Order-4 hexagonal tiling

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Order-4 hexagonal tiling
Order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 64
Schläfli symbol {6,4}
Wythoff symbol 4 | 6 2
Coxeter diagram CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
Symmetry group [6,4], (*642)
Dual Order-6 square tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

Symmetry[edit]

This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *222222 with 6 order-2 mirror intersections. In Coxeter notation can be represented as [6*,4], removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry. Adding all 6 bisectors leads to full *642 symmetry.

642 symmetry zz0.png
*222222
642 symmetry a00.png
*443
642 symmetry 0a0.png
*3222
642 symmetry 000.png
*642

Uniform colorings[edit]

There are 7 distinct uniform colorings for the order-4 hexagonal tiling. They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions and Coxeter diagrams while three of them are undercolorings.

Uniform constructions of 6.6.6.6
1 color 2 colors 3 and 2 colors 4, 3 and 2 colors
Uniform
Coloring
H2 tiling 246-1.png
(1111)
H2 tiling 266-2.png
(1212)
H2 tiling 366-5.png
(1213)
H2 tiling 366-5 undercolor.png
(1113)
Order-4 hexagonal tiling nonsimplex domain.png
(1234)
Order-4 hexagonal tiling nonsimplex domain undercolor.png
(1123)
Order-4 hexagonal tiling row coloring.png
(1122)
Symmetry [6,4]
(*642)
CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[6,6]
(*662)
CDel node c1.pngCDel split1-66.pngCDel nodeab c2.png = CDel node c1.pngCDel 6.pngCDel node c2.pngCDel 4.pngCDel node h0.png
[(6,6,3)] = [6,6,1+]
(*663)
CDel node c2.pngCDel split1-66.pngCDel branch c1.png = CDel node c2.pngCDel 6.pngCDel node c1.pngCDel 6.pngCDel node h0.png
[1+,6,6,1+]
(*3333)
CDel branch c1.pngCDel 3a3b-cross.pngCDel branch c1.png = CDel node h0.pngCDel 6.pngCDel node c1.pngCDel 6.pngCDel node h0.png = CDel node c1.pngCDel 6.pngCDel node g.pngCDel 4sg.pngCDel node g.png
Symbol {6,4} r{6,6} = {6,4}1/2 r(6,3,6) = r{6,6}1/2 r{6,6}1/4
Coxeter
diagram
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel split1-66.pngCDel nodes.png = CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h0.png CDel node.pngCDel split1-66.pngCDel branch 11.png = CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node h0.png CDel branch 11.pngCDel 3a3b-cross.pngCDel branch 11.png = CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel 6.pngCDel node g.pngCDel 4sg.pngCDel node g.png

Related polyhedra and tiling[edit]

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

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]