# Segmentotope

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.^{[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[edit | edit source]

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).

Tetrahedron
(triangular pyramid) |
Pentagonal prism | Octahedron
(triangular antiprism) |
Square cupola |
---|---|---|---|

point atop triangle | pentagon atop pentagon | triangle atop (gyrated) triangle | square atop octagon |

## 4 dimensions[edit | edit source]

*See also: List of segmentochora*

In a segmentochoron, two polyhedra or lower-dimensional polytopes 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.

The pyramids and prisms with orbiform polyhedron bases are segmentochora.

Tesseract | Octahedral pyramid | Hexagonal antifastegium |
---|---|---|

cube atop cube; cubic prism | point atop octahedron | hexagon atop (gyrated) hexagonal prism |

## External links[edit | edit source]

Klitzing, Richard. "Segmentochora."

## References[edit | edit source]

- ↑ Convex Segmentochora, R. Klitzing, Symmetry: Culture and Science Vol. 11, Nos. 1-4, 139-181, 2000