Order-8 triangular tiling

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Order-8 triangular tiling
Order-8 triangular tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 38
Schläfli symbol {3,8}
{(3,4,3)}
Wythoff symbol 8 | 3 2
4 | 3 3
Coxeter diagram CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch.pngCDel split2.pngCDel node 1.png
Symmetry group [8,3], (*832)
[(4,3,3)], (*433)
Dual Octagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

The {3,3,8} honeycomb has {3,8} vertex figures.

Symmetry[edit]

The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles:

H2 tiling 334-4.png

Related polyhedra and tilings[edit]

Finite Euclidean Compact hyperbolic Paracompact
Trigonal dihedron.png
{3,2}
Uniform polyhedron-33-t2.png
{3,3}
Uniform polyhedron-43-t2.png
{3,4}
Uniform polyhedron-53-t2.png
{3,5}
Uniform polyhedron-63-t2.png
{3,6}
Uniform tiling 73-t2.png
{3,7}
Uniform tiling 83-t2.png
{3,8}
Uniform tiling 39-t0.png
{3,9}
... H2 tiling 23i-4.png
(3,∞}

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings.

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
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}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node.png or CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel label4.pngCDel branch hh.pngCDel split2.pngCDel node h.png
Uniform tiling 83-t0.png Uniform tiling 83-t01.png Uniform tiling 83-t1.png
Uniform tiling 433-t02.png
Uniform tiling 83-t12.png
Uniform tiling 433-t012.png
Uniform tiling 83-t2.png Uniform tiling 83-t02.png Uniform tiling 83-t012.png Uniform tiling 83-snub.png Uniform tiling 433-t0.pngUniform tiling 433-t1.png Uniform tiling 433-t02.pngUniform tiling 433-t12.png Uniform tiling 433-snub1.png
Uniform tiling 433-snub2.png
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
CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 83-t2.png Ord8 triakis triang til.png Uniform dual tiling 433-t01-yellow.png Uniform dual tiling 433-t012.png Uniform tiling 83-t0.png Deltoidal trioctagonal til.png Order-3 octakis octagonal tiling.png Uniform dual tiling 433-t0.png Uniform dual tiling 433-t01.png Uniform dual tiling 433-snub.png
Spherical Hyperbolic tilings
{2,8}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 238-4.png
{3,8}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 248-4.png
{4,8}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 258-4.png
{5,8}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 268-4.png
{6,8}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 278-1.png
{7,8}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 288-1.png
{8,8}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png
... H2 tiling 28i-1.png
{∞,8}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 8.pngCDel node.png

It can also be generated from the (4 3 3) hyperbolic tilings:

Uniform (4,3,3) tilings
Symmetry: [(4,3,3)], (*433) [(4,3,3)]+, (433)
CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.png CDel 3.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.png
CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node.png CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch 01rd.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch 11.pngCDel split2.pngCDel node 1.png CDel label4.pngCDel branch hh.pngCDel split2.pngCDel node h.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h0.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
Uniform tiling 433-t0.png Uniform tiling 433-t01.png Uniform tiling 433-t1.png Uniform tiling 433-t12.png Uniform tiling 433-t2.png Uniform tiling 433-t02.png Uniform tiling 433-t012.png Uniform tiling 433-snub2.png
h{8,3}
t0{(4,3,3)}
{(4,3,3)}
r{8,3}
t0,1{(4,3,3)}
r{(3,4,3)}
h{8,3}
t1{(4,3,3)}]]
{(3,3,4)}
h2{8,3}
t1,2{(4,3,3)}
r{(4,3,3)}
{3,8}
t2{(4,3,3)}]
{(3,4,3)}
h2{8,3}
t0,2{(4,3,3)}
r{(3,3,4)}
t{3,8}
t0,1,2{(4,3,3)}
t{(3,4,3)}
s{3,8}
 
s{(3,4,3)}
Uniform duals
CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.png CDel 3.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.png
Uniform dual tiling 433-t0.png Uniform dual tiling 433-t01.png Uniform dual tiling 433-t0.png Uniform dual tiling 433-t12.png Uniform dual tiling 433-t2.png Uniform dual tiling 433-t12.png Uniform dual tiling 433-t012.png Uniform dual tiling 433-snub.png
V(3.4)3 V3.8.3.8 V(3.4)3 V3.6.4.6 V(3.3)4 V3.6.4.6 V6.6.8 V3.3.3.3.3.4

See also[edit]

References[edit]

External links[edit]