Wythoff Construction of Polyhedra
Various uniform polyhedra, including all the Platonic and Archimedean solids, can be generated by a Wythoff construction. Begin with a spherical triangle (called a Schwarz triangle) and place a single vertex inside it. Drop perpendiculars to each of the three sides of the triangle and color the regions red, blue, and yellow. Reflect the triangle in each of the three sides until the entire sphere is covered.
Snapshot 1: the great dodecahedron, one of the Kepler–Poinsot polyhedra
Snapshot 2: the omnitruncated tetrahedron or truncated octahedron
Snapshot 3: the truncated icosahedron, football, or buckyball
Snapshot 4: a nonconvex uniform star polyhedron, the great rhombihexahedron
Snapshot 5: the regular icosahedron
Snapshot 6: the truncated cube