Order-7 tetrahedral honeycomb

Order-7 tetrahedral honeycomb

Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
TypeHyperbolic regular honeycomb
Schläfli symbols{3,3,7}
Coxeter diagrams
Cells{3,3}
Faces{3}
Edge figure{7}
Vertex figure{3,7}
Dual{7,3,3}
Coxeter group[7,3,3]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-7 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,7}. It has seven tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-7 triangular tiling vertex arrangement.

Related polytopes and honeycombs

It a part of a sequence of regular polychora and honeycombs with tetrahedral cells.

{3,3,p}
Space S3 H3
Form Finite Paracompact Noncompact
Name {3,3,3}
 {3,3,4}

 {3,3,5}
 {3,3,6}

 {3,3,7}
 {3,3,8}

 ... {3,3,∞}

Image
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}


Order-8 tetrahedral honeycomb

Order-8 tetrahedral honeycomb

Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
TypeHyperbolic regular honeycomb
Schläfli symbols{3,3,8}
{3,(3,4,3)}
Coxeter diagrams
=
Cells{3,3}
Faces{3}
Edge figure{8}
Vertex figure{3,8} {(3,4,3)}
Dual{8,3,3}
Coxeter group[3,3,8]
[3,((3,4,3))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-8 tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,8}. It has eight tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an order-8 triangular tiling vertex arrangement.

Symmetry constructions

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,4,3)}, Coxeter diagram, , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,8,1+] = [3,((3,4,3))].


Infinite-order tetrahedral honeycomb

Infinite-order tetrahedral honeycomb

Rendered intersection of honeycomb with the ideal plane in Poincaré half-space model
TypeHyperbolic regular honeycomb
Schläfli symbols{3,3,∞}
{3,(3,∞,3)}
Coxeter diagrams
=
Cells{3,3}
Faces{3}
Edge figure{∞}
Vertex figure{3,∞}, {(3,∞,3)}
Dual{∞,3,3}
Coxeter group[∞,3,3]
[3,((3,∞,3))]
PropertiesRegular

In the geometry of hyperbolic 3-space, the infinite-order tetrahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,3,∞}. It has infinitely many tetrahedra {3,3} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many tetrahedra existing around each vertex in an infinite-order triangular tiling vertex arrangement.

Symmetry constructions

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(3,∞,3)}, Coxeter diagram, = , with alternating types or colors of tetrahedral cells. In Coxeter notation the half symmetry is [3,3,∞,1+] = [3,((3,∞,3))].

See also

References