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Streptohedron and Trapezohedron

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Take an -sided regular pyramid and form the union with its mirror image with respect to its base plane to get an -gonal bipyramid. Its faces are mutually congruent isosceles triangles. (With properly chosen height and =4 you get the regular octahedron as a special case.)

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Rotating its upper and lower parts by degrees yields a type of polyhedron called a streptohedron. Its faces are mutually congruent kites. It is also called a deltohedron or a trapezohedron.

In contrast to trapezohedra, which exhibit only rotational symmetry, both streptohedra and bipyramids also have planes of symmetry that pass through their -fold axes (here only the axes of rotation are shown to avoid crowding).

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Contributed by: Gábor Gévay, Lajos Szilassi, and Sándor Kabai (July 2008)
Open content licensed under CC BY-NC-SA

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The name trapezohedron can be misleading, since it is reserved for a more general type of polyhedra in geometric crystallography. Indeed, if the angle of rotation differs from the one given above, the faces become general quadrilaterals. This is the type of polyhedron that crystallographers call a trapezohedron. (For example, the trigonal trapezohedron occurs in the form of quartz crystals.)

Observe that a special case of streptohedra occurs when =3: the kites become rhombuses, giving rhombohedra; furthermore, with properly chosen height (or, the length of the threefold axis of rotation), the rhombuses become squares, yielding a cube.

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