List of Euclidean uniform tilings

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An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling Semi-regular-floor-3464.JPG
An example of uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling
Regular tilings and their duals drawn by Max Bruckner in Vielecke und Vielflache (1900) Bruckner Vielflache Fig. 96b - 98.jpg
Regular tilings and their duals drawn by Max Brückner in Vielecke und Vielflache (1900)

This table shows the 11 convex uniform tilings (regular and semiregular) of the Euclidean plane, and their dual tilings.

Contents

There are three regular and eight semiregular tilings in the plane. The semiregular tilings form new tilings from their duals, each made from one type of irregular face.

John Conway called these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.

Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex.

These 11 uniform tilings have 32 different uniform colorings . A uniform coloring allows identical sided polygons at a vertex to be colored differently, while still maintaining vertex-uniformity and transformational congruence between vertices. (Note: Some of the tiling images shown below are not color-uniform.)

In addition to the 11 convex uniform tilings, there are also 14 known nonconvex tilings, using star polygons, and reverse orientation vertex configurations. A further 28 uniform tilings are known using apeirogons. If zigzags are also allowed, there are 23 more known uniform tilings and 10 more known families depending on a parameter: in 8 cases the parameter is continuous, and in the other 2 it is discrete. The set is not known to be complete.

Laves tilings

In the 1987 book, Tilings and patterns , Branko Grünbaum calls the vertex-uniform tilings Archimedean, in parallel to the Archimedean solids. Their dual tilings are called Laves tilings in honor of crystallographer Fritz Laves. [1] [2] They're also called Shubnikov–Laves tilings after Aleksei Shubnikov. [3] John Conway called the uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra.

The Laves tilings have vertices at the centers of the regular polygons, and edges connecting centers of regular polygons that share an edge. The tiles of the Laves tilings are called planigons . This includes the 3 regular tiles (triangle, square and hexagon) and 8 irregular ones. [4] Each vertex has edges evenly spaced around it. Three dimensional analogues of the planigons are called stereohedrons.

These dual tilings are listed by their face configuration, the number of faces at each vertex of a face. For example V4.8.8 means isosceles triangle tiles with one corner with four triangles, and two corners containing eight triangles. The orientations of the vertex planigons (up to D12) are consistent with the vertex diagrams in the below sections.

Eleven planigons
TrianglesQuadrilateralsPentagonsHexagon
Tiling 6 dual face.svg
V63
CDel node.pngCDel split1.pngCDel branch.png
Tiling truncated 4a dual face.svg
V4.82
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Tiling great rhombi 3-6 dual face.svg
V4.6.12
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Tiling truncated 6 dual face.svg
V3.122
CDel 2.png
Tiling 4a dual face.svg
V44
CDel labelinfin.pngCDel branch.pngCDel 2.pngCDel branch.pngCDel labelinfin.png
Tiling 3-6 dual face.svg
V(3.6)2
CDel 2.png
Tiling small rhombi 3-6 dual face.svg
V3.4.6.4
CDel 2.png
Tiling snub 4-4 left dual face.svg
V32.4.3.4
CDel 2.png
Tiling snub 3-6 left dual face.svg
V34.6
CDel 2.png
Tiling elongated 3 dual face.svg
V33.42
CDel 2.png
Tiling 3 dual face.svg
V36
CDel 2.png

Convex uniform tilings of the Euclidean plane

All reflectional forms can be made by Wythoff constructions, represented by Wythoff symbols, or Coxeter-Dynkin diagrams, each operating upon one of three Schwarz triangle (4,4,2), (6,3,2), or (3,3,3), with symmetry represented by Coxeter groups: [4,4], [6,3], or [3[3]]. Alternated forms such as the snub can also be represented by special markups within each system. Only one uniform tiling can't be constructed by a Wythoff process, but can be made by an elongation of the triangular tiling. An orthogonal mirror construction [,2,] also exists, seen as two sets of parallel mirrors making a rectangular fundamental domain. If the domain is square, this symmetry can be doubled by a diagonal mirror into the [4,4] family.

Families:

