Deltoidal hexecontahedron

Summary

Deltoidal hexecontahedron
Deltoidal hexecontahedron
(Click here for rotating model)
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
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′′ arccos(-19-85/41)
Properties convex, face-transitive
Small rhombicosidodecahedron.png
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 one of six Catalan solids to not have a Hamiltonian path among its vertices.[2]

3D model of a deltoidal hexecontahedron

It is topologically identical to the nonconvex rhombic hexecontahedron.

Lengths and anglesEdit

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

The angle between two short edges in a single face is arccos(-5-25/20)≈118.2686774705°. The opposite angle, between long edges, is arccos(-5+95/40)≈67.783011547435° . The other two angles of each face, between a short and a long edge each, are both equal to arccos(5-25/10)≈86.97415549104°.

The dihedral angle between any pair of adjacent faces is arccos(-19-85/41)≈154.12136312578°.

TopologyEdit

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.

Cartesian coordinatesEdit

The 62 vertices of the disdyakis triacontahedron fall in three sets:

  • Twelve vertices are of the form of a regular icosahedron.
  • Twenty vertices are of the form of a   scaled regular dodecahedron.
  • Thirty vertices are of the form of a   scaled Icosidodecahedron.

These hulls are visualized in the figure below:

 

Orthogonal projectionsEdit

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

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

VariationsEdit

 
This figure from Perspectiva Corporum Regularium (1568) by Wenzel Jamnitzer can be seen as a deltoidal hexecontahedron.

The deltoidal hexecontahedron can be constructed from either the regular icosahedron or regular dodecahedron by adding vertices mid-edge, and mid-face, and creating new edges from each edge center to the face centers. Conway polyhedron notation would give these as oI, and oD, ortho-icosahedron, and ortho-dodecahedron. These geometric variations exist as a continuum along one degree of freedom.

 

Related polyhedra and tilingsEdit

 
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

When projected onto a sphere (see right), it can be seen that the edges make up the edges of an icosahedron and dodecahedron arranged in their dual positions.

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.

*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Figure
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

See alsoEdit

ReferencesEdit

  1. ^ Conway, Symmetries of things, p.284-286
  2. ^ "Archimedean Dual Graph".
  • 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)
  • http://mathworld.wolfram.com/ArchimedeanDualGraph.html

External linksEdit

  • 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.