Tuesday, November 24, 2015
In previous posts I have attempted to define polytope but succeeded only in defining interesting things such as migration, convex cover, disjoint cover, round space, cospace, significant vertex. In this post I shall try again. With regard to a significant vertex V of a set of regions R, a boundary B is significant iff two regions R0 and R1 in the pencil of V are neighbors wrt B, R0 is in R, and R1 is not in R. Wrt significant vertex V of a set of regions R, a region is significant iff it is in the intersection of R and the pencil of V. Define polytope as a collection of round spaces with region sets, such that the centers of the round spaces are the significant vertices, the boundaries of the round spaces are the significant boundaries wrt center, and the region sets are the significant regions wrt center. The fact that the round spaces share boundaries captures the facet graph and coincidences of the polytope. The sidedness of the region sets in the round spaces captures the convexities around the vertices. I believe this definition is adequate to determine whether two sets of regions from the same or different spaces are embeddings of the same polytope. However, I can think of plenty of round space collections that do not embed in any flat spaces.
Sunday, November 8, 2015
My understanding is that designing geodesic domes is nontrivial because there are so many choices. Here I present a way to find every dome up to complexity. Consider all sections of all spaces up to complexity. The cospace of the vertices on one of those sections has all planes through a point. Designate that point the center of a dome. The surface of the dome is the round space as defined in the "Coincidence" post. Note that not every region of the round space need be a facet of the dome. Some dome facets can be super-regions in the round space.