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Homogeneous mosaics on the hyperbolic plane

Examples of homogeneous mosaics
SphericalEuclideanHyperbolic
Uniform tiling 532-t0.png
{5,3}
5.5.5
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png		Uniform tiling 63-t0.png
{6.3}
6.6.6
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png		Uniform tiling 73-t0.png
{7.3}
7.7.7
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png		H2 tiling 23i-1.png

∞.∞.∞
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png
Regular mosaics on the sphere {p, q}, the Euclidean plane and the hyperbolic plane with faces in the form of regular pentagons, hexagons, heptagons and infinite angles.
Uniform tiling 532-t01.png
t {5,3}
10.10.3
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png		Uniform tiling 63-t01.png

12.12.3
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png		Uniform tiling 73-t01.png

14.14.3
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png		H2 tiling 23i-3.png

∞.∞.3
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
Truncated mosaics have 2p.2p.q vertex shapes derived from regular {p, q}
Uniform tiling 532-t1.png
r {5,3}
3.5.3.5
CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png		Uniform tiling 63-t1.png
r {6,3}
3.6.3.6
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png		Uniform tiling 73-t1.png

3.7.3.7
CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png		H2 tiling 23i-2.png

3.∞.3.∞
CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png
Quasiregular mosaics are similar to regular mosaics, but they have two types of regular polygons, alternately following around each vertex.
Uniform tiling 532-t02.png
rr {5,3}
3.4.5.4
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png		Uniform tiling 63-t02.png

3.4.6.4
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png		Uniform tiling 73-t02.png

3.4.7.4
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node 1.png		H2 tiling 23i-5.png

3.4.∞.4
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
Semi-regular mosaics have more than one type of regular polygon.
Uniform tiling 532-t012.png
tr {5,3}
4.6.10
CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png		Uniform tiling 63-t012.png

4.6.12
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png		Uniform tiling 73-t012.png

4.6.14
CDel node 1.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node 1.png		H2 tiling 23i-7.png

4.6.∞
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node 1.png
have three or more regular polygons with an even number of sides.

In hyperbolic geometry, a homogeneous (regular, quasi-regular, or semi-regular) hyperbolic mosaic is filling an hyperbolic plane with regular edge-to-edge polygons with the property of vertex transitivity (this is a mosaic transitive with respect to vertices , isogonal, i.e. there is a movement that takes any vertex to any another). It follows that all the vertices are congruent and the mosaic has a high degree of rotational and translational symmetry .

Homogeneous mosaics are uniquely determined by their , a sequence of numbers representing the number of sides of the polygons around each vertex. For example, 7.7.7 represents a heptagonal mosaic having 3 heptagons around each vertex. It is correct because all polygons are of the same size. Thus, it can be set with the Shlefly symbol {7.3}.

Homogeneous mosaics can be regular (if they are also transitive along faces and edges), quasi-regular (if they are edge-transitive, but not transitive along faces) or semi-regular (if they are not transitive either along edges or along edges). For regular triangles ( p q 2), there are two regular mosaics with the Schlefli symbols { p , q } and { q , p }.

Content

Building Withoff

 
An example of the construction of Withhoff with rectangular triangles ( r = 2) and 7 generating points. The segments for active mirrors are colored red, yellow and blue, the nodes opposite the mirrors are marked in the same color.

There are an infinite number of homogeneous mosaics based on Schwartz triangles ( p q r ), where 1 / p + 1 / q + 1 / r <1, ​​where p , q , r are the orders of reflective symmetry at the three vertices of the fundamental triangle - the symmetry group is hyperbolic group of a triangle .

Each symmetry family contains 7 homogeneous mosaics defined or the Coxeter – Dynkin diagram , 7 combinations of three active mirrors. The 8th mosaic represents the operation of , removing half of the vertices from the higher form of active mirrors.

Families with r = 2 contain regular hyperbolic mosaics defined by Coxeter groups , such as [7,3], [8,3], [9,3], ... [5,4], [6,4],. ...

Hyperbolic families with r = 3 and higher are defined by the symbols ( p q r ) and include (4 3 3), (5 3 3), (6 3 3) ... (4 4 3), (5 4 3),. .. (4 4 4) ....

Hyperbolic families ( p q r ) define compact homogeneous hyperbolic mosaics. In the limit, any of the numbers p , q or r can be replaced by the symbol ∞, which gives a paracompact hyperbolic triangle and creates homogeneous mosaics that have either infinite faces (called apeirogons or infinite angles) that converge to one imaginary point, or infinite vertex figures with an infinite number edges coming from one imaginary point.

It is possible to construct additional families of symmetries from fundamental domains that are not triangular.

Some families of homogeneous mosaics are shown below (using the Poincare model for the hyperbolic plane). Three of them - (7 3 2), (5 4 2) and (4 3 3) - and no others, are minimal in the sense that if any of the defining numbers is replaced with a smaller integer value, we get either a Euclidean or a spherical mosaic rather than hyperbolic. And vice versa, any of the numbers can be increased (even replaced by infinity) to get another hyperbolic pattern.

Each homogeneous mosaic forms a dual homogeneous mosaic , and many of them are listed below as well.

Rectangular fundamental triangles

There are infinitely many families of triangle groups ( p q 2). The article shows the correct mosaics up to p , q = 8 and homogeneous mosaics of 12 families: (7 3 2), (8 3 2), (5 4 2), (6 4 2), (7 4 2), (8 4 2), (5 5 2), (6 5 2) (6 6 2), (7 7 2), (8 6 2) and (8 8 2).

Correct Hyperbolic Mosaics

The simplest set of hyperbolic mosaics are regular mosaics { p , q }. The regular mosaic { p , q } has the dual mosaic { q , p } (the diagonals of the table are symmetric). Self-dual mosaics {3,3} , {4,4} , , etc. are located on the diagonal of the table.

(7 3 2)

The triangle group , the Coxeter group [7.3], the (* 732) contain these homogeneous mosaics.

(8 3 2)

The triangle group , the Coxeter group [8.3], the (* 832) contain these homogeneous mosaics.

(5 4 2)

The triangle group , the Coxeter group [5,4], the (* 542) contain these homogeneous mosaics.

(6 4 2)

The triangle group , the Coxeter group [6,4], the (* 642) contain these homogeneous mosaics. Since all elements are even, of the two dual homogeneous mosaics, one represents the fundamental region of mirror symmetry: * 3333, * 662, * 3232, * 443, * 222222, * 3222 and * 642, respectively. All seven mosaics can be alternated and there are dual ones for the resulting mosaics.

(7 4 2)

The triangle group , the Coxeter group [7.4], the (* 742) contain these homogeneous mosaics.

(8 4 2)

The triangle group , the Coxeter group [8.4], the (* 842) contain these homogeneous mosaics. Since all elements are even, of the two dual homogeneous mosaics, one represents the fundamental region of mirror symmetry: * 4444, * 882, * 4242, * 444, * 22222222, * 4222 and * 842, respectively. All seven mosaics can be alternated and there are dual ones for the resulting mosaics.

(5 5 2)

The triangle group , the Coxeter group [5.5], the (* 552) contain these homogeneous mosaics.

(6 5 2)

The triangle group , the Coxeter group [6.5], the (* 652) contain these homogeneous mosaics.

(6 6 2)

The triangle group , the Coxeter group [6,6], the (* 662) contain these homogeneous mosaics.

(8 6 2)

Triangle group , Coxeter group [8,6],

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