Truncated order-4 octagonal tiling

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Truncated order-4 octagonal tiling
Truncated order-4 octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.16.16
Schläfli symbol t{8,4}
tr{8,8} or
Wythoff symbol 2 8 | 8
2 8 8 |
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png or CDel node 1.pngCDel split1-88.pngCDel nodes 11.png
Symmetry group [8,4], (*842)
[8,8], (*882)
Dual Order-8 tetrakis square tiling
Properties Vertex-transitive

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

Constructions[edit]

There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1+], gives [8,8], (*882).

Two uniform constructions of 4.8.4.8
Name Tetraoctagonal Truncated octaoctagonal
Image Uniform tiling 84-t01.png Uniform tiling 88-t012.png
Symmetry [8,4]
(*842)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[8,8] = [8,4,1+]
(*882)
CDel node c1.pngCDel split1-88.pngCDel nodeab c2.png = CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node h0.png
Symbol t{8,4} tr{8,8}
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png

Dual tiling[edit]

Order-8 tetrakis square tiling.png Hyperbolic domains 882.png
The dual tiling, Order-8 tetrakis square tiling has face configuration V4.16.16, and represents the fundamental domains of the [8,8] symmetry group.

Symmetry[edit]

Truncated order-4 octagonal tiling with *882 mirror lines

The dual of the tiling represents the fundamental domains of (*882) orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images unique mirrors are colored red, green, and blue, and alternatedly colored triangles show the location of gyration points. The [8+,8+], (44×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1+,8,1+,8,1+] (4444) is the commutator subgroup of [8,8].

One larger subgroup is constructed as [8,8*], removing the gyration points of (8*4), index 16 becomes (*44444444), and its direct subgroup [8,8*]+, index 32, (44444444).

The [8,8] symmetry can be doubled by a mirror bisecting the fundamental domain, and creating *884 symmetry.

Small index subgroups of [8,8] (*882)
Index 1 2 4
Diagram 882 symmetry 000.png 882 symmetry a00.png 882 symmetry 00a.png 882 symmetry 0a0.png 882 symmetry a0b.png 882 symmetry xxx.png
Coxeter [8,8]
CDel node c1.pngCDel 8.pngCDel node c3.pngCDel 8.pngCDel node c2.png
[1+,8,8]
CDel node h0.pngCDel 8.pngCDel node c3.pngCDel 8.pngCDel node c2.png = CDel label4.pngCDel branch c3.pngCDel split2-88.pngCDel node c2.png
[8,8,1+]
CDel node c1.pngCDel 8.pngCDel node c3.pngCDel 8.pngCDel node h0.png = CDel node c1.pngCDel split1-88.pngCDel branch c3.pngCDel label4.png
[8,1+,8]
CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node c2.png = CDel label4.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel branch c2.pngCDel label4.png
[1+,8,8,1+]
CDel node h0.pngCDel 8.pngCDel node c3.pngCDel 8.pngCDel node h0.png = CDel label4.pngCDel branch c3.pngCDel 4a4b-cross.pngCDel branch c3.pngCDel label4.png
[8+,8+]
CDel node h2.pngCDel 8.pngCDel node h4.pngCDel 8.pngCDel node h2.png
Orbifold *882 *884 *4242 *4444 44×
Semidirect subgroups
Diagram 882 symmetry 0aa.png 882 symmetry aa0.png 882 symmetry a0a.png 882 symmetry 0ab.png 882 symmetry ab0.png
Coxeter [8,8+]
CDel node c1.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h2.png
[8+,8]
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node c2.png
[(8,8,2+)]
CDel node c3.pngCDel split1-88.pngCDel branch h2h2.pngCDel label2.png
[8,1+,8,1+]
CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node h0.png = CDel node c1.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h0.png = CDel node c1.pngCDel split1-88.pngCDel branch h2h2.pngCDel label4.png
= CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node h2.png = CDel label4.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel branch h2h2.pngCDel label4.png
[1+,8,1+,8]
CDel node h0.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node c2.png = CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node c2.png = CDel label4.pngCDel branch h2h2.pngCDel split2-88.pngCDel node c2.png
= CDel node h2.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node c2.png = CDel label4.pngCDel branch h2h2.pngCDel 2a2b-cross.pngCDel branch c2.pngCDel label4.png
Orbifold 8*4 2*44 4*44
Direct subgroups
Index 2 4 8
Diagram 882 symmetry aaa.png 882 symmetry abb.png 882 symmetry bba.png 882 symmetry bab.png 882 symmetry abc.png
Coxeter [8,8]+
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h2.png
[8,8+]+
CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h2.png = CDel label4.pngCDel branch h2h2.pngCDel split2-88.pngCDel node h2.png
[8+,8]+
CDel node h2.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h0.png = CDel node h2.pngCDel split1-88.pngCDel branch h2h2.pngCDel label4.png
[8,1+,8]+
CDel labelh.pngCDel node.pngCDel split1-88.pngCDel branch h2h2.pngCDel label2.png = CDel label4.pngCDel branch h2h2.pngCDel 2xa2xb-cross.pngCDel branch h2h2.pngCDel label4.png
[8+,8+]+ = [1+,8,1+,8,1+]
CDel node h4.pngCDel split1-88.pngCDel branch h4h4.pngCDel label2.png = CDel node h0.pngCDel 8.pngCDel node h0.pngCDel 8.pngCDel node h0.png = CDel node h0.pngCDel 8.pngCDel node h2.pngCDel 8.pngCDel node h0.png = CDel label4.pngCDel branch h2h2.pngCDel 4a4b-cross.pngCDel branch h2h2.pngCDel label4.png
Orbifold 882 884 4242 4444
Radical subgroups
Index 16 32
Diagram 882-m0.png 882 symmetry zz0.png 882 symmetry zza.png 882 symmetry azz.png
Coxeter [8,8*]
CDel node c1.pngCDel 8.pngCDel node g.pngCDel 8.pngCDel 3sg.pngCDel node g.png
[8*,8]
CDel node g.pngCDel 8.pngCDel 3sg.pngCDel node g.pngCDel 8.pngCDel node c2.png
[8,8*]+
CDel node h0.pngCDel 8.pngCDel node g.pngCDel 8.pngCDel 3sg.pngCDel node g.png
[8*,8]+
CDel node g.pngCDel 8.pngCDel 3sg.pngCDel node g.pngCDel 8.pngCDel node h0.png
Orbifold *44444444 44444444

Related polyhedra and tiling[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. 

See also[edit]

External links[edit]