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:
“A regular council was held with the Indians, who had come in on their ponies, and speeches were made on both sides through an interpreter, quite in the described mode,—the Indians, as usual, having the advantage in point of truth and earnestness, and therefore of eloquence. The most prominent chief was named Little Crow. They were quite dissatisfied with the white man’s treatment of them, and probably have reason to be so.”
—Henry David Thoreau (1817–1862)
“We are conscious of an animal in us, which awakens in proportion as our higher nature slumbers. It is reptile and sensual, and perhaps cannot be wholly expelled; like the worms which, even in life and health, occupy our bodies. Possibly we may withdraw from it, but never change its nature. I fear that it may enjoy a certain health of its own; that we may be well, yet not pure.”
—Henry David Thoreau (1817–1862)