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.
Symmetry
[edit | edit source]A lower symmetry form has alternating colors, and represented by cyclic symbol {(3,∞,3)}, File:CDel node 1.pngFile:CDel split1.pngFile:CDel branch.pngFile:CDel labelinfin.png. 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.
| File:Infinite-order triangular tiling.svg Alternated colored tiling |
File:Iii symmetry mirrors.png *∞∞∞ symmetry |
File:Apolleangasket symmetry.png Apollonian gasket with *∞∞∞ symmetry |
Related polyhedra and tiling
[edit | edit source]This 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 | |||||||
| File:Trigonal dihedron.svg | File:Uniform tiling 332-t2.svg | File:Uniform tiling 432-t2.svg | File:Uniform tiling 532-t2.svg | File:Uniform polyhedron-63-t2.svg | File:Order-7 triangular tiling.svg | File:H2-8-3-primal.svg | File:H2 tiling 23i-4.png | File:H2 tiling 23j12-4.png | File:H2 tiling 23j9-4.png | File:H2 tiling 23j6-4.png | File:H2 tiling 23j3-4.png |
| 3.3 | 33 | 34 | 35 | 36 | 37 | 38 | 3∞ | 312i | 39i | 36i | 33i |
Other infinite-order triangular tilings
[edit | edit source]A nonregular infinite-order triangular tiling can be generated by a recursive process from a central triangle as shown here:
See also
[edit | edit source]
- Infinite-order tetrahedral honeycomb
- List of regular polytopes
- List of uniform planar tilings
- Tilings of regular polygons
- Triangular tiling
- Uniform tilings in hyperbolic plane
References
[edit | edit source]- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapter 19, The Hyperbolic Archimedean Tessellations)
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
External links
[edit | edit source]- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
