# Deltoidal hexecontahedron

Deltoidal hexecontahedron

click for spinning version
Type Catalan
Conway notation oD or deD
Coxeter diagram
Face polygon
kite
Faces 60
Edges 120
Vertices 62 = 12 + 20 + 30
Face configuration V3.4.5.4
Symmetry group Ih, H3, [5,3], (*532)
Rotation group I, [5,3]+, (532)
Dihedral angle 154° 7' 17"
Properties convex, face-transitive

rhombicosidodecahedron
(dual polyhedron)

Net

In geometry, a deltoidal hexecontahedron (also sometimes called a trapezoidal hexecontahedron, a strombic hexecontahedron, or a tetragonal hexacontahedron[1]) is a catalan solid which looks a bit like either an overinflated dodecahedron or icosahedron. It is sometimes also called the trapezoidal hexecontahedron or strombic hexecontahedron. Its dual polyhedron is the rhombicosidodecahedron.

The 60 faces are deltoids or kites (not trapezoidal). The short and long edges of each kite are in the ratio 1.00:1.54.

It is the only Archimedean dual which does not have a Hamiltonian path among its vertices.

## Orthogonal projections

The deltoidal hexecontahedron has 3 symmetry positions located on the 3 types of vertices:

Projective Image Dual symmetry image [2] [6] [10]

## Related polyhedra and tilings

Spherical deltoidal 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 tiling is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.n.4), and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.

Dimensional family of expanded spherical polyhedra and tilings: 3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]

*832
[8,3]...

*∞32
[∞,3]

[12i,3] [9i,3] [6i,3] [3i,3]
Figure
Schläfli rr{2,3} rr{3,3} rr{4,3} rr{5,3} rr{6,3} rr{7,3} rr{8,3} rr{∞,3} rr{12i,3} rr{9i,3} rr{6i,3} rr{3i,3}
Coxeter
Dual uniform figures
Dual
Config.

V3.4.2.4

V3.4.3.4

V3.4.4.4

V3.4.5.4

V3.4.6.4

V3.4.7.4

V3.4.8.4

V3.4.∞.4
V3.4.12i.4 V3.4.9i.4 V3.4.6i.4 V3.4.3i.4
Coxeter