Wythoff construction

This construction arranges three mirrors at the sides of a triangle, like in a kaleidoscope.

However, different from a kaleidoscope, the mirrors are not parallel, but intersect at a single point.

If one places a vertex at a suitable point inside the spherical triangle enclosed by the mirrors, it is possible to ensure that the reflections of that point produce a uniform polyhedron.

For a spherical triangle ABC we have four possibilities which will produce a uniform polyhedron: The process in general also applies for higher-dimensional regular polytopes, including the 4-dimensional uniform 4-polytopes.

Uniform polytopes that cannot be created through a Wythoff mirror construction are called non-Wythoffian.