Trapezohedron

A trapezohedron is a convex polyhedron that can be understood as the dual of an antiprism. For a given integer n ≥ 2, the trapezohedron of order n has 2n kite-shaped faces (deltoids), 2n+2 vertices, and 4n edges. In the regular form, its faces are congruent kites and its symmetry is dihedral, denoted D_nd, reflecting the n-fold rotational symmetry around the central axis together with mirror symmetries.

Geometrically, the vertex set consists of two opposite poles and a belt of 2n vertices arranged around the axis. Each face is a kite bounded by two adjacent belt vertices and the two poles, so the polyhedron alternates around the belt with faces meeting along the belt edges. The dual relationship to the antiprism explains the overall structure: the two n-gonal base faces of the antiprism correspond to the two polar vertices, while the triangular side faces correspond to the kite faces of the trapezohedron.

Special cases and naming: the family includes the tetragonal deltohedron (n = 2) and the pentagonal trapezohedron (n = 5), the latter of which is the standard form used for modern d10 dice. The shape is sometimes simply called a deltohedron, reflecting its kite-shaped faces.

Context and use: trapezohedra are of interest in geometry for their symmetric, uniform-face structure and their role as duals to antiprisms. The pentagonal trapezohedron, in particular, has practical relevance in gaming as a 10-faced die. The concept also appears in studies of polyhedral duality and symmetry.