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Deltoidal hexecontahedron

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Deltoidal hexecontahedron

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)
Deltoidal hexecontahedron net
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.

Contents

  • Lengths and angles 1
  • Topology 2
  • Orthogonal projections 3
  • Related polyhedra and tilings 4
  • See also 5
  • References 6
  • External links 7

Lengths and angles

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°.

Topology

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

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

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

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} t{3,5} {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.

*n42 symmetry mutation of expanded tilings: 3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
 
[12i,3]
 
[9i,3]
 
[6i,3]
Figure
Config. 3.4.2.4 3.4.3.4 3.4.4.4 3.4.5.4 3.4.6.4 3.4.7.4 3.4.8.4 3.4.∞.4 3.4.12i.4 3.4.9i.4 3.4.6i.4

See also

References

  1. ^ Conway, Symmetries of things, p.284-286
  • (Section 3-9)  
  • The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [2] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal hexecontahedron)

External links

  • Eric W. Weisstein, DeltoidalHexecontahedron and Hamiltonian path (Catalan solid) at MathWorld
  • Deltoidal Hexecontahedron (Trapezoidal Hexecontrahedron)—Interactive Polyhedron Model
  • Example in real life—A ball almost 4 meters in diameter, from ripstop nylon, and inflated by the wind. It bounces around on the ground so that kids can play with it at kite festivals.


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