regular -gonal rings. It has congruent kite faces, and it is isohedral. Special cases: . A degenerate form of trapezohedron: a geometric tetrahedron with 6 vertices, 8 edges, and 4 degenerate kite faces that are degenerated into triangles. Its dual is a degenerate form of antiprism: also a tetrahedron. . The dual of a triangular antiprism: the kites are rhombi (or squares); hence these trapezohedra are also zonohedra. They are called rhombohedra. They are cubes scaled in the direction of a body diagonal. They are also the parallelepipeds with congruent rhombic faces. A special case of a rhombohedron is one in which the rhombi forming the faces have angles of and . It can be decomposed into two equal regular tetrahedra and a regular octahedron. Since parallelepipeds can fill space, so can a combination of regular tetrahedra and regular octahedra. . The pentagonal trapezohedron is the only polyhedron other than the Platonic solids commonly used as a die in roleplaying games such as Dungeons & Dragons. Being convex and face-transitive, it makes fair dice. Having 10 sides, it can be used in repetition to generate any decimal-based uniform probability desired. Typically, two dice of different colors are used for the two digits to represent numbers from to . Symmetry The symmetry group of an -gonal trapezohedron is , of order , except in the case of : a cube has the larger symmetry group of order , which has four versions of as subgroups. The rotation group of an -trapezohedron is , of order , except in the case of : a cube has the larger rotation group of order , which has four versions of as subgroups. Note: Every -trapezohedron with a regular zig-zag skew -gon base and congruent non-twisted kite faces has the same (dihedral) symmetry group as the dual-uniform -trapezohedron, for . One degree of freedom within symmetry from Dnd (order 4n) to Dn (order 2n) changes the congruent kites into congruent quadrilaterals with three edge lengths, called twisted kites, and the n-trapezohedron is called a twisted trapezohedron. (In the limit, one edge of each quadrilateral goes to zero length, and the n-trapezohedron becomes an n-bipyramid.) If the kites surrounding the two peaks are not twisted but are of two different shapes, the n-trapezohedron can only have Cnv (cyclic with vertical mirrors) symmetry, order 2n, and is called an unequal or asymmetric trapezohedron. Its dual is an