The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations (or honeycombs ) in hyperbolic 3-space . It is called paracompact because it has infinite cells and vertex figures , with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling .
Triangular tiling honeycomb
Type Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbol {3,6,3} h{6,3,6} h{6,3[3] } ↔ {3[3,3] }
Coxeter-Dynkin diagrams ↔ ↔ ↔
Cells {3,6}
Faces triangle {3}
Edge figure triangle {3}
Vertex figure hexagonal tiling
Dual Self-dual
Coxeter groups
Y
¯
3
{\displaystyle {\overline {Y}}_{3}}
, [3,6,3]
V
P
¯
3
{\displaystyle {\overline {VP}}_{3}}
, [6,3[3] ]
P
P
¯
3
{\displaystyle {\overline {PP}}_{3}}
, [3[3,3] ]
Properties Regular
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.
Subgroups of [3,6,3] and [6,3,6]
It has two lower reflective symmetry constructions, as an alternated order-6 hexagonal tiling honeycomb , ↔ , and as from , which alternates 3 types (colors) of triangular tilings around every edge. In Coxeter notation , the removal of the 3rd and 4th mirrors, [3,6,3* ] creates a new Coxeter group [3[3,3] ], , subgroup index 6. The fundamental domain is 6 times larger. By Coxeter diagram there are 3 copies of the first original mirror in the new fundamental domain: ↔ .
The triangular tiling honeycomb is a regular hyperbolic honeycomb in 3-space, and one of eleven paracompact honeycombs.
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}
Close
There are nine uniform honeycombs in the [3,6,3] Coxeter group family, including this regular form as well as the bitruncated form, t1,2 {3,6,3}, with all truncated hexagonal tiling facets.
More information {3,6,3}, r{3,6,3} ...
[3,6,3] family honeycombs
{3,6,3}
r{3,6,3}
t{3,6,3}
rr{3,6,3}
t0,3 {3,6,3}
2t{3,6,3}
tr{3,6,3}
t0,1,3 {3,6,3}
t0,1,2,3 {3,6,3}
Close
The honeycomb is also part of a series of polychora and honeycombs with triangular edge figures .
More information {3,p,3} polytopes, Space ...
{3,p ,3} polytopes
Space
S3
H3
Form
Finite
Compact
Paracompact
Noncompact
{3,p ,3}
{3,3,3}
{3,4,3}
{3,5,3}
{3,6,3}
{3,7,3}
{3,8,3}
... {3,∞ ,3}
Image
Cells
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞ }
Vertex figure
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞ ,3}
Close
Rectified triangular tiling honeycomb
More information , ...
Rectified triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol r{3,6,3} h2 {6,3,6}
Coxeter diagram ↔ ↔ ↔
Cells r{3,6} {6,3}
Faces triangle {3}hexagon {6}
Vertex figure triangular prism
Coxeter group
Y
¯
3
{\displaystyle {\overline {Y}}_{3}}
, [3,6,3]
V
P
¯
3
{\displaystyle {\overline {VP}}_{3}}
, [6,3[3] ]
P
P
¯
3
{\displaystyle {\overline {PP}}_{3}}
, [3[3,3] ]
Properties Vertex-transitive, edge-transitive
Close
The rectified triangular tiling honeycomb , , has trihexagonal tiling and hexagonal tiling cells, with a triangular prism vertex figure.
Truncated triangular tiling honeycomb
More information , ...
Truncated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t{3,6,3}
Coxeter diagram
Cells t{3,6} {6,3}
Faces hexagon {6}
Vertex figure tetrahedron
Coxeter group
Y
¯
3
{\displaystyle {\overline {Y}}_{3}}
, [3,6,3]
V
¯
3
{\displaystyle {\overline {V}}_{3}}
, [3,3,6]
Properties Regular
Close
The truncated triangular tiling honeycomb , , is a lower-symmetry form of the hexagonal tiling honeycomb , . It contains hexagonal tiling facets with a tetrahedral vertex figure.
Cantellated triangular tiling honeycomb
More information ...
Cantellated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol rr{3,6,3} or t0,2 {3,6,3} s2 {3,6,3}
Coxeter diagram
Cells rr{6,3} r{6,3} {}×{3}
Faces triangle {3}square {4}hexagon {6}
Vertex figure wedge
Coxeter group
Y
¯
3
{\displaystyle {\overline {Y}}_{3}}
, [3,6,3]
Properties Vertex-transitive
Close
The cantellated triangular tiling honeycomb , , has rhombitrihexagonal tiling , trihexagonal tiling , and triangular prism cells, with a wedge vertex figure.
Cantitruncated triangular tiling honeycomb
More information ...
Cantitruncated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol tr{3,6,3} or t0,1,2 {3,6,3}
Coxeter diagram
Cells tr{6,3} t{6,3} {}×{3}
Faces triangle {3}square {4}hexagon {6}dodecagon {12}
Vertex figure mirrored sphenoid
Coxeter group
Y
¯
3
{\displaystyle {\overline {Y}}_{3}}
, [3,6,3]
Properties Vertex-transitive
Close
The cantitruncated triangular tiling honeycomb , , has truncated trihexagonal tiling , truncated hexagonal tiling , and triangular prism cells, with a mirrored sphenoid vertex figure.
Runcitruncated triangular tiling honeycomb
More information ...
Runcitruncated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3 {3,6,3} s2,3 {3,6,3}
Coxeter diagrams
Cells t{3,6} rr{3,6} {}×{3} {}×{6}
Faces triangle {3}square {4}hexagon {6}
Vertex figure isosceles-trapezoidal pyramid
Coxeter group
Y
¯
3
{\displaystyle {\overline {Y}}_{3}}
, [3,6,3]
Properties Vertex-transitive
Close
The runcitruncated triangular tiling honeycomb , , has hexagonal tiling , rhombitrihexagonal tiling , triangular prism , and hexagonal prism cells, with an isosceles-trapezoidal pyramid vertex figure .
Omnitruncated triangular tiling honeycomb
More information ...
Omnitruncated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3 {3,6,3}
Coxeter diagram
Cells tr{3,6} {}×{6}
Faces square {4}hexagon {6}dodecagon {12}
Vertex figure phyllic disphenoid
Coxeter group
2
×
Y
¯
3
{\displaystyle 2\times {\overline {Y}}_{3}}
, [[ 3,6,3]]
Properties Vertex-transitive, edge-transitive
Close
The omnitruncated triangular tiling honeycomb , , has truncated trihexagonal tiling and hexagonal prism cells, with a phyllic disphenoid vertex figure.