Infinite-order triangular tiling

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Infinite-order triangular tiling

In geometry, the infinite-order triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,∞}. All vertices are ideal, located at "infinity" and seen on the boundary of the Poincaré hyperbolic disk projection.

Infinite-order triangular tiling
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Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration3
Schläfli symbol{3,}
Wythoff symbol | 3 2
Coxeter diagram
Symmetry group[,3], (*32)
DualOrder-3 apeirogonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive
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The {3,3,∞} honeycomb has {3,∞} vertex figures.

Symmetry

A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, . The tiling also represents the fundamental domains of the *∞∞∞ symmetry, which can be seen with 3 colors of lines representing 3 mirrors of the construction.

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Alternated colored tiling
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*∞∞∞ symmetry
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Apollonian gasket with *∞∞∞ symmetry
Summarize
Perspective

This tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.

More information Spherical, Euclid. ...
*n32 symmetry mutation of regular tilings: {3,n}
Spherical Euclid. Compact hyper. Paraco. Noncompact hyperbolic
3.3 33 34 35 36 37 38 3 312i 39i 36i 33i
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More information Symmetry: [∞,3], (*∞32), [∞,3]+ (∞32) ...
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.
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More information Symmetry: [(∞,3,3)], (*∞33), [(∞,3,3)]+, (∞33) ...
Paracompact hyperbolic uniform tilings in [(,3,3)] family
Symmetry: [(,3,3)], (*33) [(,3,3)]+, (33)
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(,,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
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V(3.)3 V3..3. V(3.)3 V3.6..6 V(3.3) V3.6..6 V6.6. V3.3.3.3.3.
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Other infinite-order triangular tilings

A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:

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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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