# 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. The term "segmentotope" was first used by R. Klitzing in 2000.[1] Segmentotopes are usually named with the notation "X atop Y" or "X || Y", where X and Y are the two bases.

The top and bottom bases of a convex segmentotope are orbiform polytopes. The facets that lace the bases together are segmentotopes of one dimension less. In a non-convex segmentotope, the bases aren't necessarily polytopes; they can be pseudo or even just a non-polytopal collection of elements.

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.

## 3 dimensions

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).

Examples of segmentohedra
Tetrahedron

(triangular pyramid)

Pentagonal prism Octahedron

(triangular antiprism)

Square cupola
point atop triangle pentagon atop pentagon triangle atop (gyrated) triangle square atop octagon