Segmentotope

A segmentotope is a regular-faced n-dimensional polytope whose vertices are contained in two parallel (n–1)-dimensional hyperplanes and is orbiform as well. The convex ones are thereby a subclass of the convex regular-faced polytopes.

A line segment is the only 1-dimensional segmentotope, because by definition, its vertices are in two different 0-dimensional spaces.

The triangle and square are the only 2-dimensional segmentotopes, since they are the only regular polygons for which two parallel lines (1-dimensional spaces) can be drawn that together intersect all vertices of the polygon.

Note that when S is a convex segmentochoron and S allows for a diminishing with regular faces only, then the result would be a segmentotope again. Furthermore, by means of construction, it becomes clear that the top and bottom bases (contained within the parallel hyperplanes) of a segmentotope are forced to be orbiform polytopes. The facet s that lace the bases together, however, need to be segmentotopes of one dimension less than S.

3 dimensions
All pyramid s, prism s, and antiprism s obviously are segmentohedra, since their vertices all lie on one of two parallel planes.

Convex segmentohedra can be classified as one of the following: pyramids (point atop n-gon), prisms (n-gon atop n-gon), antiprisms (n-gon atop gyro n-gon), and cupolas (n-gon atop 2n-gon). Three polyhedra have two different constructions: the tetrahedron (both a 3-gonal pyramid and a 2-gonal antiprism), the triangular prism (both a 3-gonal prism and a 2-gonal cupola) and the square pyramid (both a 4-gonal pyramid and a special construction in none of the above categories, 2-gon atop 3-gon, i.e. as the diminishing of the trigonal antiprism).

4 dimensions

 * See also: List of segmentochora

The pyramids and prisms with polyhedron bases are segmentochora, since their vertices all lie in one of two parallel 3-dimensional spaces.

Many segmentochora are named with the notation "X atop Y", or "X || Y" for short. The two polyhedra X and Y lie in parallel three-dimensional spaces, and relatively small 3D pyramids, prisms, antiprisms, and/or cupolae connect the two through the 4th dimension. In the case of the relatively simple octahedron atop cube, six square pyramids are attached to the cube's faces and their apexes connect to the vertices of the octahedron, while eight triangular pyramids (tetrahedra) are attached to the octahedron's faces and connect to the cube's vertices. Twelve digonal antiprisms (further tetrahedra) then fill in the gaps between the triangular faces of the attached pyramids, connecting the corresponding edges of the two base polyhedra.