Triangular tiling honeycomb

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
Triangular tiling honeycomb
H3 363 FC boundary.png
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 CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel split1.pngCDel branch.pngCDel node 1.pngCDel splitsplit1.pngCDel branch4.pngCDel splitsplit2.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node h0.png
Cells {3,6} Uniform tiling 63-t2.png Uniform tiling 333-t1.png
Faces triangle {3}
Edge figure triangle {3}
Vertex figure {6,3}
Uniform tiling 63-t0.png Uniform tiling 63-t12.png Uniform tiling 333-t012.png
Dual Self-dual
Coxeter groups Y3, [3,6,3]
, [3[3,3]]
Properties Regular

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.

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.

Symmetry[edit]

Subgroups of [3,6,3] and [6,3,6]

It has two lower reflective symmetry constructions, as an alternated order-6 hexagonal tiling honeycomb, CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node.pngCDel 6.pngCDel node.png, and as CDel node 1.pngCDel splitsplit1.pngCDel branch4.pngCDel splitsplit2.pngCDel node.png from CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.png, 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]], CDel node.pngCDel splitsplit1.pngCDel branch4.pngCDel splitsplit2.pngCDel node.png, 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: CDel node c2.pngCDel 3.pngCDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel node c2.pngCDel splitsplit1.pngCDel branch4 c1.pngCDel splitsplit2.pngCDel node c1.png.

Images[edit]

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

H2 tiling 2ii-4.png

Related honeycombs[edit]

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.

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}, CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png with all truncated hexagonal tiling facets.

[3,6,3] family honeycombs
{3,6,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
r{3,6,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
t{3,6,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
rr{3,6,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,3{3,6,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
2t{3,6,3}
CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
tr{3,6,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
t0,1,3{3,6,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
t0,1,2,3{3,6,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
H3 363 FC boundary.png H3 363 boundary 0100.png H3 363-1100.png H3 363-1010.png H3 363-1001.png H3 363-0110.png H3 363-1110.png H3 363-1011.png H3 363-1111.png

Rectified triangular tiling honeycomb[edit]

Rectified triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol r{3,6,3}
h2{6,3,6}
Coxeter diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel splitsplit1.pngCDel branch4 11.pngCDel splitsplit2.pngCDel node 1.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node h0.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.png
Cells r{3,6} Uniform polyhedron-63-t1.png Uniform tiling 333-t01.png
{6,3} Uniform polyhedron-63-t0.png Uniform tiling 333-t012.png
Faces triangle {3}
Hexagon {6}
Vertex figure Rectified triangular tiling honeycomb verf.png
Triangular prism
Coxeter group Y3, [3,6,3]
Properties Vertex-transitive

The rectified triangular tiling honeycomb, CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png, has alternating trihexagonal tiling and hexagonal tiling cells, with a triangular prism vertex figure.

Symmetry[edit]

A lower symmetry of this honeycomb can be constructed as a cantic order-6 hexagonal tiling honeycomb, CDel branch 10ru.pngCDel split2.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png. A second lower construction, index is CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node g.pngCDel 3sg.pngCDel node g.pngCDel node.pngCDel splitsplit1.pngCDel branch4 11.pngCDel splitsplit2.pngCDel node 1.png.

H3 363 boundary 0100.png

Truncated triangular tiling honeycomb[edit]

The truncated triangular tiling honeycomb,CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png, is the same as the hexagonal tiling honeycomb, CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png.

H3 363-1100.png H3 633 FC boundary.png

Bitruncated triangular tiling honeycomb[edit]

Bitruncated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol 2t{3,6,3}
Coxeter diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells t{6,3} Uniform polyhedron-63-t01.png
Faces Triangle {3}
Dodecagon {12}
Vertex figure Bitruncated triangular tiling honeycomb verf.png
Tetrahedron
Coxeter group Y3×2, [[3,6,3]]
Properties Vertex-transitive, edge-transitive, cell-transitive

The bitruncated triangular tiling honeycomb, CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png, has truncated hexagonal tiling cells, with a tetrahedral vertex figure.

