Trapezohedron -- from Wolfram MathWorld

An -trapezohedron, also called an antidipyramid, antibipyramid, or deltohedron (not to be confused with a deltahedron), is a solid composed of interleaved symmetric quadrilateral kites, half of which meet in a top vertex and half in a bottom vertex. A regular -trapezohedron can be constructed from two sets of points placed around two regular -gons displaced relative to one another in the direction perpendicular to the plane of the polygons and rotated relative to one another by an angle of degrees. Two additional equally spaced points are then added along symmetry axis of the polygons, one above the top circle of points and the other below the bottom one. The trapezohedron is then the convex hull of these points. The top faces are constructed from of adjacent points in the upper circle, the corresponding point between them in the lower circle, and the upper vertex, and the bottom faces analogously but with upper/lower reversed. Unfortunately, the name "trapezohedron" is not particularly well chosen since the faces can be seen to be not trapezoids but rather kites.

The -trapezohedron has vertices, edges (half short and half long), and faces.

The 3-trapezohedron (trigonal trapezohedron) is a rhombohedron having all six faces congruent. A special case is the cube (oriented along a space diagonal), corresponding to the dual of the equilateral 3-antiprism (i.e., the octahedron).

A 4-trapezohedron (tetragonal trapezohedron) appears in the upper left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

The trapezohedra are isohedra.

The canonical -trapezohedron is the dual polyhedra of the canonical (i.e., equilateral) -antiprism. In particular, the canonical -trapezohedron can be constructed from an -antiprism as the convex hull of the set of points obtained by extending the centroids of the lateral triangular faces by a factor

(1)

and the centroids of the top and bottom regular -gons by a factor of

(2)

Alternately, the canonical -trapezoid of unit midradius can be constructed from rotated circles of points at radius

(3)

at heights

(4)

with apices at vertical positions

(5)

The kites forming the faces of a canonical -trapezohedron have three equal angles nearest of the central plane

(6)

and angles at the apices of

(7)

Nets for canonical -trapezohedra are illustrated above for to 8.

For a canonical -trapezohedron normalized to have short edge length 1, the long edge length, (half-)height, inradius, midradius, surface area, and volume are given by