Order-6 cubic honeycomb

Order-6 cubic honeycomb

Perspective projection view
within Poincaré disk model
TypeHyperbolic regular honeycomb
Paracompact uniform honeycomb
Schläfli symbol {4,3,6}
{4,3[3]}
Coxeter diagram
↔
↔
Cells{4,3}
Facessquare {4}
Edge figurepentagon {6}
Vertex figuretriangular tiling {3,6}
Coxeter groupBV3, [6,3,4]
BP3, [4,3[3]]
DualOrder-4 hexagonal tiling honeycomb
PropertiesRegular, quasiregular

The order-6 cubic honeycomb is a paracompact regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. It is called paracompact because it has infinite vertex figures, with all vertices as ideal points at infinity. With schläfli symbol {4,3,6}, it is constructed from six cubes exist on each edge. Its vertex figure is an infinite triangular tiling. It is dual is the order-4 hexagonal tiling honeycomb.

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.

Images

It is similar to the 2D hyperbolic infinite-order square tiling, {4,∞} with square faces. All vertices are on the ideal surface.

Symmetry

A half symmetry construction exists as {4,3[3]}, with alternating two types (colors) of cubic cells. ↔ . Another lower symmetry, [4,3*,6], index 6 exists with a nonsimplex fundamental domain, .

This honeycomb contains that tile 2-hypercycle surfaces, similar to this paracompact tiling, :

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 related to the regular (order-4) cubic honeycomb of Euclidean 3-space, order-5 cubic honeycomb in hyperbolic space, which have 4 and 5 cubes per edge respectively.

It has a related alternation honeycomb, represented by ↔ , having hexagonal tiling and tetrahedron cells.

There are fifteen uniform honeycombs in the [6,3,4] Coxeter group family, including this regular form.

It in a sequence of regular polychora and honeycombs with cubic cells.

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

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

Rectified order-6 cubic honeycomb

Rectified order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsr{4,3,6} or t1{4,3,6}
Coxeter diagrams
↔
↔
↔
Cellsr{3,4}
{3,6}
FacesTriangle {3}
Square {4}
Vertex figure
hexagonal prism {}×{6}
Coxeter groupsBV3, [6,3,4]
DV3, [6,31,1]
[4,3[3]]
[3[ ]×[3]]
PropertiesVertex-transitive, edge-transitive

The rectified order-6 cubic honeycomb, r{4,3,6}, has cuboctahedral and triangular tiling facets, with a hexagonal prism vertex figure.

It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{4,∞}, alternating apeirogonal and square faces:

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 cubic honeycomb

Truncated order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolst{4,3,6} or t0,1{4,3,6}
Coxeter diagrams
↔
Cellst{4,3}
{3,6}
FacesTriangle {3}
octagon {8}
Vertex figure
hexagonal pyramid
Coxeter groupsBV3, [6,3,4]
[4,3[3]]
PropertiesVertex-transitive

The truncated order-6 cubic honeycomb, t{4,3,6}, has truncated octahedron and triangular tiling facets, with a hexagonal pyramid vertex figure.

It is similar to the 2D hyperbolic truncated infinite-order square tiling, t{4,∞}, with apeirogonal and octagonal (truncated square) faces:

Cantellated order-6 cubic honeycomb

Cantellated order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsrr{4,3,6} or t0,2{4,3,6}
Coxeter diagrams
↔
Cellsrr{4,3}
r{3,6}
FacesTriangle {3}
square {4}
hexagon {6}
octagon {8}
Vertex figure
triangular prism
Coxeter groupsBV3, [6,3,4]
[4,3[3]]
PropertiesVertex-transitive

The cantellated order-6 cubic honeycomb, rr{4,3,6}, has rhombicuboctahedron and trihexagonal tiling facets, with a triangular prism vertex figure.

Alternated order-6 cubic honeycomb

Alternated order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Semiregular honeycomb
Schläfli symbolh{4,3,6}
Coxeter diagram ↔
↔ ↔
↔
Cells{3,3}
{3,6}
FacesTriangle {3}
Vertex figure
trihexagonal tiling
Coxeter groupDV3, [6,31,1]
PropertiesVertex-transitive, edge-transitive, quasiregular

In 3-dimensional hyperbolic geometry, the alternated order-6 hexagonal tiling honeycomb is a uniform compact space-filling tessellations (or honeycombs). As an alternated order-6 cubic honeycomb and Schläfli symbol h{4,3,6}, with Coxeter diagram or . It can be considered a quasiregular honeycomb, alternating triangular tiling and tetrahedron around each vertex in an trihexagonal tiling vertex figure.

Symmetry

A half symmetry construction exists from {4,3[3]}, with alternating two types (colors) of cubic cells. ↔ . Another lower symmetry, [4,3*,6], index 6 exists with a nonsimplex fundamental domain, .

Related honeycombs

It has 3 related form cantic order-6 cubic honeycomb, h2{4,3,6}, , runcic order-6 cubic honeycomb, h3{4,3,6}, , runcicantic order-6 cubic honeycomb, h2,3{4,3,6}, .

Cantic order-6 cubic honeycomb

Cantic order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolh2{4,3,6}
Coxeter diagram ↔
↔ ↔
Cellst{3,3}
r{6,3}
{6,3}
FacesTriangle {3}
hexagon {6}
Vertex figure
Coxeter groupDV3, [6,31,1]
PropertiesVertex-transitive

The cantic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs) with Schläfli symbol h2{4,3,6}.

Runcic order-6 cubic honeycomb

Runcic order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolh3{4,3,6}
Coxeter diagram ↔
Cells{3,3}
{6,3}
rr{6,3}
FacesTriangle {3}
hexagon {6}
Vertex figuretriangular prism
Coxeter groupDV3, [6,31,1]
PropertiesVertex-transitive

The runcic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs). With Schläfli symbol h3{4,3,6}, with a triangular prism vertex figure.

Runcicantic order-6 cubic honeycomb

Runcicantic order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolh2,3{4,3,6}
Coxeter diagram ↔
Cells{6,3}
tr{6,3}
{3,3}
FacesTriangle {3}
square {4}
Vertex figuretetrahedron
Coxeter groupDV3, [6,31,1]
PropertiesVertex-transitive

The runcicantic order-6 cubic honeycomb is a uniform compact space-filling tessellations (or honeycombs). With Schläfli symbol h2,3{4,3,6}, with a tetrahedral vertex figure.

See also

References

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