Order-4 heptagonal tiling

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Order-4 heptagonal tiling
Order-4 heptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 74
Schläfli symbol {7,4}
r{7,7}
Wythoff symbol 4 | 7 2
2 | 7 7
Coxeter diagram CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 7.pngCDel node 1.pngCDel 7.pngCDel node.png
Symmetry group [7,4], (*742)
[7,7], (*772)
Dual Order-7 square tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 heptagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,4}.

Symmetry

This tiling represents a hyperbolic kaleidoscope of 7 mirrors meeting as edges of a regular heptagon. This symmetry by orbifold notation is called *2222222 with 7 order-2 mirror intersections. In Coxeter notation can be represented as [1+,7,1+,4], removing two of three mirrors (passing through the heptagon center) in the [7,4] symmetry.

The kaleidoscopic domains can be seen as bicolored heptagons, representing mirror images of the fundamental domain. This coloring represents the uniform tiling t1{7,7} and as a quasiregular tiling is called a heptaheptagonal tiling.

Uniform tiling 77-t1.png

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular tilings with heptagonal faces, starting with the heptagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram CDel node 1.pngCDel 7.pngCDel node.pngCDel n.pngCDel node.png, progressing to infinity.

Uniform tiling 73-t0.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 74-t0.png
{7,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 75-t0.png
{7,5}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 5.pngCDel node.png
Uniform tiling 76-t0.png
{7,6}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 77-t2.png
{7,7}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 7.pngCDel node.png

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram CDel node 1.pngCDel n.pngCDel node.pngCDel 4.pngCDel node.png, with n progressing to infinity.

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
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See also

External links

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