Uniform Tilings of The Hyperbolic Plane
For more details on this topic, see Uniform tilings in hyperbolic plane.There are infinitely many uniform tilings of convex regular polygons on the hyperbolic plane, each based on a different reflective symmetry group (p q r).
A sampling is shown here with a Poincaré disk projection.
The Coxeter-Dynkin diagram is given in a linear form, although it is actually a triangle, with the trailing segment r connecting to the first node.
Further symmetry groups exist in the hyperbolic plane with quadrilateral fundamental domains starting with (2 2 2 3), etc., that can generate new forms. As well there's fundamental domains that place vertices at infinity, such as (∞ 2 3), etc.
Right angle fundamental triangles: (p q 2)
| (p q 2) | Fund. triangles |
Parent | Truncated | Rectified | Bitruncated | Birectified (dual) |
Cantellated | Omnitruncated |
Snub |
|---|---|---|---|---|---|---|---|---|---|
| Wythoff symbol | q | p 2 | 2 q | p | 2 | p q | 2 p | q | p | q 2 | p q | 2 | p q 2 | | | p q 2 | |
| Schläfli symbol | t0{p,q} | t0,1{p,q} | t1{p,q} | t1,2{p,q} | t2{p,q} | t0,2{p,q} | t0,1,2{p,q} | s{p,q} | |
| Coxeter-Dynkin diagram | |||||||||
| Vertex figure | pq | (q.2p.2p) | (p.q.p.q) | (p.2q.2q) | qp | (p.4.q.4) | (4.2p.2q) | (3.3.p.3.q) | |
| (Hyperbolic plane) (5 4 2) |
V4.8.10 |
{5,4} |
4.10.10 |
4.5.4.5 |
5.8.8 |
{4,5} |
4.4.5.4 |
4.8.10 |
3.3.4.3.5 |
| (Hyperbolic plane) (5 5 2) |
V4.10.10 |
{5,5} |
5.10.10 |
5.5.5.5 |
5.10.10 |
{5,5} |
5.4.5.4 |
4.10.10 |
3.3.5.3.5 |
| (Hyperbolic plane) (7 3 2) |
V4.6.14 |
{7,3} |
3.14.14 |
3.7.3.7 |
7.6.6 |
{3,7} |
3.4.7.4 |
4.6.14 |
3.3.3.3.7 |
| (Hyperbolic plane) (8 3 2) |
V4.6.16 |
{8,3} |
3.16.16 |
3.8.3.8 |
8.6.6 |
{3,8} |
3.4.8.4 |
4.6.16 |
3.3.3.3.8 |
General fundamental triangles (p q r)
| Wythoff symbol (p q r) |
Fund. triangles |
q | p r | r q | p | r | p q | r p | q | p | q r | p q | r | p q r | | | p q r |
|---|---|---|---|---|---|---|---|---|---|
| Coxeter-Dynkin diagram | |||||||||
| Vertex figure | (p.r)q | (r.2p.q.2p) | (p.q)r | (q.2r.p.2r) | (q.r)p | (r.2q.p.2q) | (2p.2q.2r) | (3.r.3.q.3.p) | |
| Hyperbolic (4 3 3) |
V6.6.8 |
(3.4)3 |
3.8.3.8 |
(3.4)3 |
3.6.4.6 |
(3.3)4 |
3.6.4.6 |
6.6.8 |
3.3.3.3.3.4 |
| Hyperbolic (4 4 3) |
V6.8.8 |
(3.4)4 |
3.8.4.8 |
(4.4)3 |
3.6.4.6 |
(3.4)4 |
4.6.4.6 |
6.8.8 |
3.3.3.4.3.4 |
| Hyperbolic (4 4 4) |
V8.8.8 |
(4.4)4 |
4.8.4.8 |
(4.4)4 |
4.8.4.8 |
(4.4)4 |
4.8.4.8 |
8.8.8 |
3.4.3.4.3.4 |
Read more about this topic: Uniform Tiling
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