H3 363-0110.png

Cantellated triangular tiling honeycomb[edit]

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 CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells rr{6,3} Uniform polyhedron-63-t02.png
r{6,3} Uniform polyhedron-63-t1.png
{}×{3} Triangular prism.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Vertex figure Cantellated triangular tiling honeycomb verf.png
triangular prism
Coxeter group Y3, [3,6,3]
Properties Vertex-transitive

The cantellated triangular tiling honeycomb, CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png, has truncated hexagonal tiling and triangular prism cells, with a triangular prism vertex figure.

H3 363-1010.png

Symmetry[edit]

It can also be constructed as a half-symmetry form, cantic snub triangular tiling honeycomb, CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png, symmetry [3+,6,3].

Cantitruncated triangular tiling honeycomb[edit]

Cantitruncated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol tr{3,6,3} or t0,1,2{3,6,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells tr{6,3} Uniform polyhedron-63-t012.png
t{6,3} Uniform polyhedron-63-t01.png
{}×{6} Hexagonal prism.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Dodecagon {12}
Vertex figure Cantitruncated triangular tiling honeycomb verf.png
tetrahedron
Coxeter group Y3, [3,6,3]
Properties Vertex-transitive

The cantitruncated triangular tiling honeycomb, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png, has rhombitrihexagonal tiling and hexagonal prism cells, with a tetrahedron vertex figure.

H3 363-1110.png

Runcinated triangular tiling honeycomb[edit]

Runcinated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,3{3,6,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells {3,6} Uniform polyhedron-63-t2.png
{}×{3} Triangular prism.png
Faces Triangle {3}
Square {4}
Vertex figure Runcinated triangular tiling honeycomb verf.png
hexagonal antiprism
Coxeter group Y3, [[3,6,3]]
Properties Vertex-transitive, edge-transitive

The runcinated triangular tiling honeycomb, CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png, has hexagonal tiling and triangular prism cells, with a hexagonal antiprism vertex figure.

H3 363-1001.png

Runcitruncated triangular tiling honeycomb[edit]

Runcitruncated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbols t0,1,3{3,6,3}
s2,3{3,6,3}
Coxeter diagrams CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells t{3,6} Uniform polyhedron-63-t12.png
rr{3,6} Uniform polyhedron-63-t02.png
{}×{3} Triangular prism.png
{}×{6} Hexagonal prism.png
Faces Triangle {3}
Square {4}
Hexagon {6}
Vertex figure Runcitruncated triangular tiling honeycomb verf.png
quadrilateral pyramid
Coxeter group Y3, [3,6,3]
Properties Vertex-transitive

The runcitruncated triangular tiling honeycomb, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png, has hexagonal tiling, rhombitrihexagonal tiling, triangular prism, and dodecagonal prism cells.

H3 363-1101.png

Omnitruncated triangular tiling honeycomb[edit]

Omnitruncated triangular tiling honeycomb
Type Paracompact uniform honeycomb
Schläfli symbol t0,1,2,3{3,6,3}
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells tr{3,6} Uniform polyhedron-63-t012.png
{}×{12} Dodecagonal prism.png
Faces Square {4}
Hexagon {6}
Dodecagon {12}
Vertex figure Omnitruncated triangular tiling honeycomb verf.png
Phyllic disphenoid
Coxeter group Y3, [[3,6,3]]
Properties Vertex-transitive, edge-transitive

The omnitruncated triangular tiling honeycomb, CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png, has truncated trihexagonal tiling and hexagonal prism cells, with a tetrahedron vertex figure.

H3 363-1111.png

Runcisnub triangular tiling honeycomb[edit]

Runcisnub triangular tiling honeycomb
Type Paracompact scaliform honeycomb
Schläfli symbol s3{3,6,3}
Coxeter diagram CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Cells r{6,3} Uniform tiling 333-t02.png
{}x{3} Triangular prism.png
{3,6} Uniform tiling 333-t1.png
tricup Triangular cupola.png
Faces Triangle {3}
Square {4}
hexagon {6}
Vertex figure
Coxeter group Y3, [3+,6,3]
Properties Vertex-transitive

The runcisnub triangular tiling honeycomb, CDel node h.pngCDel 3.pngCDel node h.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png, has trihexagonal tiling, triangular tiling, triangular prism, and triangular cupola cells. It is vertex-transitive, but not uniform, since it contains Johnson solid triangular cupola cells.

See also[edit]

References[edit]