Order-6 square tiling

In geometry, the order-6 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,6}.

Order-6 square tiling
Order-6 square tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 46
Schläfli symbol {4,6}
Wythoff symbol 6 | 4 2
Coxeter diagram CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
Symmetry group [6,4], (*642)
Dual Order-4 hexagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

Symmetry

This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with six squares around every vertex. This symmetry by orbifold notation is called (*3333) with 4 order-3 mirror intersections. In Coxeter notation can be represented as [6,4*], removing two of three mirrors (passing through the square center) in the [6,4] symmetry. The *3333 symmetry can be doubled to 663 symmetry by adding a mirror bisecting the fundamental domain.

This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction t1{(4,4,3)}. A second 6-color symmetry can be constructed from a hexagonal symmetry domain.

Uniform tiling 443-t1 Order-6 square tiling nonsimplex domain
[4,6,1+] = [(4,4,3)] or (*443) symmetry
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node h0.png = CDel node 1.pngCDel split1-44.pngCDel branch.png
[4,6*] = (*222222) symmetry
CDel node 1.pngCDel 4.pngCDel node g.pngCDel 6sg.pngCDel node g.png = CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.png

Example artwork

Around 1956, M.C. Escher explored the concept of representing infinity on a two-dimensional plane. Discussions with Canadian mathematician H.S.M. Coxeter inspired Escher's interest in hyperbolic tessellations, which are regular tilings of the hyperbolic plane. Escher's wood engravings Circle Limit I–IV demonstrate this concept. The last one Circle Limit IV (Heaven and Hell), (1960) tiles repeating angels and devils by (*3333) symmetry on a hyperbolic plane in a Poincaré disk projection.

The artwork seen below has an approximate hyperbolic mirror overlay added to show the square symmetry domains of the order-6 square tiling. If you look closely, you can see one of four angels and devils around each square are drawn as back sides. Without this variation, the art would have a 4-fold gyration point at the center of each square, giving (4*3), [6,4+] symmetry.[1]

Escher circlelimit iv-with overlay
Escher circlelimit iv-with overlay

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram CDel node 1.pngCDel n.pngCDel node.pngCDel 6.pngCDel node.png, progressing to infinity.

See also

References

  1. ^ Conway, The Symmetry of Things (2008), p.224, Figure 17.4, Circle Limit IV
  • 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

Order-4-5 pentagonal honeycomb

In the geometry of hyperbolic 3-space, the order-4-5 pentagonal honeycomb a regular space-filling tessellation (or honeycomb) with Schläfli symbol {5,4,5}.

Order-4-5 square honeycomb

In the geometry of hyperbolic 3-space, the order-4-5 square honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {4,4,5}. It has five square tiling {4,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many square tiling existing around each vertex in an order-5 square tiling vertex arrangement.

Order-4 hexagonal tiling

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

Order-5 octahedral honeycomb

In the geometry of hyperbolic 3-space, the order-5 octahedral honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,4,5}. It has five octahedra {3,4} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many octahedra existing around each vertex in an order-5 square tiling vertex arrangement.

Quarter order-6 square tiling

In geometry, the quarter order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of q{4,6}. It is constructed from *3232 orbifold notation, and can be seen as a half symmetry of *443 and *662, and quarter symmetry of *642.

Rhombitetrahexagonal tiling

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

Snub order-6 square tiling

In geometry, the snub tetratritetragonal tiling or snub order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of s{(4,4,3)} or s{4,6}.

