The Gosset–Elte figures are a set of eight high dimensional quasiregular polytopes that span from 6D to 8D. They are unique due to the fact they are not derivates of the regular polytopes in those dimensions, and in fact even have their own special symmetries, with E6 (and E6×2) for 6D, E7 for 7D and E8 for 8D. They also represent the only non-linear spherical Coxeter groups that cannot form regular polytopes.
Due to their high dimensionality and strange symmetry, they are rarely thought of as "tangible shapes" and more as abstract things with some symmetry group you can perform operations on in bulk, which is generally very useful in normal polytopology but a hindrance when trying to understand these shapes from a less advanced viewpoint. This section of my website aims to change that, by visualizing them layer by layer, Quickfur style.
There are some big differences between Quickfur's visualizations and these pages' visualizations, mainly that these cannot really have any images. This means the explanations are now written in only one paragraph, with maybe one more if the shape is particularly complex (see hecatonicosoctaexon-first 421 polyzetton structure). While I originally intended to make this with facet contact diagrams when I first thought of this several months back in March, I eventually realized that would be practically impossible for anything but the visualization aids. Also, to summarize the layer structures, I've included facet counts for each layer at the end of each structure description, which Quickfur lacks for most pages.
The Gosset–Elte figures are generally indexed by their Coxeter symbol, which is a symbol of the form aijk... with a, i, j, k being integers. This is related to their Coxeter diagram representation, where the shape is formed from a chain of a unringed nodes following a ringed node, with branches of length i, j, k, etc added at the end. This results in a polytope of rank a+i+j+k+...+1. While all Coxeter symbols do form valid polytopes, only a small (but still infinite) set form finite polytopes. Most of these are either m-rectified n-simplices (0[n−m−1]m), n-orthoplices ([n−3]11), birectified n-cubes (0[n−3]11) or n-demicubes (1[n−3]1), although there are 11 exceptions. They are the eight Gosset–Elte figures plus the rectified 122 polypeton (0221), the rectified 132 polyexon (0321) and the rectified 142 polyzetton (0421). The latter three are not explored here.
These are all simple derivations of regular polytopes. They are here to provide context to higher dimensional figures when they appear as facets or vertex figures.
All of the (non-trivial) layer structures were derived from Klitzing's incidence matrices website. As some layer structures are missing from that website, they are missing here too. More specifically, the vertex-first layers of lin, the vertex- and demihepteract-first layers of bif, the octaexon- and vertex-first layers of bay and the octaexon-first layers of fy are missing. Once Klitzing adds those to his website I might add them here too.