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| [[Plane (mathematics)|Plane]] '''[[Tessellation|tilings]] by [[regular polygon]]s''' have been widely used since antiquity. The first systematic mathematical treatment was that of [[Johannes Kepler|Kepler]] in ''[[Harmonices Mundi]]''.
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| == Regular tilings ==
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| Following [[Branko Grünbaum|Grünbaum]] and Shephard (section 1.3), a tiling is said to be ''regular'' if the [[symmetry group]] of the tiling [[group action|acts transitively]] on the ''flags'' of the tiling, where a flag is a triple consisting of a mutually incident [[Vertex (geometry)#Of_a_plane_tiling|vertex]], edge and tile of the tiling. This means that for every pair of flags there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an [[edge-to-edge tiling]] by [[congruence (geometry)|congruent]] regular polygons. There must be six [[equilateral triangle]]s, four [[square (geometry)|square]]s or three regular [[hexagon]]s at a vertex, yielding the three ''[[Tessellation#Regular_and_semi-regular_tessellations|regular tessellations]]''.
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| {| class="wikitable"
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| |- align=center
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| |[[Image:Tiling_Regular_3-6_Triangular.svg|250px]]<br>3<sup>6</sup><br>[[Triangular tiling]]
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| |[[Image:Tiling_Regular_4-4_Square.svg |250px]]<br>4<sup>4</sup><br>[[Square tiling]]
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| |[[Image:Tiling_Regular_6-3_Hexagonal.svg |250px]]<br>6<sup>3</sup><br>[[Hexagonal tiling]]
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| |}
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| == Archimedean, uniform or semiregular tilings ==<!-- This section is linked from [[Archimedean tiling]] -->
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| [[Vertex-transitive|Vertex-transitivity]] means that for every pair of vertices there is a [[symmetry operation]] mapping the first vertex to the second.
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| If the requirement of flag-transitivity is relaxed to one of vertex-transitivity, while the condition that the tiling is edge-to-edge is kept, there are eight additional tilings possible, known as ''Archimedean'', ''[[uniform tessellation|uniform]]'' or ''semiregular'' tilings. Note that there are two [[mirror image]] (enantiomorphic or [[Chirality (mathematics)|chiral]]) forms of 3<sup>4</sup>.6 (snub hexagonal) tiling, both of which are shown in the following table. All other regular and semiregular tilings are achiral.
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| {| class="wikitable"
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| |- align=center
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| |[[Image:Tiling_Semiregular_3-3-3-3-6_Snub_Hexagonal.svg|250px]]<BR>3<sup>4</sup>.6<BR>[[Snub hexagonal tiling]]
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| |[[Image:Tiling_Semiregular_3-3-3-3-6_Snub_Hexagonal_Mirror.svg|250px]]<BR>3<sup>4</sup>.6<BR>[[Snub hexagonal tiling]] [[Reflection (mathematics)|reflection]]
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| |[[Image:Tiling_Semiregular_3-6-3-6_Trihexagonal.svg|250px]]<BR>3.6.3.6<BR>[[Trihexagonal tiling]]
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| |- align=center
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| |[[Image:Tiling_Semiregular_3-3-3-4-4_Elongated_Triangular.svg|250px]]<BR>3<sup>3</sup>.4<sup>2</sup><BR>[[Elongated triangular tiling]]
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| |[[Image:Tiling_Semiregular_3-3-4-3-4_Snub_Square.svg|250px]]<BR>3<sup>2</sup>.4.3.4<BR>[[Snub square tiling]]
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| |[[Image:Tiling_Semiregular_3-4-6-4_Small_Rhombitrihexagonal.svg|250px]]<BR>3.4.6.4<BR>[[Rhombitrihexagonal tiling]]
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| |- align=center
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| |[[Image:Tiling_Semiregular_4-8-8_Truncated_Square.svg|250px]]<BR>4.8<sup>2</sup><BR>[[Truncated square tiling]]
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| |[[Image:Tiling_Semiregular_3-12-12_Truncated_Hexagonal.svg|250px]]<BR>3.12<sup>2</sup><BR>[[Truncated hexagonal tiling]]
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| |[[Image:Tiling_Semiregular_4-6-12_Great_Rhombitrihexagonal.svg|250px]]<BR>4.6.12<BR>[[Truncated trihexagonal tiling]]
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| |}
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| Grünbaum and Shephard distinguish the description of these tilings as ''Archimedean'' as referring only to the local property of the arrangement of tiles around each vertex being the same, and that as ''uniform'' as referring to the global property of vertex-transitivity. Though these yield the same set of tilings in the plane, in other spaces there are Archimedean tilings which are not uniform.
