Order-4 octagonal tiling

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
Order-4 octagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 84
Schläfli symbol {8,4}
r{8,8}
Wythoff symbol 4 | 8 2
Coxeter diagram CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png or CDel node.pngCDel split1-88.pngCDel nodes 11.png
Symmetry group [8,4], (*842)
[8,8], (*882)
Dual Order-8 square tiling
Properties Vertex-transitive, edge-transitive, face-transitive

Uniform constructions

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.

Four uniform constructions of 8.8.8.8
Uniform
Coloring
H2 tiling 248-1 H2 tiling 288-2 H2 tiling 488-5 H2 tiling 488-5-4color
Symmetry [8,4]
(*842)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node c3.png
[8,8]
(*882)
CDel node c1.pngCDel 8.pngCDel node c2.pngCDel 4.pngCDel node h0.png = CDel node c2.pngCDel 8.pngCDel node c1.pngCDel 8.pngCDel node c2.png
[(8,4,8)] = [8,8,1+]
(*884)
CDel node c2.pngCDel 8.pngCDel node c1.pngCDel 8.pngCDel node h0.png = CDel node c2.pngCDel split1-88.pngCDel branch c1.pngCDel label4.png

CDel node c1.pngCDel 8.pngCDel node h0.pngCDel 4.pngCDel node c2.png = CDel label4.pngCDel branch c1.pngCDel 2a2b-cross.pngCDel nodeab c2.png

[1+,8,8,1+]
(*4444)
CDel node c1.pngCDel 8.pngCDel node g.pngCDel 4sg.pngCDel node g.png =
CDel label4.pngCDel branch c1.pngCDel 4a4b-cross.pngCDel branch c1.pngCDel label4.png
Symbol {8,4} r{8,8} r(8,4,8) = r{8,8}​12 r{8,4}​18 = r{8,8}​14
Coxeter
diagram
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node h0.png = CDel node.pngCDel split1-88.pngCDel branch 11.pngCDel label4.png

CDel node 1.pngCDel 8.pngCDel node h0.pngCDel 4.pngCDel node.png = CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png

CDel node h0.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node h0.png = CDel labelh.pngCDel node.pngCDel split1-88.pngCDel branch 11.pngCDel label4.png =
CDel node 1.pngCDel 8.pngCDel node g.pngCDel 4sg.pngCDel node g.png =CDel label4.pngCDel branch 11.pngCDel 4a4b-cross.pngCDel branch 11.pngCDel label4.png

Symmetry

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 (*28) 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.

H2chess 248e
*444
H2chess 248d
*4222
842 symmetry mirrors
*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.

Uniform tiling 88-t1 H2chess 248c

Related polyhedra and 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 CDel node 1.pngCDel 8.pngCDel node.pngCDel n.pngCDel node.png, 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 CDel node 1.pngCDel n.pngCDel node.pngCDel 4.pngCDel node.png, with n progressing to infinity.

Uniform polyhedron-34-t0
{3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 44-t0
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 54-t0
{5,4}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 64-t0
{6,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 74-t0
{7,4}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 4.pngCDel node.png
Uniform tiling 84-t0
{8,4}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
... H2 tiling 24i-1
{∞,4}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png

See also

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.

External links

Octagonal (disambiguation)

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 honeycomb

In 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 honeycomb

In 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 honeycomb

In 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 tiling

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

Rhombitetraoctagonal tiling

In 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 tiling

In 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 tiling

In 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}
Spherical Euclidean Hyperbolic tilings
Spherical square hosohedron.png Spherical square bipyramid.png Uniform tiling 44-t0.svg H2 tiling 245-1.png H2 tiling 246-1.png H2 tiling 247-1.png H2 tiling 248-1.png H2 tiling 24i-1.png
24 34 44 54 64 74 84 ...4
Regular tilings: {n,8}
Spherical Hyperbolic tilings
Spherical octagonal hosohedron.png
{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
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)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-88.pngCDel nodes.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel label4.pngCDel branch 11.pngCDel 4a4b-cross.pngCDel branch 11.pngCDel label4.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-88.pngCDel nodes 11.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-88.pngCDel nodes 11.png
= CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel branch 11.pngCDel label4.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel label4.pngCDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel label4.pngCDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel label4.pngCDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 248-1 H2 tiling 248-3 H2 tiling 248-2 H2 tiling 248-6 H2 tiling 248-4 H2 tiling 248-5 H2 tiling 248-7
{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
CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 248b H2chess 248f H2chess 248a H2chess 248e H2chess 248c H2chess 248d H2checkers 248
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)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png
= CDel label4.pngCDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-88.pngCDel nodes hh.png
CDel node.pngCDel 8.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel label4.pngCDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel label4.pngCDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-88.pngCDel nodes 10lu.png
CDel node h.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel label4.pngCDel branch hh.pngCDel 2a2b-cross.pngCDel nodes hh.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 444-t0 Uniform tiling 84-h01 Uniform tiling 443-t1 Uniform tiling 444-snub Uniform tiling 88-t0 Uniform tiling 54-t2 Uniform tiling 84-snub
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Uniform tiling 88-t1 Uniform tiling 66-t1 Uniform dual tiling 433-t0 Uniform tiling 88-t2 Uniform tiling 54-t0
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 octaoctagonal tilings
Symmetry: [8,8], (*882)
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png = CDel nodes 10ru.pngCDel split2-88.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png = CDel nodes 10ru.pngCDel split2-88.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node.png = CDel nodes.pngCDel split2-88.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-88.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-88.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node 1.png = CDel nodes 11.pngCDel split2-88.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node.png
CDel node 1.pngCDel 8.pngCDel node 1.pngCDel 8.pngCDel node 1.png = CDel nodes 11.pngCDel split2-88.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 8.pngCDel node 1.png
H2 tiling 288-1 H2 tiling 288-3 H2 tiling 288-2 H2 tiling 288-6 H2 tiling 288-4 H2 tiling 288-5 H2 tiling 288-7
{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
CDel node f1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node f1.png CDel node f1.pngCDel 8.pngCDel node f1.pngCDel 8.pngCDel node f1.png
H2chess 288b H2chess 288f H2chess 288a H2chess 288e H2chess 288c H2chess 288d H2checkers 288
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 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)
CDel node h1.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png = CDel label4.pngCDel branch 10ru.pngCDel split2-88.pngCDel node.png CDel node h.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node h1.pngCDel 8.pngCDel node.png = CDel nodes 11.pngCDel 4a4b-cross.pngCDel nodes.png CDel node.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node h.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node h1.png = CDel node.pngCDel split1-88.pngCDel branch 01ld.png CDel node h.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node h.png = CDel nodes hh.pngCDel split2-88.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 8.pngCDel node.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 8.pngCDel node h.png = CDel nodes hh.pngCDel split2-88.pngCDel node h.png
= CDel node h0.pngCDel 4.pngCDel node h.pngCDel 8.pngCDel node h.png
Uniform tiling 88-h0 Uniform tiling 444-t0 Uniform tiling 88-h0 Uniform tiling 443-t1 Uniform tiling 88-snub
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
CDel node fh.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node.png CDel node.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node fh.png CDel node.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 8.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 8.pngCDel node fh.png
Uniform tiling 88-t1 Uniform tiling 66-t1
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8

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