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词条 Order-6 cubic honeycomb
释义

  1. Images

  2. Symmetry

  3. Related polytopes and honeycombs

      Rectified order-6 cubic honeycomb    Truncated order-6 cubic honeycomb    Cantellated order-6 cubic honeycomb    Alternated order-6 cubic honeycomb    Symmetry    Related honeycombs  Cantic order-6 cubic honeycomb  Runcic order-6 cubic honeycomb  Runcicantic order-6 cubic honeycomb 

  4. See also

  5. References

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 diagramnode_1|4|node|3|node|6|node}}
{{CDD|node_1|4|node|split1|branch}} ↔ {{CDD|node_1|4|node|3|node|6|node_h0}}
{{CDD|node_1|ultra|node|split1|branch|uaub|nodes_11}} ↔ {{CDD|node_1|4|node_g|3sg|node_g|6|node}}
Cells{4,3}
Facessquare {4}
Edge figurepentagon {6}
Vertex figuretriangular tiling {3,6}
Coxeter groupBV}}3, [6,3,4]
{{overline|BP}}3, [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 has six cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. It is dual is the order-4 hexagonal tiling honeycomb.

{{Honeycomb}}

Images


One cell viewed outside of Poincare sphere model
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. {{CDD|node_1|4|node|3|node|6|node_h0}} ↔ {{CDD|node_1|4|node|split1|branch}}. Another lower symmetry, [4,3*,6], index 6 exists with a nonsimplex fundamental domain, {{CDD|node_1|ultra|node|split1|branch|uaub|nodes_11}}.

This honeycomb contains {{CDD|node|3|node|ultra|node_1}} that tile 2-hypercycle surfaces, similar to this paracompact tiling, {{CDD|node|3|node|infin|node_1}}:

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.

{{Regular_paracompact_H3_honeycombs}}

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 {{CDD|node_h1|4|node|3|node|6|node}} ↔ {{CDD|nodes_10ru|split2|node|6|node}}, having hexagonal tiling and tetrahedron cells.

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

{{634 family}}

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

{{Cubic cell tessellations}}

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

{{Triangular tiling vertex figure tessellations small}}

Rectified order-6 cubic honeycomb

Rectified order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolsr{4,3,6} or t1{4,3,6}
Coxeter diagramsnode|4|node_1|3|node|6|node}}
{{CDD|nodes_11|split2|node|6|node}} ↔ {{CDD|node_h0|4|node_1|3|node|6|node}}
{{CDD|node|4|node_1|split1|branch}} ↔ {{CDD|node|4|node_1|3|node|6|node_h0}}
{{CDD|node_1|split1|branch|split2|node_1}} ↔ {{CDD|node_h0|4|node_1|3|node|6|node_h0}}
Cellsr{3,4}
{3,6}
FacesTriangle {3}
Square {4}
Vertex figure
hexagonal prism {}×{6}
Coxeter groupsBV}}3, [6,3,4]
{{overline|DV}}3, [6,31,1]
[4,3[3]]
[3[ ]×[3]]
PropertiesVertex-transitive, edge-transitive

The rectified order-6 cubic honeycomb, r{4,3,6}, {{CDD||node|4|node_1|3|node|6|node}} has cuboctahedral and triangular tiling facets, with a hexagonal prism vertex figure.

It is similar to the 2D hyperbolic tetraapeirogonal tiling, r{4,∞}, {{CDD||node|4|node_1|infin|node}} alternating apeirogonal and square faces:

{{Hexagonal tiling vertex figure tessellations}}{{-}}

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 diagramsnode_1|4|node_1|3|node|6|node}}
{{CDD|node_1|4|node_1|split1|branch}} ↔ {{CDD|node_1|4|node_1|3|node|6|node_h0}}
Cellst{4,3}
{3,6}
FacesTriangle {3}
octagon {8}
Vertex figure
hexagonal pyramid
Coxeter groupsBV}}3, [6,3,4]
[4,3[3]]
PropertiesVertex-transitive