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| == Combinations of regular polygons that can meet at a vertex ==
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| For Euclidean tilings, the [[internal angle]]s of the polygons meeting at a vertex must add to 360 degrees. A regular <math>n\,\!</math>-gon has internal angle <math>\left(1-\frac{2}{n}\right)180</math> degrees. There are seventeen combinations of regular polygons whose internal angles add up to 360 degrees, each being referred to as a ''species'' of vertex; in four cases there are two distinct cyclic orders of the polygons, yielding twenty-one ''types'' of vertex. Only eleven of these can occur in a uniform tiling of regular polygons. In particular, if three polygons meet at a vertex and one has an odd number of sides, the other two polygons must be the same. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd.
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| With 3 polygons at a vertex:
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| * 3.7.42 (cannot appear in any tiling of regular polygons)
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| * 3.8.24 (cannot appear in any tiling of regular polygons)
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| * 3.9.18 (cannot appear in any tiling of regular polygons)
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| * 3.10.15 (cannot appear in any tiling of regular polygons)
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| * 3.12<sup>2</sup> - semi-regular, [[truncated hexagonal tiling]]
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| * 4.5.20 (cannot appear in any tiling of regular polygons)
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| * 4.6.12 - semi-regular, [[truncated trihexagonal tiling]]
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| * 4.8<sup>2</sup> - semi-regular, [[truncated square tiling]]
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| * 5<sup>2</sup>.10 (cannot appear in any tiling of regular polygons)
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| * 6<sup>3</sup> - regular, [[hexagonal tiling]]
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| Below are diagrams of such vertices:
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| <gallery>
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| File:Regular polygons meeting at vertex 3 3 7 42.svg|3.7.42
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| File:Regular polygons meeting at vertex 3 3 8 24.svg|3.8.24
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| File:Regular polygons meeting at vertex 3 3 9 18.svg|3.9.18
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| File:Regular polygons meeting at vertex 3 3 10 15.svg|3.10.15
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| File:Regular polygons meeting at vertex 3 3 12 12.svg|3.12.12
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| File:Regular polygons meeting at vertex 3 4 5 20.svg|4.5.20
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| File:Regular polygons meeting at vertex 3 4 6 12.svg|4.6.12
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| File:Regular polygons meeting at vertex 3 4 8 8.svg|4.8.8
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| File:Regular polygons meeting at vertex 3 5 5 10.svg|5.5.10
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| File:Regular polygons meeting at vertex 3 6 6 6.svg|6.6.6
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| </gallery>
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| With 4 polygons at a vertex:
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| * 3<sup>2</sup>.4.12 - not [[Uniform polyhedron|uniform]], has two different types of vertices 3<sup>2</sup>.4.12 and 3<sup>6</sup>
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| * 3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4
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| * 3<sup>2</sup>.6<sup>2</sup> - not uniform, occurs in two patterns with vertices 3<sup>2</sup>.6<sup>2</sup>/3<sup>6</sup> and 3<sup>2</sup>.6<sup>2</sup>/3.6.3.6.
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| * 3.6.3.6 - semi-regular, [[trihexagonal tiling]]
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| * 4<sup>4</sup> - regular, [[square tiling]]
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| * 3.4<sup>2</sup>.6 - not uniform, has vertices 3.4<sup>2</sup>.6 and 3.6.3.6.
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| * 3.4.6.4 - semi-regular, [[rhombitrihexagonal tiling]]
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| Below are diagrams of such vertices:
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| <gallery>
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| File:Regular polygons meeting at vertex 4 3 3 4 12.svg|3.3.4.12
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| File:Regular polygons meeting at vertex 4 3 4 3 12.svg|3.4.3.12
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| File:Regular polygons meeting at vertex 4 3 3 6 6.svg|3.3.6.6
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| File:Regular polygons meeting at vertex 4 3 6 3 6.svg|3.6.3.6
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| File:Regular polygons meeting at vertex 4 4 4 4 4.svg|4.4.4.4
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| File:Regular polygons meeting at vertex 4 3 4 4 6.svg|3.4.4.6
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| File:Regular polygons meeting at vertex 4 3 4 6 4.svg|3.4.6.4
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| </gallery>
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| With 5 polygons at a vertex:
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| * 3<sup>4</sup>.6 - semi-regular, [[Snub hexagonal tiling]], comes in two [[chirality (mathematics)|enantiomorphic]] forms. The vertex figures of the two enantiomorphs are the same, but the resulting tilings are different.
