Order-4 octagonal tiling

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Order-4 octagonal tiling
Order-4 octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 84
Schläfli symbol {8,4}
r{8,8}
Wythoff symbol 4 | 8 2
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png or CDel node.pngCDel split1-88.pngCDel nodes 11.png
Symmetry group [8,4], (*842)
[8,8], (*882)
Dual Order-8 square tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}, its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.

Uniform constructions[edit]

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.

Four uniform constructions of 8.8.8.8
Uniform
Coloring
H2 tiling 248-1.png H2 tiling 288-2.png H2 tiling 488-5.png H2 tiling 488-5-4color.png
Symmetry [8,4]
(*842)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[8,8]
(*882)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel node c2.pngCDel 8.pngCDel node c1.pngCDel 8.pngCDel node c2.png
[(8,4,8)] = [8,8,1+]
(*884)
CDel node c2.pngCDel 8.pngCDel node c1.pngCDel 8.pngCDel node h0.png = CDel node c2.pngCDel split1-88.pngCDel branch c1.pngCDel label4.png

CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel label4.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel nodeab c2.png

[1+,8,8,1+]
(*4444)
CDel node c1.pngCDel 8.pngCDel node g.pngCDel 4sg.pngCDel node g.png =
CDel label4.pngCDel branch c1.pngCDel 4a4b-cross.pngCDel branch c1.pngCDel label4.png
Symbol {8,4} r{8,8} r(8,4,8) = r{8,8}​12 r{8,4}​18 = r{8,8}​14
Coxeter
diagram
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node h0.png = CDel node.pngCDel split1-88.pngCDel branch 11.pngCDel label4.png

CDel node 1.pngCDel 8.pngCDel node h0.pngCDel 4.pngCDel node.png = CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png

CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node h0.png = CDel labelh.pngCDel node.pngCDel split1-88.pngCDel branch 11.pngCDel label4.png =
CDel node 1.pngCDel 8.pngCDel node g.pngCDel 4sg.pngCDel node g.png =CDel label4.pngCDel branch 11.pngCDel 4a4b-cross.pngCDel branch 11.pngCDel label4.png

Symmetry[edit]

This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon; this symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.

H2chess 248e.png
*444
H2chess 248d.png
*4222
842 symmetry mirrors.png
*832

The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain; this coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.

Uniform tiling 88-t1.png H2chess 248c.png

Related polyhedra and tiling[edit]

This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel n.pngCDel node.png, progressing to infinity.

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.

Uniform polyhedron-34-t0.png
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0.png
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 54-t0.png
{5,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 64-t0.png
{6,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 74-t0.png
{7,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 84-t0.png
{8,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
... H2 tiling 24i-1.png
{∞,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png

See also[edit]

References[edit]

  • 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)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.

External links[edit]