Cantic octagonal tiling

Cantic octagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration3.6.4.6
Schläfli symbolh2{8,3}
Wythoff symbol4 3 | 3
Coxeter diagram =
Symmetry group[(4,3,3)], (*433)
DualOrder-4-3-3 t12 dual tiling
PropertiesVertex-transitive

In geometry, the tritetratrigonal tiling or shieldotritetragonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t1,2(4,3,3). It can also be named as a cantic octagonal tiling, h2{8,3}.

Dual tiling

*n33 orbifold symmetries of cantic tilings: 3.6.n.6
Symmetry
*n32
[1+,2n,3]
= [(n,3,3)]		 Spherical Euclidean Compact Hyperbolic Paracompact
*233
[1+,4,3]
= [3,3]		 *333
[1+,6,3]
= [(3,3,3)]		 *433
[1+,8,3]
= [(4,3,3)]		 *533
[1+,10,3]
= [(5,3,3)]		 *633...
[1+,12,3]
= [(6,3,3)]		 *∞33
[1+,∞,3]
= [(∞,3,3)]
Coxeter
Schläfli		 =
h2{4,3}		 =
h2{6,3}		 =
h2{8,3}		 =
h2{10,3}		 =
h2{12,3}		 =
h2{∞,3}
Cantic
figure
Vertex 3.6.2.6 3.6.3.6 3.6.4.6 3.6.5.6 3.6.6.6 3.6.∞.6

Domain
Wythoff 2 3 | 3 3 3 | 3 4 3 | 3 5 3 | 3 6 3 | 3 ∞ 3 | 3
Dual
figure
Face V3.6.2.6 V3.6.3.6 V3.6.4.6 V3.6.5.6 V3.6.6.6 V3.6.∞.6

References

  • 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

  • Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
  • Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
  • Hyperbolic and Spherical Tiling Gallery
  • KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
  • Hyperbolic Planar Tessellations, Don Hatch
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