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| * 3<sup>3</sup>.4<sup>2</sup> - semi-regular, [[Elongated triangular tiling]]
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| * 3<sup>2</sup>.4.3.4 - semi-regular, [[Snub square tiling]]
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| Below are diagrams of such vertices:
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| <gallery>
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| File:Regular polygons meeting at vertex 5 3 3 3 3 6.svg|3.3.3.3.6
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| File:Regular polygons meeting at vertex 5 3 3 3 4 4.svg|3.3.3.4.4
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| File:Regular polygons meeting at vertex 5 3 3 4 3 4.svg|3.3.4.3.4
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| </gallery>
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| With 6 polygons at a vertex:
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| * 3<sup>6</sup> - regular, [[Triangular tiling]]
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| Below is a diagram of such a vertex:
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| <gallery>
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| File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|3.3.3.3.3.3
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| </gallery>
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| == Other edge-to-edge tilings ==
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| Any number of non-uniform (sometimes called demiregular) edge-to-edge tilings by regular polygons may be drawn. Here are four examples:
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| {| class=wikitable
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| |- align=center
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| ||[[File:dem3366bc.png|150px]]<br/>3<sup>2</sup>.6<sup>2</sup> and 3<sup>6</sup>
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| ||[[File:dem3366rbc.gif|150px]]<br/>3<sup>2</sup>.6<sup>2</sup> and 3.6.3.6
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| ||[[File:Academ Periodic tiling by dodecagons and other convex regular polygons.svg|150px]]<br/>3<sup>2</sup>.4.12 and 3<sup>6</sup>
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| ||[[File:Academ Periodic tiling by hexagons and other regular polygons.svg|150px]]<br/>3.4<sup>2</sup>.6 and 3.6.3.6
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| |}
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| Such periodic tilings may be classified by the number of [[Group action#Orbits and stabilizers|orbits]] of vertices, edges and tiles. If there are <math>n</math> orbits of vertices, a tiling is known as <math>n</math>-uniform or <math>n</math>-isogonal; if there are <math>n</math> orbits of tiles, as <math>n</math>-isohedral; if there are <math>n</math> orbits of edges, as <math>n</math>-isotoxal. The examples above are four of the twenty 2-uniform tilings. {{harvtxt|Chavey|1989}} lists all those edge-to-edge tilings by regular polygons which are at most 3-uniform, 3-isohedral or 3-isotoxal.
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| == Tilings that are not edge-to-edge ==
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| {| class=wikitable
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| |-
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| |+ [[Wallpaper group|Periodic ]] tilings with '''[[Regular polygon#Regular_star_polygons|star polygons]]'''
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| |- align=center
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| |[[File:Academ_Squares_and_regular_star_octagons_in_a_periodic_tiling.svg|240px]]<br/>[[Octagram]]s and squares
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| |[[File:Academ_Periodic_tiling_by_star_dodecagons_and_equilateral_triangles.svg|240px]]<br/>[[Dodecagram]]s and equilateral triangles
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| |-
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| | colspan="2"|
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| Two tilings by regular polygons of two kinds. Two elements of the same kind are [[Congruence (geometry)|congruent]].<br/>Every element which is [[Convex (set)|not convex]] is a [[stellation]] of a regular polygon with stripes.
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| |}
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| {{clear}}
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| {| class=wikitable width=480
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| |- align=center
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| |+ Periodic tilings by '''convex''' polygons
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| |-
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| |[[File:Academ_Regular_hexagons_and_equilateral_triangles_in_periodic_tiling.svg|240px]]<br/>Six triangles surround every hexagon.<br/>No pair of triangles has a common boundary, if their sides have a length lower than<br/>the side length of hexagons.
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| |[[File:A_Pythagorean_tiling_View_7.svg|240px]]<br/>[[Pythagorean tiling|A tiling by squares]] of two different sizes, manifestly periodic by overlaying an appropriate grid. The present grid divides every large tile into four congruent polygons: [[Dissection puzzle|possible puzzle]] pieces to prove [[Pythagorean theorem|the Pythagorean theorem]].
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| |}
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| Regular polygons can also form plane tilings that are not edge-to-edge. Such tilings may also be known as uniform if they are vertex-transitive; there are eight families of such uniform tilings, each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles.