Truncated order-6 hexagonal tiling

In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically constructed as a cantic order-6 square tiling, h2{4,6}

Truncated order-6 square tiling

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact
Uniform tiling 432-t0
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform tiling 44-t0
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 245-4
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 246-4
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 247-4
{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 248-4
{4,8}...
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png
H2 tiling 24i-4
{4,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
Regular tilings {n,6}
Spherical Euclidean Hyperbolic tilings
Spherical hexagonal hosohedron.png
{2,6}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tiling 63-t2.svg
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 246-4.png
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 256-4.png
{5,6}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 266-4.png
{6,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 267-1.png
{7,6}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 268-1.png
{8,6}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png
... H2 tiling 26i-1.png
{∞,6}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 6.pngCDel node.png
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel branch 11.pngCDel 3a3b-cross.pngCDel branch 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-66.pngCDel nodes 11.png
= CDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
= CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 246-1 H2 tiling 246-3 H2 tiling 246-2 H2 tiling 246-6 H2 tiling 246-4 H2 tiling 246-5 H2 tiling 246-7
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 246b H2chess 246f H2chess 246a H2chess 246e H2chess 246c H2chess 246d H2checkers 246
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-66.pngCDel branch hh.pngCDel label2.png
CDel node.pngCDel 6.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-66.pngCDel nodes 10lu.png
CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel 2xa2xb-cross.pngCDel branch hh.pngCDel label2.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 443-t0 Uniform tiling 64-h02 Uniform tiling 64-h1 Uniform tiling 443-snub2 Uniform tiling 66-t0 Uniform tiling 3.4.4.4.4 Uniform tiling 64-snub
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443)
[(4,4,3+)]
(3*22)
[(4,1+,4,3)]
(*3232)
CDel branch 01rd.pngCDel split2-44.pngCDel node.png CDel branch 01rd.pngCDel split2-44.pngCDel node 1.png CDel branch.pngCDel split2-44.pngCDel node 1.png CDel branch 10ru.pngCDel split2-44.pngCDel node 1.png CDel branch 10ru.pngCDel split2-44.pngCDel node.png CDel branch 11.pngCDel split2-44.pngCDel node.png CDel branch 11.pngCDel split2-44.pngCDel node 1.png CDel branch hh.pngCDel split2-44.pngCDel node h.png CDel branch hh.pngCDel split2-44.pngCDel node.png CDel branch.pngCDel split2-44.pngCDel node h.png CDel branch 10ru.pngCDel split2-44.pngCDel node h.png
CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h0.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
Uniform tiling 443-t0 Uniform tiling 443-t01 Uniform tiling 443-t1 Uniform tiling 443-t12 Uniform tiling 443-t2 Uniform tiling 443-t02 Uniform tiling 443-t012 Uniform tiling 443-snub1 Uniform tiling 64-h1 Uniform tiling 66-t2 Uniform tiling verf 34664
h{6,4}
t0(4,4,3)
h2{6,4}
t0,1(4,4,3)
{4,6}1/2
t1(4,4,3)
h2{6,4}
t1,2(4,4,3)
h{6,4}
t2(4,4,3)
r{6,4}1/2
t0,2(4,4,3)
t{4,6}1/2
t0,1,2(4,4,3)
s{4,6}1/2
s(4,4,3)
hr{4,6}1/2
hr(4,3,4)
h{4,6}1/2
h(4,3,4)
q{4,6}
h1(4,3,4)
Uniform duals
Uniform tiling 66-t1 Ord64 qreg rhombic til Order4 hexakis hexagonal til Uniform tiling 66-t0
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6
Uniform tilings in symmetry *3222
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png 64
Uniform tiling 64-t0
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 01.png 6.6.4.4
Uniform tiling 6.6.4.4 (green)
CDel branch 01.pngCDel 2a2b-cross.pngCDel nodes 01.png (3.4.4)2
Uniform tiling 3.4.4.3.4.4
CDel branch hh.pngCDel 2a2b-cross.pngCDel nodes 01.png 4.3.4.3.3.3
Uniform tiling 4.3.4.3.3.3
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 10.png 6.6.4.4
Uniform tiling 6.6.4.4
CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png 6.4.4.4
Uniform tiling 4.4.4.6
CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 11.png 3.4.4.4.4
Uniform tiling 3.4.4.4.4 (green)
CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png (3.4.4)2
Uniform tiling 64-h1
CDel branch 01.pngCDel 2a2b-cross.pngCDel nodes 11.png 3.4.4.4.4
Uniform tiling 3.4.4.4.4
CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png 46
Uniform tiling 64-t2

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