| Infinite-order triangular tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 3∞ |
| Schläfli symbol | {3,∞} |
| Wythoff symbol | ∞ | 3 2 |
| Coxeter diagram |       
|
| Symmetry group | [∞,3], (*∞32) |
| Dual | Order-3 apeirogonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
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.
Symmetry
editA 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.
 Alternated colored tiling |
 *∞∞∞ symmetry |
 Apollonian gasket with *∞∞∞ symmetry |
Related polyhedra and tiling
editThis tiling is topologically related as part of a sequence of regular polyhedra with Schläfli symbol {3,p}.
| *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 |
| 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 | ||||||||||
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
|
|
|
|
|
|
||
| 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.∞ | |
| 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.∞ | ||||
Other infinite-order triangular tilings
editA nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:
See also
edit
Wikimedia Commons has media related to Infinite-order triangular tiling.
References
edit- 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.