Deltoidal hexecontahedron

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(-5-2√5/20)≈118.2686774705°. The opposite angle, between long edges, is arccos(-5+9√5/40)≈67.783011547435° . The other two angles of each face, between a short and a long edge each, are both equal to arccos(5-2√5/10)≈86.97415549104°.

The dihedral angle between any pair of adjacent faces is arccos(-19-8√5/41)≈154.12136312578°.

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.

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

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.

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.

*n42 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

• Deltoidal icositetrahedron

• 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 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal hexecontahedron)
• http://mathworld.wolfram.com/ArchimedeanDualGraph.html

• 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.