Order-6 tetrahedral honeycomb

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Order-6 tetrahedral honeycomb

Perspective projection view
within Poincaré disk model
TypeHyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbols{3,3,6}
{3,3[3]}
Coxeter diagrams
=
CellsTetrahedron {3,3}
FacesTriangle {3}
Edge figureHexagon {6}
Vertex figureTriangular tiling {3,6}
DualHexagonal tiling honeycomb, {6,3,3}
Coxeter groups{{\bar {V}}}_{3}, [6,3,3]
{{\bar {P}}}_{3}, [3,3[3]]
PropertiesRegular

In the geometry of hyperbolic 3-space, the order-6 tetrahedral honeycomb a regular space-filling tessellation (or honeycomb). With Schläfli symbol {3,3,6}. It has six tetrahedra {3,3} around each edge. All vertices are ideal vertices with infinitely many tetrahedra existing around each ideal vertex in an triangular tiling vertex arrangement. [1]

Symmetry constructions

Subgroup relations

It has a second construction as a uniform honeycomb, Schläfli symbol {3,3[3]}, with alternating types or colors of tetrahedral cells.

Related polytopes and honeycombs

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

It is similar to the 2-dimensional hyperbolic tiling, infinite-order triangular tiling, {3,∞}, for having all ideal vertices made of regular simplices.

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual hexagonal tiling honeycomb, {6,3,3}.

The rectified order-6 tetrahedral honeycomb, t1{3,3,6} has tetrahedron and triangular tiling cells connected in a hexagonal prism vertex figure:

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

{3,3,p}
Space S3 H3
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,∞}

It a part of a sequence of honeycombs with triangular tiling vertex figures.

{p,3,6}
Space H3
Name {3,3,6}
 {4,3,6}
 {5,3,6}
 {6,3,6}
Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}

See also

References

  1. ↑ Coxeter The Beauty of Geometry, 1999, Chapter 10, Table III
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