Friday, October 30, 2015

Vertex Cospace Polytope

In n dimensions, each set of n boundaries is a vertex. As in the algorithm to construct an affine plane through regions, we can interpret vertices as planes to obtain a space called a cospace. Cospaces are not strictly linear. Some nonempty regions can be degenerate in the sense that more than n planes can pass through the same point in n dimensions. For example, if we construct a cospace from the vertices that lie on a particular boundary, all planes in the cospace pass through the same point. I propose that any polytope, no matter the nature of its facets, convexities, or colinearities, can be identified by the cospace of the polytope's significant vertices. A vertex is significant if some polyant of each proper superpencil is improper wrt the regions in the polytope. A superpencil is a pencil of a subset of the boundaries of a pencil. A pencil of a set of boundaries is the set of regions with neighbors wrt the boundaries. A region is a neighbor wrt boundaries if it's sidednesses are the same, except opposite for the boundaries. A polyant is the regions on the same side of a set of boundaries. A polyant of a pencil is the intersection between pencil and polyant. 

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