Truncated triheptagonal tiling

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Truncated triheptagonal tiling
Truncated triheptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex figure 4.6.14
Schläfli symbol tr{7,3}
Wythoff symbol 2 7 3 |
Coxeter diagram CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Symmetry group [7,3], (*732)
Dual Order 3-7 kisrhombille
Properties Vertex-transitive

In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of tr{7,3}.

Uniform colorings[edit]

There is only one uniform coloring of a truncated triheptagonal tiling. (Naming the colors by indices around a vertex: 123.)

Symmetry[edit]

Each triangle in this dual tiling, order 3-7 kisrhombille, represent a fundamental domain of the Wythoff construction for the symmetry group [7,3].

Order-3 heptakis heptagonal tiling.png H2checkers 237.png
The dual tiling is called an order-3 bisected heptagonal tiling, made as a complete bisection of the heptagonal tiling, here shown with triangles with alternating colors.

Related polyhedra and tilings[edit]

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram CDel node 1.pngCDel p.pngCDel node 1.pngCDel 3.pngCDel node 1.png. For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

Dimensional family of omnitruncated polyhedra and tilings: 4.6.2n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
P6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Coxeter
Schläfli
CDel node 1.pngCDel 2.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{2,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{3,3}
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{4,3}
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{5,3}
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{6,3}
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{7,3}
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{8,3}
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
tr{∞,3}
Omnitruncated
figure
Spherical truncated trigonal prism.png Uniform tiling 332-t012.png Uniform tiling 432-t012.png Uniform tiling 532-t012.png Uniform polyhedron-63-t012.png H2 tiling 237-7.png H2 tiling 238-7.png H2 tiling 23i-7.png
Vertex figure 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞
Dual figures
Coxeter CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Omnitruncated
duals
Hexagonale bipiramide.png Tetrakishexahedron.jpg Disdyakisdodecahedron.jpg Disdyakistriacontahedron.jpg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Order-3 heptakis heptagonal tiling.png Order-3 octakis octagonal tiling.png H2checkers 23i.png
Face
configuration
V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞


From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 73-t0.png Uniform tiling 73-t01.png Uniform tiling 73-t1.png Uniform tiling 73-t12.png Uniform tiling 73-t2.png Uniform tiling 73-t02.png Uniform tiling 73-t012.png Uniform tiling 73-snub.png
{7,3} t{7,3} r{7,3} 2t{7,3}=t{3,7} 2r{7,3}={3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 73-t2.png Ord7 triakis triang til.png Order73 qreg rhombic til.png Order3 heptakis heptagonal til.png Uniform tiling 73-t0.png Deltoidal triheptagonal til.png Order-3 heptakis heptagonal tiling.png Ord7 3 floret penta til.png
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7

References[edit]

See also[edit]

External links[edit]