In hyperbolic 3-space , the order-6 tetrahedral honeycomb is a paracompact regular space-filling tessellation (or honeycomb ). It is paracompact because it has vertex figures composed of an infinite number of faces, and has all vertices as ideal points at infinity. With Schläfli symbol {3,3,6}, the order-6 tetrahedral honeycomb has six ideal tetrahedra around each edge. All vertices are ideal , with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure .[1]
Order-6 tetrahedral honeycomb
Perspective projection view within Poincaré disk model
Type Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols {3,3,6} {3,3[3] }
Coxeter diagrams ↔
Cells {3,3}
Faces triangle {3}
Edge figure hexagon {6}
Vertex figure triangular tiling
Dual Hexagonal tiling honeycomb
Coxeter groups
V
¯
3
{\displaystyle {\overline {V}}_{3}}
, [3,3,6]
P
¯
3
{\displaystyle {\overline {P}}_{3}}
, [3,3[3] ]
Properties Regular, quasiregular
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.
Subgroup relations
The order-6 tetrahedral honeycomb has a second construction as a uniform honeycomb, with Schläfli symbol {3,3[3] }. This construction contains alternating types, or colors, of tetrahedral cells. In Coxeter notation , this half symmetry is represented as [3,3,6,1+ ] ↔ [3,((3,3,3))], or [3,3[3] ]: ↔ .
The order-6 tetrahedral honeycomb is similar to the two-dimensional infinite-order triangular tiling , {3,∞ }. Both tessellations are regular, and only contain triangles and ideal vertices.
The order-6 tetrahedral honeycomb is also a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
More information 11 paracompact regular honeycombs ...
11 paracompact regular honeycombs
{6,3,3}
{6,3,4}
{6,3,5}
{6,3,6}
{4,4,3}
{4,4,4}
{3,3,6}
{4,3,6}
{5,3,6}
{3,6,3}
{3,4,4}
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This honeycomb is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the hexagonal tiling honeycomb .
More information [6,3,3] family honeycombs, {6,3,3} ...
[6,3,3] family honeycombs
{6,3,3}
r{6,3,3}
t{6,3,3}
rr{6,3,3}
t0,3 {6,3,3}
tr{6,3,3}
t0,1,3 {6,3,3}
t0,1,2,3 {6,3,3}
{3,3,6}
r{3,3,6}
t{3,3,6}
rr{3,3,6}
2t{3,3,6}
tr{3,3,6}
t0,1,3 {3,3,6}
t0,1,2,3 {3,3,6}
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The order-6 tetrahedral honeycomb is part of a sequence of regular polychora and honeycombs with tetrahedral cells .
More information {3,3,p} polytopes, Space ...
{3,3,p} polytopes
Space
S3
H3
Form
Finite
Paracompact
Noncompact
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,∞ }
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It is also part of a sequence of honeycombs with triangular tiling vertex figures .
More information Form, Paracompact ...
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}
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Rectified order-6 tetrahedral honeycomb
More information , ...
Rectified order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb Semiregular honeycomb
Schläfli symbols r{3,3,6} or t1 {3,3,6}
Coxeter diagrams ↔
Cells r{3,3} {3,6}
Faces triangle {3}
Vertex figure hexagonal prism
Coxeter groups
V
¯
3
{\displaystyle {\overline {V}}_{3}}
, [3,3,6]
P
¯
3
{\displaystyle {\overline {P}}_{3}}
, [3,3[3] ]
Properties Vertex-transitive, edge-transitive
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The rectified order-6 tetrahedral honeycomb , t1 {3,3,6} has octahedral and triangular tiling cells arranged in a hexagonal prism vertex figure .
Perspective projection view within Poincaré disk model
More information Space, H3 ...
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{5,3}
r{6,3}
r{7,3}
r{∞ ,3}
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Truncated order-6 tetrahedral honeycomb
More information , ...
Truncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t{3,3,6} or t0,1 {3,3,6}
Coxeter diagrams ↔
Cells t{3,3} {3,6}
Faces triangle {3}hexagon {6}
Vertex figure hexagonal pyramid
Coxeter groups
V
¯
3
{\displaystyle {\overline {V}}_{3}}
, [3,3,6]
P
¯
3
{\displaystyle {\overline {P}}_{3}}
, [3,3[3] ]
Properties Vertex-transitive
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The truncated order-6 tetrahedral honeycomb , t0,1 {3,3,6} has truncated tetrahedron and triangular tiling cells arranged in a hexagonal pyramid vertex figure .
Cantellated order-6 tetrahedral honeycomb
More information , ...
Cantellated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols rr{3,3,6} or t0,2 {3,3,6}
Coxeter diagrams ↔
Cells r{3,3} r{3,6} {}x{6}
Faces triangle {3}square {4}hexagon {6}
Vertex figure isosceles triangular prism
Coxeter groups
V
¯
3
{\displaystyle {\overline {V}}_{3}}
, [3,3,6]
P
¯
3
{\displaystyle {\overline {P}}_{3}}
, [3,3[3] ]
Properties Vertex-transitive
Close
The cantellated order-6 tetrahedral honeycomb , t0,2 {3,3,6} has cuboctahedron , trihexagonal tiling , and hexagonal prism cells arranged in an isosceles triangular prism vertex figure .
Cantitruncated order-6 tetrahedral honeycomb
More information , ...
Cantitruncated order-6 tetrahedral honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols tr{3,3,6} or t0,1,2 {3,3,6}
Coxeter diagrams ↔
Cells tr{3,3} t{3,6} {}x{6}
Faces square {4}hexagon {6}
Vertex figure mirrored sphenoid
Coxeter groups
V
¯
3
{\displaystyle {\overline {V}}_{3}}
, [3,3,6]
P
¯
3
{\displaystyle {\overline {P}}_{3}}
, [3,3[3] ]
Properties Vertex-transitive
Close
The cantitruncated order-6 tetrahedral honeycomb , t0,1,2 {3,3,6} has truncated octahedron , hexagonal tiling , and hexagonal prism cells connected in a mirrored sphenoid vertex figure .