Five-dimensional Regular Polytopes and Higher
In five dimensions, a regular polytope can be named as where is the hypercell (or teron) type, is the cell type, is the face type, and is the face figure, is the edge figure, and is the vertex figure.
A 5-polytope has been called a polyteron, and if infinite (i.e. a honeycomb) a tetracomb.
- A vertex figure (of a 5-polytope) is a polychoron, seen by the arrangement of neighboring vertices to each vertex.
- An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.
- A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.
A regular polytope exists only if and are regular polychora.
The space it fits in is based on the expression:
-
- : Spherical 4-space tessellation or 5-space polytope
- : Euclidean 4-space tessellation
- : hyperbolic 4-space tessellation
Enumeration of these constraints produce 3 convex polytopes, zero nonconvex polytopes, 3 4-space tessellations, and 5 hyperbolic 4-space tessellations. There are no non-convex regular polytopes in five dimensions or higher.
Higher-dimensional polytopes have sometimes received names. 6-polytopes have sometimes been called polypeta, 7-polytopes polyexa, 8-polytopes polyzetta, and 9-polytopes polyyotta.
Read more about this topic: List Of Regular Polytopes
Famous quotes containing the words regular and/or higher:
“While you’re playing cards with a regular guy or having a bite to eat with him, he seems a peaceable, good-humoured and not entirely dense person. But just begin a conversation with him about something inedible, politics or science, for instance, and he ends up in a deadend or starts in on such an obtuse and base philosophy that you can only wave your hand and leave.”
—Anton Pavlovich Chekhov (1860–1904)
“In the mountains of truth you will never climb in vain: either you will already get further up today or you will exercise your strength so that you can climb higher tomorrow.”
—Friedrich Nietzsche (1844–1900)