The Wythoff construction produces a uniform polyhedron or uniform tiling from a Coxeter group. A generating point is chosen in a fundamental triangle bounded by three reflection mirrors. For the ordinary uniform construction, the point is equidistant from all mirrors that do not contain it. In the polyhedral case, its group orbit under the Coxeter group gives the vertices, while reflected copies of suitable edges generate the edges and faces.
The mirrors which contain the generating point determine which reflections fix it and are encoded by a Wythoff symbol. Choosing the prescribed equidistant positions in the fundamental triangle produces regular, truncated, rectified, and other uniform members associated with the same symmetry group. Arbitrary positions generally give unequal edge lengths.
