Order-8 triangular tiling

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In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

Uniform colorings

The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles:

H2 tiling 334-4.png

Symmetry

Octagonal tiling with *444 mirror lines,
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From [(4,4,4)] symmetry, there are 15 small index subgroups (7 unique) 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 fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, [(1+,4,1+,4,1+,4)] (222222) is the commutator subgroup of [(4,4,4)].

A larger subgroup is constructed [(4,4,4*)], index 8, as (2*2222) with gyration points removed, becomes (*22222222).

The symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as 832 symmetry, by 3 bisecting mirrors per domain.

Small index subgroups of [(4,4,4)] (*444)
Index 1 2 4
Diagram 444 symmetry mirrors.png 444 symmetry a00.png 444 symmetry 0a0.png 444 symmetry 00a.png 444 symmetry ab0.png 444 symmetry xxx.png
Coxeter [(4,4,4)]
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 		[(1+,4,4,4)]
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 		[(4,1+,4,4)]
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 		[(4,4,1+,4)]
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 		[(1+,4,1+,4,4)]
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 		[(4+,4+,4)]
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Orbifold *444 *4242 2*222 222×
Diagram 444 symmetry 0bb.png 444 symmetry b0b.png 444 symmetry bb0.png 444 symmetry 0b0.png 444 symmetry a0b.png
Coxeter [(4,4+,4)]
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 		[(4,4,4+)]
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 		[(4+,4,4)]
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 		[(4,1+,4,1+,4)]
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 		[(1+,4,4,1+,4)]
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Orbifold 4*22 2*222
Direct subgroups
Index 2 4 8
Diagram 444 symmetry aaa.png 444 symmetry abb.png 444 symmetry bab.png 444 symmetry bba.png 444 symmetry abc.png
Coxeter [(4,4,4)]+
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 		[(4,4+,4)]+
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 		[(4,4,4+)]+
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 		[(4+,4,4)]+
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 		[(4,1+,4,1+,4)]+
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Orbifold 444 4242 222222
Radical subgroups
Index 8 16
Diagram 444 symmetry 0zz.png 444 symmetry z0z.png 444 symmetry zz0.png 444 symmetry azz.png 444 symmetry zaz.png 444 symmetry zza.png
Coxeter [(4,4*,4)] [(4,4,4*)] [(4*,4,4)] [(4,4*,4)]+ [(4,4,4*)]+ [(4*,4,4)]+
Orbifold *22222222 22222222

Related polyhedra and tilings

The {3,3,8} honeycomb has {3,8} vertex figures.


From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.


It can also be generated from the (4 3 3) hyperbolic tilings:


See also

  • Order-8 tetrahedral honeycomb
  • Tilings of regular polygons
  • List of uniform planar tilings
  • List of regular polytopes

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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. 

External links



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