between the vertices on 3- and 4-fold symmetry axes (between yellow and red vertices in the images below). Dyakis dodecahedron A variant with pyritohedral symmetry is called a dyakis dodecahedron or diploid. It is common in crystallography.A dyakis dodecahedron can be created by enlarging 24 of the 48 faces of a disdyakis dodecahedron. A tetartoid can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron. Stellation The great triakis octahedron is a stellation of the deltoidal icositetrahedron. Related polyhedra and tilings The deltoidal icositetrahedron is a member of a family of duals to the uniform polyhedra related to the cube and regular octahedron. When projected onto a sphere (see right), it can be seen that the edges make up the edges of a cube and regular octahedron arranged in their dual positions. It can also be seen that the 3- and 4-fold corners can be made to have the same distance to the center. In that case the resulting icositetrahedron will no longer have a rhombicuboctahedron for a dual, since the centers of the square and triangle faces of a rhombicuboctahedron are at different distances from its center. This polyhedron is a term of a sequence of topologically related deltoidal polyhedra with face configuration V3.4.n.4; this sequence continues with tilings of the Euclidean and hyperbolic planes. These face-transitive figures have (*n32) reflectional symmetry. See also Deltoidal hexecontahedron Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube. "The Haunter of the Dark", a story by H.P. Lovecraft, whose plot involves this figure. Pseudo-deltoidal icositetrahedron References (Section 3-9) (The thirteen semiregular convex polyhedra and their duals, Page 23, Deltoidal icositetrahedron) The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron) External links Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model Catalan solids