Wikipedia:WikiProject Uniform Polytopes

The uniform 4₂₁ polytope, shown inside its Petrie polygon as an orthogonal projection, defined by its Coxeter–Dynkin diagram:

Uniform operators on the regular cube and octahedron

Welcome to WikiProject Uniform Polytopes, the WikiProject which aims to improve Wikipedia's coverage of uniform polytopes! A uniform polytope is an extension to the regular polytopes, including regular and semiregular polytopes, being vertex transitive, and having regular polygon faces.

Regular polytopes are identified by Schläfli symbols. Uniform polytopes are (nearly all) identified by Coxeter–Dynkin diagrams, existing in Coxeter group symmetry families. These are called Wythoffian, being generated by reflection symmetry called a Wythoff construction. Most of the remaining nonwythoffian forms are generated by operational modifications on the reflective forms.

News

• All links to Richard Klitzing's site have to be updated as the address moved from ogre.nu to bendwavy.org in January 2011.

Tasks

• Create a template navigator:
  • Template:Polytopes - started with list of polytopes by dimension

Members

1. Tom Ruen (talk) - I referenced all the wiki articles I know related to this topic, still quite unsure what the immediate goals for this project should be.
2. Mateus Zica - i'm here to help!
3. Double sharp - I will usually stick to the polychora, polytera, and polypeta but may sometimes venture out into higher dimensions.
4. OfficialURL (talk) – As with WikiProject Polyhedra, I'll translate articles to Spanish.

Articles

• Polytope terminology:
  • Vertex, Edge, Face, Cell, Hypercell, Peak, Ridge, Facet, Euler characteristic
  • Dual polyhedron, Conway polyhedron notation, Chiral polytope

• Regular polytope
  • List of regular polytopes
  • Schläfli symbol
  • Simplex, Hypercube, Orthoplex, Hypercubic honeycomb

• Uniform polyhedron terminology:
  • Coxeter–Dynkin diagram, Wythoff symbol, Vertex figure
  • Schwarz triangle, Goursat tetrahedron, density (polytope)
  • Vertex-transitive, Edge-transitive, Face-transitive, Cell-transitive, Wythoff construction
  • Petrie polygon, Schlegel diagram, Polyhedral net, Uniform coloring
  • Vertex configuration, Face configuration
  • Rectification, Truncation, Bitruncation, Cantellation, Runcination, Sterication, Pentellation, Expansion, Snub
  • demihypercube, Semiregular k 21 polytope, Uniform_2_k1_polytope, Uniform_1_k2_polytope
  • Alternated hypercubic honeycomb

Uniform polytopes/tessellations by dimension

2-polytopes: Regular polygon Star polyhedron Tessellations: Apeirogon 3-polytopes/2-tilings: Regular polyhedra: Platonic_solid (regular, convex) Kepler–Poinsot_polyhedron (regular, nonconvex) Hosohedron and Dihedron (regular, degenerate) List of spherical symmetry groups Uniform polyhedron Archimedean solid (uniform, convex) Semiregular polyhedron Catalan solid (Uniform duals, convex) Uniform star polyhedron (uniform, nonconvex) Prismatic uniform polyhedron, Prism (geometry) and Antiprism (prismatic) Duals Bipyramid, Trapezohedron Tessellations: Uniform tiling List of uniform tilings Hyperbolic tessellations: Uniform tilings in hyperbolic plane

4-polytopes/3-honeycombs: Regular 4-polytope (regular, convex and star) Uniform 4-polytope (uniform, convex) Duoprism (uniform, prismatic, convex) Tessellations: Convex uniform honeycombs (infinite, uniform, convex) Hyperbolic tessellations: Convex uniform honeycombs in hyperbolic_space (hyperbolic, infinite, uniform, convex) Paracompact uniform honeycombs 5-polytopes: Uniform 5-polytope (uniform, convex) 6-polytopes: Uniform 6-polytope (uniform, convex) Uniform 7-polytopes (convex) Uniform 8-polytopes (convex) Uniform 9-polytopes (convex) Uniform 10-polytopes (convex)

Geometers

Geometers who have worked on regular and uniform polytopes:

Plato (428–348 BC) Archimedes (287–212 BC) Johannes Kepler (1571–1630) Leonhard Euler (1707–1783) Louis Poinsot (1777–1859) Ludwig Schläfli (1814–1895) Eugène Charles Catalan (1814–1894) Arthur Cayley (1821–1895) Joseph Bertrand (1822–1900) Edmund Hess (1843–1903)

Pieter Hendrik Schoute (1846–1923) Alicia Boole Stott (1860–1940) W. A. Wythoff (1865–1939) Thorold Gosset (1869–1962) Alfredo Andreini (1870–1943) D. M. Y. Sommerville (1879–1934) M. C. Escher (1898–1972) H. S. M. Coxeter (1907–2003) J. C. P. Miller (1906–1981)

Magnus Wenninger (1919–) Michael S. Longuet-Higgins (1925–) Branko Grünbaum (1929–) Norman Johnson (1930–) John Horton Conway (1937–) Keith Critchlow George Olshevsky (1946–) George W. Hart (1955–) Jeffrey Weeks

Related pages

• User:Tomruen/list of uniform polychora
• User:Tomruen/Uniform operator conversion - Johnson vs Bower names
• User:Tomruen/List of Coxeter groups
• User:Tomruen/Uniform polyteron verf
• Talk:Vertex_figure/polychoron
• User:Tomruen/Gosset-Elte figures
• User:Tomruen/Convex_uniform_tetracomb
• User:Tomruen/References
• User:Double sharp/Polyhedra
• User:Double sharp/Polychora
• User:Double sharp/Polytera (these three give lists with full names)
• List of uniform polyhedra by Schwarz triangle
• User:Double sharp/List of uniform tilings by Schwarz triangle
• User:Double sharp/List of uniform polychora by Goursat tetrahedron

Related WikiProjects

• Wikipedia:WikiProject Mathematics
• Wikipedia:WikiProject Polyhedra