Deltoidal hexecontahedron

Deltoidal hexecontahedron
Type	Catalan solid
Faces	60 kites
Edges	120
Vertices	62
Symmetry group	icosahedral symmetry ${\displaystyle \mathrm {I} _{\mathrm {h} }}$
Properties	convex, face-transitive
Net

3D model of a deltoidal hexecontahedron

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]

It is topologically identical to the nonconvex rhombic hexecontahedron.

Lengths and angles

The 60 faces are deltoids or kites. The short and long edges of each kite are in the ratio ${\textstyle 1:{\frac {7+{\sqrt {5}}}{6}}}$ ≈ 1:1.539344663...

The angle between two short edges in a single face is ${\textstyle \arccos({\frac {-5-2{\sqrt {5}}}{20}})}$ ≈ 118.2686774705°. The opposite angle, between long edges, is ${\textstyle \arccos({\frac {-5+9{\sqrt {5}}}{40}})}$ ≈ 67.783011547435°. The other two angles of each face, between a short and a long edge each, are both equal to ${\textstyle \arccos({\frac {5-2{\sqrt {5}}}{10}})}$ ≈ 86.97415549104°.

The dihedral angle between any pair of adjacent faces is ${\textstyle \arccos({\frac {-19-8{\sqrt {5}}}{41}})}$ ≈ 154.12136312578°.

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.

Cartesian coordinates

The 62 vertices of the deltoidal hexecontahedron fall in three sets centered on the origin:

• Twelve vertices are of the form of a unit circumradius regular icosahedron.
• Twenty vertices are of the form of a ${\displaystyle {\frac {3}{11}}{\sqrt {15-{\frac {6}{\sqrt {5}}}}}\approx 0.9571}$ scaled regular dodecahedron.
• Thirty vertices are of the form of a ${\displaystyle 3{\sqrt {1-{\frac {2}{\sqrt {5}}}}}\approx 0.9748}$ scaled Icosidodecahedron.

These hulls are visualized in the figure below:

Orthogonal projections

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

Variations

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 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

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 also

• Deltoidal icositetrahedron

References

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-Strauss, ISBN   978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal hexecontahedron)
• http://mathworld.wolfram.com/ArchimedeanDualGraph.html

External links

• Weisstein, Eric W., "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.