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Contents

   



(Top)
 


1 Symmetry  





2 Related polyhedra and tiling  





3 See also  





4 References  





5 External links  














Truncated triapeirogonal tiling







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Truncated triapeirogonal tiling
Truncated triapeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.6.∞
Schläfli symbol tr{∞,3} or
Wythoff symbol 2 ∞ 3 |
Coxeter diagram or
Symmetry group [∞,3], (*∞32)
Dual Order 3-infinite kisrhombille
Properties Vertex-transitive

Ingeometry, the truncated triapeirogonal tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.

Symmetry[edit]

Truncated triapeirogonal tiling with mirrors

The dual of this tiling represents the fundamental domains of [∞,3], *∞32 symmetry. There are 3 small index subgroup constructed from [∞,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

A special index 4 reflective subgroup, is [(∞,∞,3)], (*∞∞3), and its direct subgroup [(∞,∞,3)]+, (∞∞3), and semidirect subgroup [(∞,∞,3+)], (3*∞).[1] Given [∞,3] with generating mirrors {0,1,2}, then its index 4 subgroup has generators {0,121,212}.

An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).

Small index subgroups of [∞,3], (*∞32)
Index 1 2 3 4 6 8 12 24
Diagrams
Coxeter
(orbifold)
[∞,3]
=
(*∞32)
[1+,∞,3]
=
(*∞33)
[∞,3+]

(3*∞)
[∞,∞]

(*∞∞2)
[(∞,∞,3)]

(*∞∞3)
[∞,3*]
=
(*∞3)
[∞,1+,∞]

(*(∞2)2)
[(∞,1+,∞,3)]

(*(∞3)2)
[1+,∞,∞,1+]

(*∞4)
[(∞,∞,3*)]

(*∞6)
Direct subgroups
Index 2 4 6 8 12 16 24 48
Diagrams
Coxeter
(orbifold)
[∞,3]+
=
(∞32)
[∞,3+]+
=
(∞33)
[∞,∞]+

(∞∞2)
[(∞,∞,3)]+

(∞∞3)
[∞,3*]+
=
(∞3)
[∞,1+,∞]+

(∞2)2
[(∞,1+,∞,3)]+

(∞3)2
[1+,∞,∞,1+]+

(∞4)
[(∞,∞,3*)]+

(∞6)

Related polyhedra and tiling[edit]

Paracompact uniform tilings in [∞,3] family
  • t
  • e
  • Symmetry: [∞,3], (*∞32) [∞,3]+
    (∞32)
    [1+,∞,3]
    (*∞33)
    [∞,3+]
    (3*∞)

    =

    =

    =
    =
    or
    =
    or

    =
    {∞,3} t{∞,3} r{∞,3} t{3,∞} {3,∞} rr{∞,3} tr{∞,3} sr{∞,3} h{∞,3} h2{∞,3} s{3,∞}
    Uniform duals
    V∞3 V3.∞.∞ V(3.∞)2 V6.6.∞ V3 V4.3.4.∞ V4.6.∞ V3.3.3.3.∞ V(3.∞)3 V3.3.3.3.3.∞

    This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

    *n32 symmetry mutation of omnitruncated tilings: 4.6.2n
  • t
  • e
  • Sym.
    *n32
    [n,3]
    Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
    *232
    [2,3]
    *332
    [3,3]
    *432
    [4,3]
    *532
    [5,3]
    *632
    [6,3]
    *732
    [7,3]
    *832
    [8,3]
    *∞32
    [∞,3]
     
    [12i,3]
     
    [9i,3]
     
    [6i,3]
     
    [3i,3]
    Figures
    Config. 4.6.4 4.6.6 4.6.8 4.6.10 4.6.12 4.6.14 4.6.16 4.6.∞ 4.6.24i 4.6.18i 4.6.12i 4.6.6i
    Duals
    Config. V4.6.4 V4.6.6 V4.6.8 V4.6.10 V4.6.12 V4.6.14 V4.6.16 V4.6.∞ V4.6.24i V4.6.18i V4.6.12i V4.6.6i

    See also[edit]

    References[edit]

    1. ^ Norman W. Johnson and Asia Ivic Weiss, Quadratic Integers and Coxeter Groups, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336 [1]
    • 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. LCCN 99035678.
  • External links[edit]


    Retrieved from "https://en.wikipedia.org/w/index.php?title=Truncated_triapeirogonal_tiling&oldid=1189601991"

    Categories: 
    Apeirogonal tilings
    Hyperbolic tilings
    Isogonal tilings
    Truncated tilings
    Uniform tilings
    Hidden category: 
    Commons category link is on Wikidata
     



    This page was last edited on 12 December 2023, at 21:57 (UTC).

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