Deltoidal hexecontahedron

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Deltoidal hexecontahedron
Deltoidal hexecontahedron
click for spinning version
Type Catalan
Conway notation oD or deD
Coxeter diagram CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png
Face polygon DU27 facets.png
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
Small rhombicosidodecahedron.png
(dual polyhedron)
Deltoidal hexecontahedron 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 is the dual polyhedron of the rhombicosidodecahedron, an Archimedean solid. It is the only Catalan solid which does not have a Hamiltonian path among its vertices.

Lengths and angles[edit]

The 60 faces are deltoids or kites. The short and long edges of each kite are in the ratio 6:(7+√5) ≈ 1:1.539344663...

The angle between two short edges is 118.22°. The opposite angle, between long edges, is 67.76°. The other two angles, between a short and a long edge each, are both 87.01°.

The dihedral angle between all faces is 154.12°.


Topologically, the deltoidal hexecontahedron is identical to the nonconvex rhombic hexecontahedron. The deltoidal hexecontahedron can be derived from a dodecahedron (or icosahedron) by pushing the face centers, edge centers and vertices out to different radii from the body center. The radii are chosen so that the resulting shape has planar kite faces each such that vertices go to degree-3 corners, faces to degree-five corners, and edge centers to degree-four points.

Orthogonal projections[edit]

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

Orthogonal projections
[2] [6] [10]
Image Dual dodecahedron t02 f4.png Dual dodecahedron t02 A2.png Dual dodecahedron t02 H3.png
Dodecahedron t02 f4.png Dodecahedron t02 A2.png Dodecahedron t02 H3.png

Related polyhedra and tilings[edit]

Spherical deltoidal hexecontahedron
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3. V3.4.5.4 V4.6.10 V3.

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.

*n42 symmetry mutation of expanded tilings: 3.4.n.4
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
Figure Spherical triangular prism.png Uniform tiling 332-t02.png Uniform tiling 432-t02.png Uniform tiling 532-t02.png Uniform polyhedron-63-t02.png H2 tiling 237-5.png H2 tiling 238-5.png H2 tiling 23i-5.png H2 tiling 23j12-5.png H2 tiling 23j9-5.png H2 tiling 23j6-5.png
Config. 3.4.∞.4 3.4.12i.4 3.4.9i.4 3.4.6i.4

See also[edit]


  1. ^ Conway, Symmetries of things, p.284-286
  • 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)
  • 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 286, tetragonal hexecontahedron)

External links[edit]