Segmentotope

A segmentotope is a polytope in N dimensions whose vertices are only on two parallel hyperplanes of N-1 dimensions. The vertices must all lie on the surface of an N-dimensional hypersphere, and the segmentotope must have only one edge length.

Lower dimensions
A line can be thought of as a 1-dimensional segmentotope, since by definition, its vertices are in two different 0-dimensional spaces.

The triangle and square are 2-dimensional segmentotopes, since two parallel lines (1-dimensional spaces) can be drawn that together intersect all vertices of each shape.

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

4 dimensions
The pyramids, prisms, and antiprisms 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." The two polyhedra X and Y lie in parallel three-dimensional spaces, and relatively small 3D pyramids and prisms 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 tetrahedra then fill in the gaps between the triangular faces of the attached pyramids.