The [4,4] group family

Uniform tilings
(Platonic and Archimedean)
Vertex figure and dual face
Wythoff symbol(s)
Symmetry group
Coxeter diagram(s)
Dual-uniform tilings
(called Laves or Catalan tilings)
Tiling 4a simple.svg
Square tiling (quadrille)
Tiling 4a vertfig.svg Tiling 4a dual face.svg
4.4.4.4 (or 44)
4 | 2 4
p4m, [4,4], (*442)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png
Tiling 4b simple.svg
self-dual (quadrille)
Tiling truncated 4a simple.svg
Truncated square tiling (truncated quadrille)
Tiling truncated 4a vertfig.svg Tiling truncated 4a dual face.svg
4.8.8
2 | 4 4
4 4 2 |
p4m, [4,4], (*442)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node 1.png or CDel node 1.pngCDel split1-44.pngCDel nodes 11.png
Tiling truncated 4a dual simple.svg
Tetrakis square tiling (kisquadrille)
Tiling snub 4-4 left simple.svg
Snub square tiling (snub quadrille)
Tiling snub 4-4 left vertfig.svg Tiling snub 4-4 left dual face.svg
3.3.4.3.4
| 4 4 2
p4g, [4+,4], (4*2)
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png or CDel node h.pngCDel split1-44.pngCDel nodes hh.png
Tiling snub 4-4 left dual simple.svg
Cairo pentagonal tiling (4-fold pentille)

The [6,3] group family

Platonic and Archimedean tilings Vertex figure and dual face
Wythoff symbol(s)
Symmetry group
Coxeter diagram(s)
Dual Laves tilings
Tiling 6 simple.svg
Hexagonal tiling (hextille)
Tiling 6 vertfig.svg Tiling 6 dual face.svg
6.6.6 (or 63)
3 | 6 2
2 6 | 3
3 3 3 |
p6m, [6,3], (*632)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel split1.pngCDel branch 11.png
Tiling 3 simple.svg
Triangular tiling (deltille)
Tiling 3-6 simple.svg
Trihexagonal tiling (hexadeltille)
Tiling 3-6 vertfig.svg Tiling 3-6 dual face.svg
(3.6)2
2 | 6 3
3 3 | 3
p6m, [6,3], (*632)
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel branch 10ru.pngCDel split2.pngCDel node 1.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Tiling 3-6 dual simple.svg
Rhombille tiling (rhombille)
Tiling truncated 6 simple.svg
Truncated hexagonal tiling (truncated hextille)
Tiling truncated 6 vertfig.svg Tiling truncated 6 dual face.svg
3.12.12
2 3 | 6
p6m, [6,3], (*632)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
Tiling truncated 6 dual simple.svg
Triakis triangular tiling (kisdeltille)
Tiling 3 simple.svg
Triangular tiling (deltille)
Tiling 3 vertfig.svg Tiling 3 dual face.svg
3.3.3.3.3.3 (or 36)
6 | 3 2
3 | 3 3
| 3 3 3
p6m, [6,3], (*632)
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node 1.pngCDel split1.pngCDel branch.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel split1.pngCDel branch hh.png
Tiling 6 simple.svg
Hexagonal tiling (hextille)
Tiling small rhombi 3-6 simple.svg
Rhombitrihexagonal tiling (rhombihexadeltille)
Tiling small rhombi 3-6 vertfig.svg Tiling small rhombi 3-6 dual face.svg
3.4.6.4
3 | 6 2
p6m, [6,3], (*632)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Tiling small rhombi 3-6 dual simple.svg
Deltoidal trihexagonal tiling (tetrille)
Tiling great rhombi 3-6 simple.svg
Truncated trihexagonal tiling (truncated hexadeltille)
Tiling great rhombi 3-6 vertfig.svg Tiling great rhombi 3-6 dual face.svg
4.6.12
2 6 3 |
p6m, [6,3], (*632)
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Tiling great rhombi 3-6 dual simple.svg
Kisrhombille tiling (kisrhombille)
Tiling snub 3-6 left simple.svg
Snub trihexagonal tiling (snub hextille)
Tiling snub 3-6 left vertfig.svg Tiling snub 3-6 left dual face.svg
3.3.3.3.6
| 6 3 2
p6, [6,3]+, (632)
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Tiling snub 3-6 left dual simple.svg
Floret pentagonal tiling (6-fold pentille)

Non-Wythoffian uniform tiling

Platonic and Archimedean tilings Vertex figure and dual face
Wythoff symbol(s)
Symmetry group
Coxeter diagram
Dual Laves tilings
Tiling elongated 3 simple.svg
Elongated triangular tiling (isosnub quadrille)
Tiling elongated 3 vertfig.svg Tiling elongated 3 dual face.svg
3.3.3.4.4
2 | 2 (2 2)
cmm, [∞,2+,∞], (2*22)
CDel node.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png
CDel node h.pngCDel infin.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel infin.pngCDel node 1.png
Tiling elongated 3 dual simple.svg
Prismatic pentagonal tiling (iso(4-)pentille)

Uniform colorings

There are a total of 32 uniform colorings of the 11 uniform tilings:

