Wednesday, April 5, 2017

Classifying Polytopes

The properties of polytopes that I chose to keep invariant are discontinuities, flatness, colinearity, and convexity. To indicate that two points on a polytope are colinear, or cohyperplanar, I collect the points into boundaries and intersections between boundaries, such that two points are coplanar iff they are in the same boundary. The discontinuities in a polytope occur only where boundaries intersect. To understand convexity, note that intersections between halfspaces are convex. Thus, if a discontinuity is concave, it consists of more than one halfspace intersection of the same intersecting boundaries. I call the halfspace intersections polyants, and specify them as maps from boundary to side. Thus, in an n dimensional space, a vertex has 2^n polyants, and an edge has 2^(n-1) polyants. If more than one polyant of a vertex has points near the vertex in the polytope, then the vertex is not convex, and similarly for edges, and so on. Note that the polytope has only the empty polyant, and the single boundaries each have two polyants. Wrt a boundary in its domain, a polyant is significant iff points in the polyant near the boundary are near points both in and not in the polytope. In fact, a two boundary polyant is significant iff one of the boundaries is significant in the section of the polytope by the other boundary. Thus, a polytope is a graph of polyants. Since a polyant is specified by boundaries and sides, and a graph is a map from polyant to set of polyant, equivalent polytopes are found by permuting the boundaries and mirroring sides across boundaries.

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