In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.
Order-4 octagonal tiling | |
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Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 8^{4} |
Schläfli symbol | {8,4} r{8,8} |
Wythoff symbol | 4 | 8 2 |
Coxeter diagram | or |
Symmetry group | [8,4], (*842) [8,8], (*882) |
Dual | Order-8 square tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive |
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1^{+}], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4^{*}], leaves remaining mirrors *4444 symmetry.
Uniform Coloring |
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---|---|---|---|---|
Symmetry | [8,4] (*842) |
[8,8] (*882) = |
[(8,4,8)] = [8,8,1^{+}] (*884) = = |
[1^{+},8,8,1^{+}] (*4444) = |
Symbol | {8,4} | r{8,8} | r(8,4,8) = r{8,8}^{1}⁄_{2} | r{8,4}^{1}⁄_{8} = r{8,8}^{1}⁄_{4} |
Coxeter diagram |
=
= |
= = = |
This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*2^{8}) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8^{*},4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.
*444 |
*4222 |
*832 |
The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.
This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram , progressing to infinity.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
{3,4} |
{4,4} |
{5,4} |
{6,4} |
{7,4} |
{8,4} |
... | {∞,4} |
Octagonal refers to the property of being like an octagon.
Octagonal may also refer to:
Octagonal (horse) (1992–2016), New Zealand racehorse that raced in Australia
Octagonal tiling
Truncated octagonal tiling
Truncated order-4 octagonal tiling
Order-6 octagonal tiling
Order-8 octagonal tiling
Truncated order-8 octagonal tiling
Snub octagonal tiling
Octagonal number
Centered octagonal number
Octagonal polyhedra
Octagonal prism
Octagonal antiprism
Octagonal prismatic prism
Octagonal bipyramid
Octagonal trapezohedron
Octagonal polychoron
Octagonal antiprismatic prism
List of octagonal buildings and structures
Octagonal barn (disambiguation)
Octagonal house
Octagonal School (disambiguation)
Octagonal Building (disambiguation)
Octagonal deadhouse
Order-3-4 heptagonal honeycombIn the geometry of hyperbolic 3-space, the order-3-4 heptagonal honeycomb or 7,3,4 honeycomb a regular space-filling tessellation (or honeycomb). Each infinite cell consists of a heptagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
Order-4-3 pentagonal honeycombIn the geometry of hyperbolic 3-space, the order-4-3 pentagonal honeycomb or 5,4,3 honeycomb is a regular space-filling tessellation (or honeycomb). Each infinite cell is an order-4 pentagonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
Order-8-3 triangular honeycombIn the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb (or 3,8,3 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,3}.
Order-8 square tilingIn geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.
Rhombitetraoctagonal tilingIn geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.
Truncated order-4 octagonal tilingIn geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.
Truncated tetraoctagonal tilingIn geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.
*n42 symmetry mutation of regular tilings: {n,4} | |||||||
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Spherical | Euclidean | Hyperbolic tilings | |||||
2^{4} | 3^{4} | 4^{4} | 5^{4} | 6^{4} | 7^{4} | 84 | ...∞^{4} |
Regular tilings: {n,8} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Spherical | Hyperbolic tilings | ||||||||||
{2,8} |
{3,8} |
{4,8} |
{5,8} |
{6,8} |
{7,8} |
{8,8} |
... | {∞,8} |
Uniform octagonal/square tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
[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 | |||||||||||
V8^{4} | 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} | V8^{8} | V4.4^{4} | V3.3.4.3.8 |
Uniform octaoctagonal tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry: [8,8], (*882) | |||||||||||
= = |
= = |
= = |
= = |
= = |
= = |
= = | |||||
{8,8} | t{8,8} |
r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} | |||||
Uniform duals | |||||||||||
V8^{8} | V8.16.16 | V8.8.8.8 | V8.16.16 | V8^{8} | V4.8.4.8 | V4.16.16 | |||||
Alternations | |||||||||||
[1^{+},8,8] (*884) |
[8^{+},8] (8*4) |
[8,1^{+},8] (*4242) |
[8,8^{+}] (8*4) |
[8,8,1^{+}] (*884) |
[(8,8,2^{+})] (2*44) |
[8,8]^{+} (882) | |||||
= | = | = | = = |
= = | |||||||
h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} | |||||
Alternation duals | |||||||||||
V(4.8)^{8} | V3.4.3.8.3.8 | V(4.4)4 | V3.4.3.8.3.8 | V(4.8)^{8} | V4^{6} | V3.3.8.3.8 |
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