Uniform Polyhedra and their duals

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Uniform Polyhedra and their Duals

Uniform polyhedra have regular faces, and identical vertex figures, meaning that each vertex is surrounded by faces is the same way. The faces needn't be the same though, and the polyhedra need not be convex. The set includes the Platonic solids, the Kepler-Poinsot solids, the Archimedean solids, and prisms and antiprisms, which are shown on their own pages. What remains are the nonconvex equivalents of the Archimedean solids, and this is what appears below. These are just the models I have made, but not the complete set.

A uniform polyhedron may be completely specified by its vertex description, which describes the sequence of faces around each vertex. For example, 8.8.3 indicates a truncated cube, because each vertex is surrounded by two regular octagons and an equilateral triangle. Some uniform polyhedra also make use of nonconvex regular polygons, such as pentagrams. A regular pentagram (5-pointed star) is described as 5/2, meaning that its 5 sides go around the polygon's centre twice before repeating. Uniform polyhedra make use of pentagrams (5/2), octagrams (8/3) and decagrams (10/3) in addition to other convex regular polygons. One such example is 8/3.3.8/3.4 which specifies the great cubicuboctahedron, having a cycle of faces around each vertex as follows: a regular octagram, equilateral triangle, another octagram, and a square.

These models were made using nets generated by Great Stella and can also be made using Stella4D. When you load a uniform polyhedron in Stella, it is generated on-the-fly just from the vertex description. That is, the final geometry is NOT already stored anywhere in the program.

Click on the images below to see a bigger picture and find more photos and information about them.