The Symmetriad : a journey of discovery through the land of the polychora

I devised and implemented a method for constructing regular and semiregular 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 yields an object that exhibits the symmetries specified by G. If the point is chosen well, the object is guaranteed to be regular or semiregular, and many interesting regular and semiregular objects arise this way. By starting with the symmetry group, I can use the group structure both to simplify the actual graphics involved with displaying the object, and to illustrate various aspects of its structure. For example, subgroups of the symmetry group (and their cosets) correspond to substructures of the object. Conversely, by displaying such symmetric objects and their various substructures, I find that I can elucidate the structure of the symmetry group that gives rise to them. I have written The Symmetriad, the computer system whose name this document has inherited, and used it to explore 3- and 4-dimensional symmetric objects and their symmetry groups. The 3-dimensional objects are already well understood, but they serve to illustrate the techniques used on the 4-dimensional objects and make them more comprehensible. Four dimensions offers a treasure trove of intriguing structures, many of which have no ready 3D analogue. These are what I will show you here.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2005

Author dc:creator

• Radul, Alexey

Advisor dc:contributor.advisor

• Gerald Jay Sussman.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/33337

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/33337