List of convex uniform tilings

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An example uniform tiling in the Archeological Museum of Seville, Sevilla, Spain: rhombitrihexagonal tiling

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

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 calls the uniform duals Archimedean tilings, in parallel to the Archimedean 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 nonconvex tilings, using star polygons, and reverse orientation vertex configurations.

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 Shubnikov, Alekseĭ Vasilʹevich.[3] John Conway calls 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.

Eleven planigons
Triangles Quadrilaterals Pentagons Hexagon
Alchemy fire symbol.svg
V6.6.6
CDel node.pngCDel split1.pngCDel branch.png
Tiling face 4-8-8.svg
V4.8.8
CDel node.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Tiling face 4-6-12.svg
V4.6.12
CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
Tiling face 3-12-12.svg
V3.12.12
CDel 2.png
Regular quadrilateral.svg
V4.4.4.4
CDel labelinfin.pngCDel branch.pngCDel 2.pngCDel branch.pngCDel labelinfin.png
Tiling face 3-6-3-6.svg
V3.6.3.6
CDel 2.png
Tiling face 3-4-6-4.svg
V3.4.6.4
CDel 2.png
Tiling face 3-3-4-3-4.svg
V3.3.4.3.4
CDel 2.png
Tiling face 3-3-3-3-6.svg
V3.3.3.3.6
CDel 2.png
Tiling face 3-3-3-4-4.svg
V3.3.3.4.4
CDel 2.png
Hexagon.svg
V3.3.3.3.3.3
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:

  • (4,4,2), [math]\displaystyle{ {\tilde{BC}}_2 }[/math], [4,4] - Symmetry of the regular square tiling
    • [math]\displaystyle{ {\tilde{I}}_1^2 }[/math], [∞,2,∞]
  • (6,3,2), [math]\displaystyle{ {\tilde{G}}_2 }[/math], [6,3] - Symmetry of the regular hexagonal tiling and triangular tiling.
    • (3,3,3), [math]\displaystyle{ {\tilde{A}}_2 }[/math], [3[3]]

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)

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

Non-Wythoffian uniform tiling

Platonic and Archimedean tilings Vertex figure and dual face
Wythoff symbol(s)
Symmetry group
Coxeter diagram
Dual Laves tilings

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.png80px80px80px80px80px80px80pxUniform triangular tiling 111213.png
  2. Square tiling - 9 colorings: 7 wythoffian, 2 nonwythoffian
    • Square tiling uniform coloring 1.png80px80px80px80px80px80px80pxSquare tiling uniform coloring 9.png
  3. Hexagonal tiling - 3 colorings, all wythoffian
    • Uniform tiling 63-t0.png80pxUniform tiling 333-t012.png
  4. Trihexagonal tiling - 2 colorings, both wythoffian
    • Uniform polyhedron-63-t1.pngUniform tiling 333-t01.png
  5. Snub square tiling - 2 colorings, both alternated wythoffian
    • Uniform tiling 44-h01.pngUniform tiling 44-snub.png
  6. Truncated square tiling - 2 colorings, both wythoffian
    • Uniform tiling 44-t12.pngUniform tiling 44-t012.png
  7. Truncated hexagonal tiling - 1 coloring, wythoffian
    • Uniform tiling 63-t01.png
  8. Rhombitrihexagonal tiling - 1 coloring, wythoffian
    • Uniform tiling 63-t02.png
  9. Truncated trihexagonal tiling - 1 coloring, wythoffian
    • Uniform tiling 63-t012.svg
  10. Snub hexagonal tiling - 1 coloring, alternated wythoffian
    • Uniform tiling 63-snub.png
  11. Elongated triangular tiling - 3 coloring, nonwythoffian
    • Elongated triangular tiling 1.png

See also

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. The Symmetries of things, Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Euclidean Plane Tessellations, p. 288
  3. Encyclopaedia of Mathematics: Orbit - Rayleigh Equation edited by Michiel Hazewinkel, 1991
  4. Hazewinkel, Michiel, ed. (2001), "Planigon", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Planigon&oldid=31578 

Further reading

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (Chapter 19, Archimedean tilings, table 19.1, Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table).
  • H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, Uniform polyhedra, Phil. Trans. 1954, 246 A, 401–50.
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X.  (Section 2–3 Circle packings, plane tessellations, and networks, pp 34–40).
  • Asaro, et. al. "Uniform edge-c-colorings of the Archimedean Tilings", [2].
  • Grünbaum, Branko & Shepard, Geoffrey (Nov. 1977). "Tilings by Regular polygons", Vol. 50, No. 5.
  • Dale Seymour and Jill Britton, Introduction to Tessellations, 1989, ISBN 978-0866514613, pp. 50–57, 71-74

External links