Octagonal tiling

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Octagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	83
Schläfli symbol	{8,3} t{4,8}
Wythoff symbol	3 | 8 2 2 8 | 4 4 4 4 |
Coxeter diagram
Symmetry group	[8,3], (*832) [8,4], (*842) [(4,4,4)], (*444)
Dual	Order-8 triangular tiling
Properties	Vertex-transitive, edge-transitive, face-transitive

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

Uniform colorings[]

Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.

Regular	Truncations
{8,3}	t{4,8}	t{4[3]} = =
Dual tiling
{3,8} =	=	= =

Related polyhedra and tilings[]

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}.

*n32 symmetry mutation of regular tilings: {n,3} v
Spherical				Euclidean	Compact hyperb.		Paraco.	Noncompact hyperbolic
{2,3}	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	{∞,3}	{12i,3}	{9i,3}	{6i,3}	{3i,3}

And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.

n82 symmetry mutations of regular tilings: 8ⁿ

• v

Space	Spherical	Compact hyperbolic						Paracompact
Tiling
Config.	8.8	83	84		86		88	...

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

Uniform octagonal/triangular tilings v
Symmetry: [8,3], (*832)							[8,3]+ (832)	[1+,8,3] (*443)		[8,3+] (3*4)
{8,3}	t{8,3}	r{8,3}	t{3,8}	{3,8}	rr{8,3} s2{3,8}	tr{8,3}	sr{8,3}	h{8,3}	h2{8,3}	s{3,8}
								or	or
Uniform duals
V83	V3.16.16	V3.8.3.8	V6.6.8	V38	V3.4.8.4	V4.6.16	V34.8	V(3.4)3	V8.6.6	V35.4

Uniform octagonal/square tilings v
[8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
= = =	=	= = =	=	= =	=
{8,4}	t{8,4}	r{8,4}	2t{8,4}=t{4,8}	2r{8,4}={4,8}	rr{8,4}	tr{8,4}
Uniform duals
V84	V4.16.16	V(4.8)2	V8.8.8	V48	V4.4.4.8	V4.8.16
Alternations
[1+,8,4] (*444)	[8+,4] (8*2)	[8,1+,4] (*4222)	[8,4+] (4*4)	[8,4,1+] (*882)	[(8,4,2+)] (2*42)	[8,4]+ (842)
=	=	=	=	=	=
h{8,4}	s{8,4}	hr{8,4}	s{4,8}	h{4,8}	hrr{8,4}	sr{8,4}
Alternation duals
V(4.4)4	V3.(3.8)2	V(4.4.4)2	V(3.4)3	V88	V4.44	V3.3.4.3.8

Uniform (4,4,4) tilings v
Symmetry: [(4,4,4)], (*444)							[(4,4,4)]+ (444)	[(1+,4,4,4)] (*4242)	[(4+,4,4)] (4*22)
t0(4,4,4) h{8,4}	t0,1(4,4,4) h2{8,4}	t1(4,4,4) {4,8}1/2	t1,2(4,4,4) h2{8,4}	t2(4,4,4) h{8,4}	t0,2(4,4,4) r{4,8}1/2	t0,1,2(4,4,4) t{4,8}1/2	s(4,4,4) s{4,8}1/2	h(4,4,4) h{4,8}1/2	hr(4,4,4) hr{4,8}1/2
Uniform duals
V(4.4)4	V4.8.4.8	V(4.4)4	V4.8.4.8	V(4.4)4	V4.8.4.8	V8.8.8	V3.4.3.4.3.4	V88	V(4,4)3

See also[]

Wikimedia Commons has media related to Order-3 octagonal tiling.

• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

References[]

External links[]

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch