Uniform tilings in hyperbolic plane

There are an infinite number of uniform tilings on the hyperbolic plane based on the (p q r) where 1/p + 1/q + 1/r < 1, where p,q,r are each orders of reflection symmetry at three points of the fundamental domain triangle – the symmetry group is a hyperbolic triangle group.

Each family contains up to 8 uniform tilings, defined by a Wythoff symbol or Coxeter-Dynkin diagram, 7 representing combinations of 3 active mirrors, and an 8th representing an alternation operation, deleting alternate vertices from the highest form with all mirrors active.

Families with r=2 contain regular hyperbolic tilings, defined by a Coxeter group such as [7,3], [8,3], [9,3], ... [5,4], [6,4], ....

Hyperbolic families with r=3 or higher are given by (p q r) and include (4 3 3), (5 3 3), (6 3 3) ... (4 4 3), (5 4 3), ... (4 4 4)....

Three families of uniform tilings are shown below (using the Poincaré disk model for the hyperbolic plane). These three, and no others, are minimal in the sense that if any of their defining numbers is replaced by a smaller integer the resulting pattern is either Euclidean or spherical rather than hyperbolic; conversely, any of the numbers can be increased (even to infinity) to generate other hyperbolic patterns.

Each uniform tiling generates a dual uniform tiling, also given below.

The [7,3] (7 3 2) group family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group Coxeter-Dynkin diagram	Dual tilings
Order-3 heptagonal tiling	7.7.7	Order-7 triangular tiling
	3 | 7 2
	[7,3]
Order-3 truncated heptagonal tiling	3.14.14	Order-7 triakis triangular tiling
	2 3 | 7
	[7,3]
Triheptagonal tiling	3.7.3.7	Order-7-3 rhombille tiling
	2 | 7 3
	[7,3]
Order-7 truncated triangular tiling	7.6.6	Order-3 heptakis heptagonal tiling
	2 7 | 3
	[7,3]
Order-7 triangular tiling	37	Order-3 heptagonal tiling
	7 | 3 2
	[7,3]
Rhombitriheptagonal tiling	3.4.7.4	Deltoidal triheptagonal tiling
	3 | 7 2
	[7,3]
Truncated triheptagonal tiling	4.6.14	Order-3 bisected heptagonal tiling
	2 7 3 |
	[7,3]
Order-3 snub heptagonal tiling	3.3.3.3.7	Order-7-3 floret pentagonal tiling
	| 7 3 2
	[7,3]

The [5,4] (5 4 2) group family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group Coxeter-Dynkin diagram	Dual tilings
Order-4 pentagonal tiling	5.5.5.5	Order-5 square tiling
	4 | 5 2
	[5,4]
Truncated pentagonal tiling	4.10.10	Order-5 tetrakis square tiling
	2 4 | 5
	[5,4]
tetrapentagonal tiling	4.5.4.5	280px Order-5-4 rhombille tiling
	2 | 5 4
	[5,4]
Order-5 truncated square tiling	8.8.5	Order-4 pentakis pentagonal tiling
	2 5 | 4
	[5,4]
Order-5 square tiling	45	Order-4 pentagonal tiling
	5 | 4 2
	[5,4]
Rhombitetrapentagonal tiling	4.4.5.4	Deltoidal tetrapentagonal tiling
	4 | 5 2
	[5,4]
Truncated tetrapentagonal tiling	4.8.10	Order-4 bisected pentagonal tiling
	2 5 4 |
	[5,4]
Order-4 snub pentagonal tiling	3.3.4.3.5	Order-5-4 floret pentagonal tiling
	| 5 4 2
	[5,4]

(4 3 3) family

Uniform hyperbolic tilings	Vertex figure Wythoff symbol(s) Symmetry group Coxeter-Dynkin diagram	Dual tilings
Order-4-3-3_t0 tiling	(3.4)3	Order-4-3-3_t0 dual tiling
	3 | 3 4
	(4 3 3)
Order-4-3-3_t01 tiling	3.8.3.8	Order-4-3-3_t01 dual tiling
	3 3 | 4
	(4 3 3)
Order-4-3-3_t12 tiling	3.6.4.6	Order-4-3-3_t12 dual tiling
	4 3 | 3
	(4 3 3)
Order-4-3-3_t2 tiling	(3.3)4	Order-4-3-3_t2 dual tiling
	4 | 3 3
	(4 3 3)
Order-4-3-3_t012 tiling	6.6.8	Order-4-3-3_t012 dual tiling
	4 3 3 |
	(4 3 3)
Order-4-3-3_snub tiling	3.3.3.3.3.4	Order-4-3-3_snub dual tiling
	| 4 3 3
	(4 3 3)

External links

• Hatch, Don. "Hyperbolic Planar Tessellations". http://www.plunk.org/~hatch/HyperbolicTesselations/. Retrieved 2010-08-19. 
• Eppstein, David. "The Geometry Junkyard: Hyperbolic Tiling". http://www.ics.uci.edu/~eppstein/junkyard/hypertile.html. Retrieved 2010-08-19. 
• Joyce, David. "Hyperbolic Tessellations". http://aleph0.clarku.edu/~djoyce/poincare/poincare.html. Retrieved 2010-08-19.