Nature

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Nature is a concept created by Jonathan Bowers to differentiate polytopes that have co-spatial elements. An n-polytope's nature is either tame, feral, or wild, depending on two properties:

  • There are three or more d-elements that share both a (d+1)-space and a (d-1)-element for some d
  • Some of 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.

For instance, a feral polyhedron will have at least three coplanar edges meeting at at least one of its vertices, but no face coplanar to said edges. A polyhedron with at least one feral face is automatically feral (or wild), as d could be the rank of any of its elements, not just the rank one less than its own.

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 3 coplanar edges meeting at a vertex, but no two of them are in the same face. And its vertex figure has three collinear points, but there is no actual line there.

Wild[edit | edit source]

The triangular toroprism is wild. An edge between two triangular faces lies in the same plane as a square face, and shares a vertex with that square face too.

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. (So a face and an edge are in the same plane and share a vertex, but that edge isn't part of that face.)

Wild polyhedra have vertex figures with three collinear points and an edge connecting two of them.