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''C&R: Cundy & Rollet's notation'' <br/>
''GJ-H: Notation of GomJau-Hogg''
== Combinations of regular polygons that can meet at a vertex ==
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.
With 3 polygons at a vertex:
* 3.7.42 (cannot appear in any tiling of regular polygons)
* 3.8.24 (cannot appear in any tiling of regular polygons)
* 3.9.18 (cannot appear in any tiling of regular polygons)
* 3.10.15 (cannot appear in any tiling of regular polygons)
* 3.12<sup>2</sup> - semi-regular, [[truncated hexagonal tiling]]
* 4.5.20 (cannot appear in any tiling of regular polygons)
* 4.6.12 - semi-regular, [[truncated trihexagonal tiling]]
* 4.8<sup>2</sup> - semi-regular, [[truncated square tiling]]
* 5<sup>2</sup>.10 (cannot appear in any tiling of regular polygons)
* 6<sup>3</sup> - regular, [[hexagonal tiling]]
Below are diagrams of such vertices:
<gallery>
File:Regular polygons meeting at vertex 3 3 7 42.svg|3.7.42
File:Regular polygons meeting at vertex 3 3 8 24.svg|3.8.24
File:Regular polygons meeting at vertex 3 3 9 18.svg|3.9.18
File:Regular polygons meeting at vertex 3 3 10 15.svg|3.10.15
File:Regular polygons meeting at vertex 3 3 12 12.svg|3.12.12
File:Regular polygons meeting at vertex 3 4 5 20.svg|4.5.20
File:Regular polygons meeting at vertex 3 4 6 12.svg|4.6.12
File:Regular polygons meeting at vertex 3 4 8 8.svg|4.8.8
File:Regular polygons meeting at vertex 3 5 5 10.svg|5.5.10
File:Regular polygons meeting at vertex 3 6 6 6.svg|6.6.6
</gallery>
With 4 polygons at a vertex:
* 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>
* 3.4.3.12 - not uniform, has two different types of vertices 3.4.3.12 and 3.3.4.3.4
* 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.
* 3.6.3.6 - semi-regular, [[trihexagonal tiling]]
* 4<sup>4</sup> - regular, [[square tiling]]
* 3.4<sup>2</sup>.6 - not uniform, has vertices 3.4<sup>2</sup>.6 and 3.6.3.6.
* 3.4.6.4 - semi-regular, [[rhombitrihexagonal tiling]]
Below are diagrams of such vertices:
<gallery>
File:Regular polygons meeting at vertex 4 3 3 4 12.svg|3.3.4.12
File:Regular polygons meeting at vertex 4 3 4 3 12.svg|3.4.3.12
File:Regular polygons meeting at vertex 4 3 3 6 6.svg|3.3.6.6
File:Regular polygons meeting at vertex 4 3 6 3 6.svg|3.6.3.6
File:Regular polygons meeting at vertex 4 4 4 4 4.svg|4.4.4.4
File:Regular polygons meeting at vertex 4 3 4 4 6.svg|3.4.4.6
File:Regular polygons meeting at vertex 4 3 4 6 4.svg|3.4.6.4
</gallery>
With 5 polygons at a vertex:
* 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.
* 3<sup>3</sup>.4<sup>2</sup> - semi-regular, [[Elongated triangular tiling]]
* 3<sup>2</sup>.4.3.4 - semi-regular, [[Snub square tiling]]
Below are diagrams of such vertices:
<gallery>
File:Regular polygons meeting at vertex 5 3 3 3 3 6.svg|3.3.3.3.6
File:Regular polygons meeting at vertex 5 3 3 3 4 4.svg|3.3.3.4.4
File:Regular polygons meeting at vertex 5 3 3 4 3 4.svg|3.3.4.3.4
</gallery>
With 6 polygons at a vertex:
* 3<sup>6</sup> - regular, [[Triangular tiling]]
Below is a diagram of such a vertex:
<gallery>
File:Regular polygons meeting at vertex 6 3 3 3 3 3 3.svg|3.3.3.3.3.3
</gallery>
== Archimedean, uniform or semiregular tilings ==<!-- This section is linked from [[Archimedean tiling]] -->
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