Wythoff Construction of Polyhedra

symmetry type

tetrahedral

octahedral

icosahedral

convex only

barycentric coordinates of vertex

x

0.55

y

0.

z

1.

Schwarz triangle

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.

Details

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

External Links

Uniform Polyhedron (Wolfram MathWorld)

Schwarz Triangle (Wolfram MathWorld)

Permanent Citation

Adam P. Goucher

"Wythoff Construction of Polyhedra"
http://demonstrations.wolfram.com/WythoffConstructionOfPolyhedra/
Wolfram Demonstrations Project
Published: August 29, 2012