Triheptagonal tiling

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Triheptagonal tiling
Triheptagonal tiling.svg
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (3.7)2
Schläfli symbol r{7,3} or
Wythoff symbol 7 3
Coxeter diagram CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel split1-73.pngCDel nodes.png
Symmetry group [7,3], (*732)
Dual Order-7-3 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.

Contents

Compare to trihexagonal tiling with vertex configuration 3.6.3.6.

Images

Uniform tiling 73-t1 klein.png
Klein disk model of this tiling preserves straight lines, but distorts angles
7-3 rhombille tiling.svg
The dual tiling is called an Order-7-3 rhombille tiling, made from rhombic faces, alternating 3 and 7 per vertex.

7-3 Rhombille

7-3 rhombille tiling
7-3 rhombille tiling.svg
Faces Rhombi
Coxeter diagram CDel node.pngCDel 3.pngCDel node f1.pngCDel 7.pngCDel node.png
Symmetry group [7,3], *732
Rotation group [7,3]+, (732)
Dual polyhedron Triheptagonal tiling
Face configuration V3.7.3.7
Properties edge-transitive face-transitive

In geometry, the 7-3 rhombille tiling is a tessellation of identical rhombi on the hyperbolic plane. Sets of three and seven rhombi meet two classes of vertices.

Order 7-3 rhombic tiling in the Band Model.png
7-3 rhombile tiling in band model

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:

Quasiregular tilings: (3.n)2
Sym.
*n32
[n,3]
Spherical Euclid.Compact hyperb.Paraco.Noncompact hyperbolic
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
 
*832
[8,3]...
 
*32
[,3]
 
[12i,3][9i,3][6i,3]
Figure
Quasiregular fundamental domain.png
Uniform tiling 332-t1-1-.png Uniform tiling 432-t1.png Uniform tiling 532-t1.png Uniform tiling 63-t1.svg Triheptagonal tiling.svg H2-8-3-rectified.svg H2 tiling 23i-2.png H2 tiling 23j12-2.png H2 tiling 23j9-2.png H2 tiling 23j6-2.png
Figure
Half quasiregular fundamental domain.png
Uniform tiling 332-t02.png Uniform tiling 333-t12.png H2 tiling 334-3.png H2 tiling 33i-3.png
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.)2 (3.12i)2(3.9i)2(3.6i)2
Schläfli r{3,3}r{3,4}r{3,5}r{3,6}r{3,7}r{3,8}r{3,}r{3,12i}r{3,9i}r{3,6i}
Coxeter
CDel node.pngCDel n.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel labelp.pngCDel branch 11.pngCDel split2.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel ultra.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node.pngCDel branch 11.pngCDel split2.pngCDel node.pngCDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.pngCDel labelinfin.pngCDel branch 11.pngCDel split2.pngCDel node.png
Dual uniform figures
Dual
conf.
Uniform tiling 432-t0.png
V(3.3)2
Spherical rhombic dodecahedron.png
V(3.4)2
Spherical rhombic triacontahedron.png
V(3.5)2
Rhombic star tiling.png
V(3.6)2
7-3 rhombille tiling.svg
V(3.7)2
H2-8-3-rhombic.svg
V(3.8)2
Ord3infin qreg rhombic til.png
V(3.)2

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.pngCDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
Heptagonal tiling.svg Truncated heptagonal tiling.svg Triheptagonal tiling.svg Truncated order-7 triangular tiling.svg Order-7 triangular tiling.svg Rhombitriheptagonal tiling.svg Truncated triheptagonal tiling.svg Snub triheptagonal tiling.svg
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
CDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Order-7 triangular tiling.svg Order-7 triakis triangular tiling.svg 7-3 rhombille tiling.svg Heptakis heptagonal tiling.svg Heptagonal tiling.svg Deltoidal triheptagonal tiling.svg 3-7 kisrhombille.svg 7-3 floret pentagonal tiling.svg
V73 V3.14.14V3.7.3.7V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7
Dimensional family of quasiregular polyhedra and tilings: 7.n.7.n
Symmetry
*7n2
[n,7]
Hyperbolic...ParacompactNoncompact
*732
[3,7]
*742
[4,7]
*752
[5,7]
*762
[6,7]
*772
[7,7]
*872
[8,7]...
*72
[,7]
 
[iπ/λ,7]
Coxeter CDel node.pngCDel 3.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node.pngCDel 5.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node.pngCDel 6.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node.pngCDel 8.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node.pngCDel infin.pngCDel node 1.pngCDel 7.pngCDel node.pngCDel node.pngCDel ultra.pngCDel node 1.pngCDel 7.pngCDel node.png
Quasiregular
figures
configuration
Triheptagonal tiling.svg
3.7.3.7
H2 tiling 247-2.png
4.7.4.7
H2 tiling 257-2.png
7.5.7.5
H2 tiling 267-2.png
7.6.7.6
H2 tiling 277-2.png
7.7.7.7
H2 tiling 278-2.png
7.8.7.8
H2 tiling 25i-2.png
7..7.
 
7..7.

See also

Related Research Articles

Truncated trihexagonal tiling

In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}.

Heptagonal tiling

In geometry, a heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schlafli symbol of {7,3}, having three regular heptagons around each vertex.

Order-7 triangular tiling

In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.

Truncated triheptagonal tiling

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}.

Truncated heptagonal tiling

In geometry, the truncated heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are one triangle and two tetradecagons on each vertex. It has Schläfli symbol of t{7,3}. The tiling has a vertex configuration of 3.14.14.

Rhombitriheptagonal tiling

In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has Schläfli symbol rr{7, 3}. It can be seen as constructed as a rectified triheptagonal tiling, r{7,3}, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling.

Snub triheptagonal tiling

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one heptagon on each vertex. It has Schläfli symbol of sr{7,3}. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol sr{7,4}.

Truncated order-7 triangular tiling

In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball with heptagons in place of pentagons. It has Schläfli symbol of t{3,7}.

Octagonal tiling

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

Snub trioctagonal tiling

In geometry, the order-3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of sr{8,3}.

Trioctagonal tiling

In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling. There are two triangles and two octagons alternating on each vertex. It has Schläfli symbol of r{8,3}.

Truncated octagonal tiling

In geometry, the Truncated octagonal tiling is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex. It has Schläfli symbol of t{8,3}.

Order-4 heptagonal tiling

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.

Order-7 heptagonal tiling

In geometry, the order-7 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,7}, constructed from seven heptagons around every vertex. As such, it is self-dual.

Truncated order-7 heptagonal tiling

In geometry, the truncated order-7 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{7,7}, constructed from one heptagons and two tetrakaidecagons around every vertex.

Snub heptaheptagonal tiling

In geometry, the snub heptaheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular heptagons and three equilateral triangles around every vertex.

Truncated pentahexagonal tiling

In geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon on each vertex. It has Schläfli symbol of t0,1,2{6,5}. Its name is somewhat misleading: literal geometric truncation of pentahexagonal tiling produces rectangles instead of squares.

Truncated hexaoctagonal tiling

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

Snub hexaoctagonal tiling

In geometry, the snub hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are three triangles, one hexagon, and one octagon on each vertex. It has Schläfli symbol of sr{8,6}.

Order-6 apeirogonal tiling

In geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}.

References