The Symmetriad: A Journey of Discovery Through the Land of the Polychora

Abstract

I devised and implemented a method for constructing regular andsemiregular geometric objects in n-dimensional Euclidean space.Given a finite reflection group (a Coxeter group) G, there is a standard way to give G a group action on n-space.Reflecting a point through this group action yieldsan object that exhibits the symmetries specified by G. If the pointis chosen well, the object is guaranteed to be regular or semiregular,and many interesting regular and semiregular objectsarise this way. By starting with the symmetry group, I can use thegroup structure both to simplify the actual graphics involved withdisplaying the object, and to illustrate various aspects of itsstructure. For example, subgroups of the symmetry group (and theircosets) correspond to substructures of the object. Conversely, bydisplaying such symmetric objects and their various substructures, Ifind that I can elucidate the structure of the symmetry group thatgives rise to them.I have written The Symmetriad, the computer system whose name thisdocument has inherited, and used it to explore 3- and 4-dimensionalsymmetric objects and their symmetry groups. The 3-dimensionalobjects are already well understood, but they serve to illustrate thetechniques used on the 4-dimensional objects and make them morecomprehensible. Four dimensions offers a treasure trove of intriguingstructures, many of which have no ready 3D analogue. These are what Iwill show you here.