# Segmentotope

(Redirected from Segmentoteron)

A segmentotope is an orbiform, rank-n polytope whose vertices are contained in two parallel (n–1)-dimensional hyperplanes. The convex ones are thereby a subclass of the convex regular-faced polytopes. The term "segmentotope" was first used by R. Klitzing in 2000. 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).