The truncated order-6 cubic honeycomb, t{4,3,6}, {{CDD||node_1|4|node_1|3|node|6|node}} 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,∞}, {{CDD||node_1|4|node_1|infin|node}} 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 diagramsnode_1|4|node|3|node_1|6|node}}
{{CDD|node_1|4|node|split1|branch_11}} ↔ {{CDD|node_1|4|node|3|node_1|6|node_h0}}
Cellsrr{4,3}
r{3,6}
FacesTriangle {3}
square {4}
hexagon {6}
octagon {8}
Vertex figure
triangular prism
Coxeter groupsBV}}3, [6,3,4]
[4,3[3]]
PropertiesVertex-transitive

The cantellated order-6 cubic honeycomb, rr{4,3,6}, {{CDD||node_1|4|node|3|node_1|6|node}} 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 diagramnode_h1|4|node|3|node|6|node}} ↔ {{CDD|nodes_10ru|split2|node|6|node}}
{{CDD|node_h1|4|node|split1|branch}} ↔ {{CDD|node_h1|4|node|3|node|6|node_h0}} ↔ {{CDD|node_1|split1|branch|split2|node}}
{{CDD|node_h|ultra|node|split1|branch|uaub|nodes_hh}} ↔ {{CDD|node_h|4|node_g|3sg|node_g|6|node}}
Cells{3,3}
{3,6}
FacesTriangle {3}
Vertex figure
trihexagonal tiling
Coxeter groupDV}}3, [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 {{CDD|node_h1|4|node|3|node|6|node}} or {{CDD|nodes_10ru|split2|node|6|node}}. It can be considered a quasiregular honeycomb, alternating triangular tiling and tetrahedron around each vertex in a trihexagonal tiling vertex figure.

Symmetry

A half symmetry construction exists from {4,3[3]}, with alternating two types (colors) of cubic cells. {{CDD|node_h1|4|node|3|node|6|node_h0}} ↔ {{CDD|node_h1|4|node|split1|branch}}. Another lower symmetry, [4,3*,6], index 6 exists with a nonsimplex fundamental domain, {{CDD|node_h|ultra|node|split1|branch|uaub|nodes_hh}}.

Related honeycombs

{{Quasiregular polychora and honeycombs}}

It has 3 related form cantic order-6 cubic honeycomb, h2{4,3,6}, {{CDD|node_h1|4|node|3|node_1|6|node}}, runcic order-6 cubic honeycomb, h3{4,3,6}, {{CDD|node_h1|4|node|3|node|6|node_1}}, runcicantic order-6 cubic honeycomb, h2,3{4,3,6}, {{CDD|node_h1|4|node|3|node_1|6|node_1}}.

{{-}}

Cantic order-6 cubic honeycomb

Cantic order-6 cubic honeycomb
TypeParacompact uniform honeycomb
Schläfli symbolh2{4,3,6}
Coxeter diagramnode_h1|4|node|3|node_1|6|node}} ↔ {{CDD|nodes_10ru|split2|node_1|6|node}}
{{CDD|node_h1|4|node|3|node_1|6|node_h0}} ↔ {{CDD|node_h1|4|node|split1|branch_11}} ↔ {{CDD|node_1|split1|branch_11|split2|node}}
Cellst{3,3}
r{6,3}
{6,3}
FacesTriangle {3}
hexagon {6}
Vertex figure
Coxeter groupDV}}3, [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 diagramnode_h1|4|node|3|node|6|node_1}} ↔ {{CDD|nodes_10ru|split2|node|6|node_1}}
Cells{3,3}
{6,3}
rr{6,3}
FacesTriangle {3}
hexagon {6}
Vertex figuretriangular prism
Coxeter groupDV}}3, [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 diagramnode_h1|4|node|3|node_1|6|node_1}} ↔ {{CDD|nodes_10ru|split2|node_1|6|node_1}}
Cells{6,3}
tr{6,3}
{3,3}
FacesTriangle {3}
square {4}
Vertex figuretetrahedron
Coxeter groupDV}}3, [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

  • Convex uniform honeycombs in hyperbolic space

References

  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. {{isbn|0-486-61480-8}}. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, {{LCCN|99035678}}, {{isbn|0-486-40919-8}} (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
  • Jeffrey R. Weeks The Shape of Space, 2nd edition {{isbn|0-8247-0709-5}} (Chapter 16-17: Geometries on Three-manifolds I,II)
  • Norman Johnson Uniform Polytopes, Manuscript
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
    • N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups

1 : Honeycombs (geometry)
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