Friday, March 24, 2017

Regularity

Till now, I have focussed on irregular spaces and polytopes without coincidental boundaries, but spaces with missing regions have been a nagging possibility. For example, if a space is caught in the process of migrating, then the migration itself is like a sidedness space with a missing region, or an affine space with more than the dimension's worth of boundaries through a point. Parallelism is also a a form of degenerate space. Imagine a migration of a round space, where the migrating region is an outside region in a flat rotation space of the round space. Now consider embeddings of polytopes into spaces. Since an embedding is a subset of the regions in the space, one can specify the embedding as the degenerate space consisting of just the embedded regions. Restoring the degenerate space to a linear space is one to many, but each restoration is a valid embedding of the same polytope. Now note that regular polytopes contain parallel sides. Irregular and regular polytopes are both degenerate spaces; the only distinction is whether there is a partial restoration to a space missing outside regions.