  1. Triangular tiling – 9 uniform colorings, 4 wythoffian, 5 nonwythoffian
    • Uniform tiling 63-t2.svg   Uniform tiling 333-t1.svg   Uniform tiling 333-snub.png   Uniform tiling 63-h12.png   Uniform triangular tiling 111222.png   Uniform triangular tiling 112122.png   Uniform triangular tiling 111112.png   Uniform triangular tiling 111212.png   Uniform triangular tiling 111213.png  
  2. Square tiling – 9 colorings: 7 wythoffian, 2 nonwythoffian
    • Square tiling uniform coloring 1.svg   Square tiling uniform coloring 2.png   Square tiling uniform coloring 7.png   Square tiling uniform coloring 8.png   Square tiling uniform coloring 3.png   Square tiling uniform coloring 6.png   Square tiling uniform coloring 4.png   Square tiling uniform coloring 5.png   Square tiling uniform coloring 9.png  
  3. Hexagonal tiling – 3 colorings, all wythoffian
    • Uniform tiling 63-t0.svg   Uniform tiling 63-t12.svg   Uniform tiling 333-t012.svg  
  4. Trihexagonal tiling – 2 colorings, both wythoffian
    • Uniform tiling 63-t1.svg   Uniform tiling 333-t01.png  
  5. Snub square tiling – 2 colorings, both alternated wythoffian
    • Uniform tiling 44-h01.png   Uniform tiling 44-snub.svg  
  6. Truncated square tiling – 2 colorings, both wythoffian
    • Uniform tiling 44-t12.svg   Uniform tiling 44-t012.svg  
  7. Truncated hexagonal tiling – 1 coloring, wythoffian
    • Uniform tiling 63-t01.svg  
  8. Rhombitrihexagonal tiling – 1 coloring, wythoffian
    • Uniform tiling 63-t02.svg  
  9. Truncated trihexagonal tiling – 1 coloring, wythoffian
    • Uniform tiling 63-t012.svg  
  10. Snub hexagonal tiling – 1 coloring, alternated wythoffian
    • Uniform tiling 63-snub.svg  
  11. Elongated triangular tiling – 1 coloring, nonwythoffian
    • Elongated triangular tiling 1.png  

See also

Related Research Articles

<span class="mw-page-title-main">Convex uniform honeycomb</span> Spatial tiling of convex uniform polyhedra

In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.

<span class="mw-page-title-main">Euclidean tilings by convex regular polygons</span> Subdivision of the plane into polygons that are all regular

Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi.

<span class="mw-page-title-main">Uniform polyhedron</span> Isogonal polyhedron with regular faces

In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular, quasi-regular, or semi-regular. The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra.

<span class="mw-page-title-main">Triangular tiling</span> Regular tiling of the plane

In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.

<span class="mw-page-title-main">Truncated hexagonal tiling</span>

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.

<span class="mw-page-title-main">Truncated trihexagonal tiling</span>

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}.

<span class="mw-page-title-main">Truncated square tiling</span>

In geometry, the truncated square tiling is a semiregular tiling by regular polygons of the Euclidean plane with one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon. It has Schläfli symbol of t{4,4}.

<span class="mw-page-title-main">Rhombitrihexagonal tiling</span> Semiregular tiling of the Euclidean plane

In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.

<span class="mw-page-title-main">Snub trihexagonal tiling</span>

In geometry, the snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.

<span class="mw-page-title-main">Elongated triangular tiling</span>

In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.

<span class="mw-page-title-main">Cubic honeycomb</span> Only regular space-filling tessellation of the cube

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a cubille.

<span class="mw-page-title-main">Bitruncated cubic honeycomb</span>

The bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

<span class="mw-page-title-main">Quarter cubic honeycomb</span>

The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.

<span class="mw-page-title-main">Snub triheptagonal tiling</span>

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles and 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}.

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

In geometry, a uniform honeycomb or uniform tessellation or infinite uniform polytope, is a vertex-transitive honeycomb made from uniform polytope facets. All of its vertices are identical and there is the same combination and arrangement of faces at each vertex. Its dimension can be clarified as n-honeycomb for an n-dimensional honeycomb.

<span class="mw-page-title-main">Octagonal tiling</span>

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}.

<span class="mw-page-title-main">Snub trioctagonal tiling</span>

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}.

<span class="mw-page-title-main">Truncated octagonal tiling</span>

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}.

<span class="mw-page-title-main">Planigon</span> Convex polygon which can tile the plane by itself

In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself. In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons.

References

  1. Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns . W. H. Freeman and Company. pp.  59, 96. ISBN   0-7167-1193-1.
  2. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (April 18, 2008). "Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Euclidean Plane Tessellations". The Symmetries of Things. A K Peters / CRC Press. p. 288. ISBN   978-1-56881-220-5. Archived from the original on September 19, 2010.
  3. Encyclopaedia of Mathematics: Orbit - Rayleigh Equation, 1991
  4. Ivanov, A. B. (2001) [1994], "Planigon", Encyclopedia of Mathematics , EMS Press

Further reading