Order-6 tetrahedral honeycomb

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
↔
Cells{3,3}
FacesTriangle {3}
Edge figureHexagon {6}
Vertex figureTriangular tiling {3,6}

DualHexagonal tiling honeycomb, {6,3,3}
Coxeter groups, [6,3,3]
, [3,3[3]]
PropertiesRegular, quasiregular

In the geometry of hyperbolic 3-space, the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb). It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. 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]

A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.

Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.

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. In Coxeter notation the half symmetry is [3,3,6,1+] ↔ [3,((3,3,3))] or [3,3[3]]: ↔ .

It is similar to the 2-dimensional hyperbolic tiling, infinite-order triangular tiling, {3,∞}, having all triangle faces, and all ideal vertices.

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

633 honeycombs

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}.

Tetrahedral cell honeycombs

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

Triangular tiling vertex figures

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

Hyperbolic uniform honeycombs: {p,3,6} and {p,3[3]}
Form Paracompact Noncompact
Name {3,3,6}
{3,3[3]}		 {4,3,6}
{4,3[3]}		 {5,3,6}
{5,3[3]}		 {6,3,6}
{6,3[3]}		 {7,3,6}
{7,3[3]}		 {8,3,6}
{8,3[3]}		 ... {∞,3,6}
{∞,3[3]}






Image
Cells
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}

Rectified order-6 tetrahedral honeycomb

Rectified order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Semiregular honeycomb
Schläfli symbolsr{3,3,6} or t1{3,3,6}
Coxeter diagrams
↔
Cells{3,4}
{3,6}
Vertex figure
Hexagonal prism { }×{6}
Coxeter groups, [6,3,3]
, [3,3[3]]
PropertiesVertex-transitive, edge-transitive

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


Perspective projection view within Poincaré disk model
r{p,3,6}
Space H3
Form Paracompact Noncompact
Name r{3,3,6}
 r{4,3,6}
 r{5,3,6}
 r{6,3,6}
 r{7,3,6}
 ... r{∞,3,6}
Image
Cells

{3,6}
r{3,3}
r{4,3}
r{6,3}
r{6,3}
r{∞,3}
r{∞,3}

Truncated order-6 tetrahedral honeycomb

Truncated order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolst{3,3,6} or t0,1{3,3,6}
Coxeter diagrams
↔
Cellst{3,3}
{3,6}
Vertex figure
Hexagonal pyramid { }v{6}
Coxeter groups, [6,3,3]
, [3,3[3]]
PropertiesVertex-transitive

The truncated order-6 tetrahedral honeycomb, t0,1{3,3,6} has truncated tetrahedra and triangular tiling cells connected in a hexagonal prism vertex figure.

Bitruncated order-6 tetrahedral honeycomb

Cantellated order-6 tetrahedral honeycomb

Cantellated order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsrr{3,3,6} or t0,2{3,3,6}
Coxeter diagrams
↔
Cellsr{3,3}
r{3,6}
{}x{6}
Vertex figure
tetrahedron
Coxeter groups, [6,3,3]
, [3,3[3]]
PropertiesVertex-transitive

The cantellated order-6 tetrahedral honeycomb, t0,2{3,3,6} has cuboctahedron and trihexagonal tiling cells connected in a tetrahedron vertex figure.

Cantitruncated order-6 tetrahedral honeycomb

Cantitruncated order-6 tetrahedral honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolstr{3,3,6} or t0,1,2{3,3,6}
Coxeter diagrams
↔
Cellstr{3,3}
r{3,6}
{}x{6}
Vertex figure
tetrahedron
Coxeter groups, [6,3,3]
, [3,3[3]]
PropertiesVertex-transitive

The cantitruncated order-6 tetrahedral honeycomb, t0,1,2{3,3,6} has truncated cuboctahedron and trihexagonal tiling cells connected in an octahedron vertex figure.

See also

References

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