Order 3-7 kisrhombille

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Order 3-7 kisrhombille
Order-3 heptakis heptagonal tiling.png
Type Dual semiregular hyperbolic tiling
Coxeter diagram CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 7.pngCDel node f1.png
Faces Right triangle
Face configuration V4.6.14
Symmetry group [7,3], (*732)
Rotation group [7,3]+, (732)
Dual Truncated triheptagonal tiling
Properties face-transitive

In geometry, the order 3-7 kisrhombille tiling is a semiregular dual tiling of the hyperbolic plane. It is constructed by congruent right triangles with 4, 6, and 14 triangles meeting at each vertex.

The image shows a Poincaré disk model projection of the hyperbolic plane.

It is labeled V4.6.14 because each right triangle face has three types of vertices: one with 4 triangles, one with 6 triangles, and one with 14 triangles. It is the dual tessellation of the truncated triheptagonal tiling which has one square and one heptagon and one tetrakaidecagon at each vertex.

Naming[edit]

The name 3-7 kisrhombille is given by Conway, seeing it as a 3-7 rhombic tiling, divided by a kis operator, adding a center point to each rhombus, and dividing into four triangles.

Symmetry[edit]

There are no mirror removal subgroups of [7,3]. The only small index subgroup is the alternation, [7,3]+, (732).

Subgroup domains of [7,3] by mirror removal
Fundamental
domains
H2checkers 237.png
Reflective subgroup index 1
Coxeter
(orbifold)
[7,3]
(*732)
Rotation subgroup index 2
Coxeter
(orbifold)
[7,3]+
(732)

Related polyhedra and tilings[edit]

Three isohedral (regular or quasiregular) tilings can be constructed from this tiling by combining triangles:

Projections centered on different triangle points
Poincaré
disk
model
Hyperbolic domains 732.png Hyperbolic domains 732b.png Hyperbolic domains 732c.png
Center Heptagon Triangle Rhombic
Klein
disk
model
Hyperbolic domains klein 732.png Hyperbolic domains klein 732b.png Hyperbolic domains klein 732c.png
Related
tiling
Uniform tiling 73-t0.png Uniform tiling 73-t2.png Order73 qreg rhombic til.png
Heptagonal tiling Triangular tiling Rhombic tiling
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

It is topologically related to a polyhedra sequence; see discussion. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and are the reflection domains for the (2,3,n) triangle groups – for the heptagonal tiling, the important (2,3,7) triangle group.

See also the uniform tilings of the hyperbolic plane with (2,3,7) symmetry.

The kisrhombille tilings can be seen as from the sequence of rhombille tilings, starting with the cube, with faces divided or kissed at the corners by a face central point.

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.∞
Visualization of the map (2,3,∞) → (2,3,7) by morphing the associated tilings.[1]

Just as the (2,3,7) triangle group is a quotient of the modular group (2,3,∞), the associated tiling is the quotient of the modular tiling, as depicted in the video at right.

References[edit]

See also[edit]