Three-dimensional Regular Polytopes
In three dimensions, polytopes are called polyhedra:
A regular polyhedron with Schläfli symbol has a regular face type, and regular vertex figure .
A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.
Existence of a regular polyhedron is constrained by an inequality, related to the vertex figure's angle defect:
- : Polyhedron (existing in Euclidean 3-space)
- : Euclidean plane tiling
- : Hyperbolic plane tiling
By enumerating the permutations, we find 5 convex forms, 4 nonconvex forms and 3 plane tilings, all with polygons and limited to:, and .
Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.
Read more about this topic: List Of Regular Polytopes
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