
dual
figure

V3.3.3.3.2

V3.3.3.3.3

V3.3.3.3.4

V3.3.3.3.5

V3.3.3.3.6

V3.3.3.3.7
V3.3.3.3.8
V3.3.3.3.∞
Coxeter