Infinite-order truncated triangular tiling
Truncated infinite-order triangular tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration.6.6
Schläfli symbolt{3,}
Wythoff symbol2 | 3
Coxeter diagram
Symmetry group[,3], (*32)
Dualapeirokis apeirogonal tiling
PropertiesVertex-transitive

In geometry, the truncated infinite-order triangular tiling is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{3,∞}.

Symmetry

Truncated infinite-order triangular tiling with mirror lines, .

The dual of this tiling represents the fundamental domains of *∞33 symmetry. There are no mirror removal subgroups of [(∞,3,3)], but this symmetry group can be doubled to ∞32 symmetry by adding a mirror.

Small index subgroups of [(∞,3,3)], (*∞33)
Type Reflectional Rotational
Index 1 2
Diagram
Coxeter
(orbifold)
[(∞,3,3)]

(*∞33)
[(∞,3,3)]+

(∞33)

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (6.n.n), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: n.6.6
Sym.
*n42
[n,3]
Spherical Euclid. Compact Parac. 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]
Truncated
figures
Config. 2.6.6 3.6.6 4.6.6 5.6.6 6.6.6 7.6.6 8.6.6 .6.6 12i.6.6 9i.6.6 6i.6.6
n-kis
figures
Config. V2.6.6 V3.6.6 V4.6.6 V5.6.6 V6.6.6 V7.6.6 V8.6.6 V.6.6 V12i.6.6 V9i.6.6 V6i.6.6
Paracompact uniform tilings in [,3] family
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
V3 V3.. V(3.)2 V6.6. V3 V4.3.4. V4.6. V3.3.3.3. V(3.)3 V3.3.3.3.3.
Paracompact hyperbolic uniform tilings in [(,3,3)] family
Symmetry: [(,3,3)], (*33) [(,3,3)]+, (33)
(,,3) t0,1(,3,3) t1(,3,3) t1,2(,3,3) t2(,3,3) t0,2(,3,3) t0,1,2(,3,3) s(,3,3)
Dual tilings
V(3.)3 V3..3. V(3.)3 V3.6..6 V(3.3) V3.6..6 V6.6. V3.3.3.3.3.

See also

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. LCCN 99035678.
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