Order-3 apeirogonal tiling
| Order-3 apeirogonal tiling | |
|---|---|
| Order-3 apeirogonal tiling Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | ∞3 |
| Schläfli symbol | {∞,3} t{∞,∞} t(∞,∞,∞) |
| Wythoff symbol | 3 | ∞ 2 2 ∞ | ∞ ∞ ∞ ∞ | |
| Coxeter diagram | File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel split2-ii.pngFile:CDel node 1.png |
| Symmetry group | [∞,3], (*∞32) [∞,∞], (*∞∞2) [(∞,∞,∞)], (*∞∞∞) |
| Dual | Infinite-order triangular tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle. The order-2 apeirogonal tiling represents an infinite dihedron in the Euclidean plane as {∞,2}.
Images
Each apeirogon face is circumscribed by a horocycle, which looks like a circle in a Poincaré disk model, internally tangent to the projective circle boundary.
Uniform colorings
Like the Euclidean hexagonal tiling, there are 3 uniform colorings of the order-3 apeirogonal tiling, each from different reflective triangle group domains:
Symmetry
The dual to this tiling represents the fundamental domains of [(∞,∞,∞)] (*∞∞∞) symmetry. There are 15 small index subgroups (7 unique) constructed from [(∞,∞,∞)] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled as ∞∞2 symmetry by adding a mirror bisecting the fundamental domain. Dividing a fundamental domain by 3 mirrors creates a ∞32 symmetry. A larger subgroup is constructed [(∞,∞,∞*)], index 8, as (∞*∞∞) with gyration points removed, becomes (*∞∞).
Related polyhedra and tilings
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.
| *n32 symmetry mutation of regular tilings: {n,3} | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
| File:Spherical trigonal hosohedron.svg | File:Uniform tiling 332-t0.png | File:Uniform tiling 432-t0.png | File:Uniform tiling 532-t0.png | File:Uniform polyhedron-63-t0.png | File:Heptagonal tiling.svg | File:H2-8-3-dual.svg | File:H2-I-3-dual.svg | File:H2 tiling 23j12-1.png | File:H2 tiling 23j9-1.png | File:H2 tiling 23j6-1.png | File:H2 tiling 23j3-1.png |
| {2,3} | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} | {12i,3} | {9i,3} | {6i,3} | {3i,3} |
See also
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes
- Hexagonal tiling honeycomb, similar {6,3,3} honeycomb in H3.
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.
