 | ||
In geometry, a deltoidal hexecontahedron (also sometimes called a trapezoidal hexecontahedron, a strombic hexecontahedron, or a tetragonal hexacontahedron) 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.
Contents
Lengths and angles
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 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:
Related polyhedra and tilings
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.