Nature

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Nature is a concept created by Jonathan Bowers to differentiate polytopes that have co-spatial elements. There are three categories: Tame, Feral, and Wild that a d-dimensional polytope can have, determined from two properties:

  • There are three (or more) d-elements that share a (d+1) space and a (d-1) element
  • Those d-elements also share a (d+1) element

The category can be determined from looking at the (d-3) element figures of a d-dimensional polytope.

Nature Bowers.png

Tame[edit | edit source]

A polytope is said to be tame if it satisfies neither of the two properties. For example, a polyhedron where no three edges are in the same plane and meet at a vertex. A tame polyhedron will have no vertex figures with 3 collinear points. All convex polytopes are tame, and every uniform polyhedron is tame.

Feral[edit | edit source]

A polytope is said to be feral if it satisfies the first case, but not the second. In other words, there are coplanar elements, but they have no (d+1) elements between them. For example, a feral polyhedron has vertex figures with three colinear points, but there is no actual line there.

Wild[edit | edit source]

Any polytope that satisfies both properties is said to be wild. For example, a polyhedron with three coplanar edges meeting at a vertex, with two of them belonging to the same face. Wild polyhedra have vertex figures with three collinear points and an edge connecting two of them.