Higher Dimensional Periodic Table Of Regular And Semi-Regular Polytopes: Fid-Bau Portal

Higher Dimensional Periodic Table Of Regular And Semi-Regular Polytopes

Lalvani, Haresh

This paper presents a higher-dimensional periodic table of regular and semi-regular n-dimensional polytopes. For regular n-dimensional polytopes, designated by their Schlafli symbol {p,q,r,…u,v,w}, the table is an (n-1)-dimensional hypercubic lattice in which each polytope occupies a different vertex of the lattice. The values of p,q,r,…u,v,w also establish the corresponding n-dimensional Cartesian co-ordinates (p,q,r,…u,v,w) of their respective positions in the hypercubic lattice. The table is exhaustive and includes all known regular polytopes in Euclidean, spherical and hyperbolic spaces, in addition to others candidate polytopes which do not appear in the literature.

For n-dimensional semi-regular polytopes, each vertex of this hypercubic lattice branches into analogous n-dimensional cubes, where each n-cube encompasses a family with a distinct semi-regular polytope occupying each vertex of each n-cube. The semi-regular polytopes are obtained by varying the location of a vertex within the fundamental region of the polytope. Continuous transformations within each family are a natural fallout of this variable vertex location. Extensions of this method to less regular space structures and to derivation of architectural form are in progress and provide a way to develop an integrated index for space structures. Besides the economy in computational processing of space structures, integrated indices based on unified morphologies are essential for establishing a meta-structural knowledge base for architecture.