The Symmetriad

You can skip straight to the pictures if you wish, but since I prefer doing things in order, I start with the explanation.

The Symmetriad is a computer program to compute and display highly symmetric objects. It can handle objects with finite reflection symmetries in up to four dimensions. I developed the Symmetriad on top of code by (and with much aid from) Rebecca Frankel.

The objects the Symmetriad is after are the (generalized) Platonic and Archimedean solids. The Platonic solids are the familiar cube, tetrahedron, octahedron, dodecahedron and icosahedron. They are characterized by being convex, having identical vertices, and having identical regular polygons for faces. The Archimedean solids are characterized by a relaxation of these rules: The vertices must remain identical, and the faces must remain regular, but the faces need no longer be the same. For example, if you cut the corners off a cube at the right distance, you can make the Archimedean solid called the truncated cube, that will have six regular octagonal faces (the cube's six squares with corners cut off) and eight regular triangular faces (where the cube's corners used to be).

The Symmetriad handles these solids, and a generalization thereof to four dimensions. The technical term is convex semiregular solids: Convex solids with identical vertices and edges of equal length (which implies regular polygonal faces), but whose faces and 3-cells need not be the same. If this subject intrigues you, or if you just want to look at pretty pictures, you are welcome to read my master's thesis. It contains a picture gallery of graphics made with the Symmetriad starting on page 81, and the rest of the document is for the incurably curious --- those who want to understand what they are seeing. The document assumes a basic understanding of group theory, geometry, and linear algebra, or a willingness to take my theorems on faith.

Since the publication of my thesis, I was able to improve both the performance and graphical versatility (i.e. moving beyond the wireframes you'll find in my thesis) of the Symmetriad. Here, then, is a gallery of some of my favorite pictures that the Symmetriad has allowed me to make. The image files are named, in my notation (see thesis), for the objects shown in them. My apologies for the actual graphics quality --- finding a viewer that could handle 4D coordinates at all was hard enough without having to worry about how nice the edges of curves look.