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| == The hyperbolic plane ==
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| {{main|Uniform tilings in hyperbolic plane}}
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| These [[tessellations]] are also related to regular and semiregular polyhedra and tessellations of the [[hyperbolic geometry|hyperbolic plane]]. Semiregular polyhedra are made from regular polygon faces, but their angles at a point add to less than 360 degrees. Regular polygons in hyperbolic geometry have angles smaller than they do in the plane. In both these cases, that the arrangement of polygons is the same at each vertex does not mean that the polyhedron or tiling is vertex-transitive.
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| Some regular tilings of the hyperbolic plane (Using Poincaré disc model projection)
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| {|
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| |align=center|[[Image:Hyperspace_tiling_4-5.png|150px]]
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| |align=center|[[Image:Hyperspace_tiling_5-4.png|150px]]
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| |align=center|[[Image:Hyperbolic tiling 3-7.png|150px]]
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| |align=center|[[Image:Hyperbolic tiling 7-3.png|150px]]
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| |align=center|[[Image:Hyperbolic tiling truncated 3-7.png|150px]]
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| |-
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| |align=center|[[Image:Hyperbolic tiling rectified 3-7.png|150px]]
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| |align=center|[[Image:Hyperbolic tiling truncated 7-3.png|150px]]
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| |align=center|[[Image:Hyperbolic_tiling_cantellated_3-7.png|150px]]
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| |align=center|[[Image:Hyperbolic tiling omnitruncated 3-7.png|150px]]
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| |align=center|[[Image:Hyperbolic tiling snub 3-7.png|150px]]
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| |}
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| == See also ==
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| <div style="-moz-column-count:2; column-count:2;">
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| * [[List of uniform tilings]]
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| * [[Wythoff symbol]]
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| * [[Tessellation]]
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| * [[Wallpaper group]]
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| * [[Regular polyhedron]] (the [[Platonic solid]]s)
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| * [[Semiregular polyhedron]] (including the [[Archimedean solid]]s)
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| * [[Hyperbolic geometry]]
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| * [[Penrose tiling]]
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| * [[Tiling with rectangles]]
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| </div>
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| == References ==
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| * {{cite book | author=[[Branko Grünbaum|Grünbaum, Branko]]; [[G.C. Shephard|Shephard, G. C.]] | title=Tilings and Patterns | publisher=W. H. Freeman and Company | year=1987 | isbn=0-7167-1193-1}}
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| * {{cite journal | first=D. |last=Chavey | title=Tilings by Regular Polygons—II: A Catalog of Tilings | url=https://www.beloit.edu/computerscience/faculty/chavey/catalog/ | journal=Computers & Mathematics with Applications | year=1989 | volume=17 | pages=147–165 | doi=10.1016/0898-1221(89)90156-9|ref=harv}}
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| * [[Duncan MacLaren Young Sommerville|D. M. Y. Sommerville]], ''An Introduction to the Geometry of '''n''' Dimensions.'' New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
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| == External links ==
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| Euclidean and general tiling links:
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| * {{cite web
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| | author = Dutch, Steve
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| | title = Uniform Tilings
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| | url = http://www.uwgb.edu/dutchs/symmetry/uniftil.htm
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| | accessdate = 2006-09-09}}
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| * {{cite web
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| | author = Mitchell, K
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| | title = Semi-Regular Tilings
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| | url = http://people.hws.edu/mitchell/tilings/Part1.html
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| | accessdate = 2006-09-09}}
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| * {{MathWorld | urlname=SemiregularTessellation | title=Semiregular tessellation}}
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| ** {{MathWorld | urlname=DemiregularTessellation | title=Demiregular tessellation}}
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| * {{cite web|title=Tilings of semi-regular polygons|url=http://library.thinkquest.org/16661/of.regular.polygons/}}
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| Hyperbolic tiling links:
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| * {{cite web
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| | author = Eppstein, David
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| | authorlink = David Eppstein
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| | title = The Geometry Junkyard: Hyperbolic Tiling
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| | url = http://www.ics.uci.edu/~eppstein/junkyard/hypertile.html
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| | accessdate = 2006-09-09}}
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| * {{cite web
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| | author = Hatch, Don
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| | title = Hyperbolic Planar Tessellations
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| | url = http://www.plunk.org/~hatch/HyperbolicTesselations/
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| | accessdate = 2012-11-28}}
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| * {{cite web
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| | author = Joyce, David
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| | title = Hyperbolic Tessellations
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| | url = http://aleph0.clarku.edu/~djoyce/poincare/poincare.html
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| | accessdate = 2006-09-09}}
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| [[Category:Euclidean plane geometry]]
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| [[Category:Tessellation]]
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