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
Famous quotes containing the words uniform and/or plane:
“The Federal Constitution has stood the test of more than a hundred years in supplying the powers that have been needed to make the Central Government as strong as it ought to be, and with this movement toward uniform legislation and agreements between the States I do not see why the Constitution may not serve our people always.”
—William Howard Taft (1857–1930)
“Even though I had let them choose their own socks since babyhood, I was only beginning to learn to trust their adult judgment.. . . I had a sensation very much like the moment in an airplane when you realize that even if you stop holding the plane up by gripping the arms of your seat until your knuckles show white, the plane will stay up by itself. . . . To detach myself from my children . . . I had to achieve a condition which might be called loving objectivity.”
—Anonymous Parent of Adult Children. Ourselves and Our Children, by Boston Women’s Health Book Collective, ch. 5 (1978)