Order-6 cubic honeycomb

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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
Cellscube {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

The order-6 cubic honeycomb is a regular space-filling tessellations (or honeycombs) in hyperbolic 3-space. 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.

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.

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.

{4,3,p}
Space S3 E3 H3
Name
 {4,3,3}
 {4,3,4}
 {4,3,5}
 {4,3,6}
Image
Vertex
figure
{3,3}
{3,4}
{3,5}
{3,6}

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

  • Convex uniform honeycombs in hyperbolic space

References

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