List of regular polytopes and compounds
Name Schläfli Symbol { p , q , r } Cell type { p , q } Face type { p } Edge figure { r } Vertex figure { q , r } χ Dual
Order-6 tetrahedral honeycomb { 3,3,6 } { 3,3 } { 3 } { 6 } { 3,6 } 0 { 6,3,3 }
Hexagonal tiling honeycomb { 6,3,3 } { 6,3 } { 6 } { 3 } { 3,3 } 0 { 3,3,6 }
Order-4 octahedral honeycomb { 3,4,4 } { 3,4 } { 3 } { 4 } { 4,4 } 0 { 4,4,3 }
Square tiling honeycomb { 4,4,3 } { 4,4 } { 4 } { 3 } { 4,3 } 0 { 3,3,4 }
Triangular tiling honeycomb { 3,6,3 } { 3,6 } { 3 } { 3 } { 6,3 } 0 Self-dual
Order-6 cubic honeycomb { 4,3,6 } { 4,3 } { 4 } { 4 } { 3,4 } 0 { 6,3,4 }
Order-4 hexagonal tiling honeycomb { 6,3,4 } { 6,3 } { 6 } { 4 } { 3,4 } 0 { 4,3,6 }
Order-4 square tiling honeycomb { 4,4,4 } { 4,4 } { 4 } { 4 } { 4,4 } 0 { 4,4,4 }
Order-6 dodecahedral honeycomb { 5,3,6 } { 5,3 } { 5 } { 5 } { 3,5 } 0 { 6,3,5 }
Order-5 hexagonal tiling honeycomb { 6,3,5 } { 6,3 } { 6 } { 5 } { 3,5 } 0 { 5,3,6 }
Order-6 hexagonal tiling honeycomb { 6,3,6 } { 6,3 } { 6 } { 6 } { 3,6 } 0 Self-dual
Apeirotopes -- Three dimensions ( 4-apeirotopes )
There are ten flat regular honeycombs of hyperbolic 3-space : [ 19 ] ( previously listed above as tessellations ) 4 are compact : { 3,5,3 } , { 4,3,5 } , { 5,3,4 } , and { 5,3,5 } while 6 are paracompact : { 3,3,6 } , { 6,3,3 } , { 3,4,4 } , { 4,4,3 } , { 3,6,3 } , { 4,3,6 } , { 6,3,4 } , { 4,4,4 } , { 5,3,6 } , { 6,3,5 } , and { 6,3,6 } . 4 compact regular honeycombs { 5,3,4 } { 5,3,5 } { 4,3,5 } { 3,5,3 } 4 of 11 paracompact regular honeycombs { 3,4,4 } { 3,6,3 } { 4,4,3 } { 4,4,4 } Tessellations of hyperbolic 3-space can be called hyperbolic honeycombs . There are 15 hyperbolic honeycombs in H3 , 4 compact and 11 paracompact . 4 compact regular honeycombs Name SchläfliSymbol { p , q , r } Coxeter Celltype { p , q } Facetype { p } Edgefigure { r } Vertexfigure { q , r } χ Dual Icosahedral honeycomb { 3,5,3 } { 3,5 } { 3 } { 3 } { 5,3 } 0 Self-dual Order-5 cubic honeycomb { 4,3,5 } { 4,3 } { 4 } { 5 } { 3,5 } 0 { 5,3,4 } Order-4 dodecahedral honeycomb { 5,3,4 } { 5,3 } { 5 } { 4 } { 3,4 } 0 { 4,3,5 } Order-5 dodecahedral honeycomb { 5,3,5 } { 5,3 } { 5 } { 5 } { 3,5 } 0 Self-dual There are also 11 paracompact H3 honeycombs ( those with infinite ( Euclidean ) cells and/or vertex figures ) : { 3,3,6 } , { 6,3,3 } , { 3,4,4 } , { 4,4,3 } , { 3,6,3 } , { 4,3,6 } , { 6,3,4 } , { 4,4,4 } , { 5,3,6 } , { 6,3,5 } , and { 6,3,6 } .
This page lists the regular polytopes and regular polytope compounds in Euclidean, spherical and hyperbolic spaces. The Schläfli symbol describes every regular tessellation of an n-sphere, Euclidean and hyperbolic spaces. A Schläfli symbol describing an n-polytope equivalently describes a tessellation of an (n − 1)-sphere. In addition, the symmetry of a regular polytope or tessellation is expressed as a Coxeter group, which Coxeter expressed identically to the Schläfli symbol, except delimiting by square brackets, a notation that is called Coxeter notation. Another related symbol is the Coxeter-Dynkin diagram which represents a symmetry group with no rings, and the represents regular polytope or tessellation with a ring on the first node. For example, the cube has Schläfli symbol {4,3}, and with its octahedral symmetry, [4,3] or , it is represented by Coxeter diagram . The regular polytopes are grouped by dimension and subgrouped by convex, nonconvex and infinite forms. Nonconvex forms use the same vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one-lower-dimensional Euclidean space. Infinite forms can be extended to tessellate a hyperbolic space. Hyperbolic space is like normal space at a small scale, but parallel lines diverge at a distance. This allows vertex figures to have negative angle defects, like making a vertex with seven equilateral triangles and allowing it to lie flat.