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 | |
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Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 4^{6} |
Schläfli symbol | {4,6} |
Wythoff symbol | 6 | 4 2 |
Coxeter diagram | |
Symmetry group | [6,4], (*642) |
Dual | Order-4 hexagonal tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive |
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 t_{1}{(4,4,3)}. A second 6-color symmetry can be constructed from a hexagonal symmetry domain.
[4,6,1^{+}] = [(4,4,3)] or (*443) symmetry = |
[4,6^{*}] = (*222222) symmetry = |
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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]}
This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4^{n}).
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 , progressing to infinity.
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 honeycombIn 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 tilingIn geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.
Order-5 octahedral honeycombIn 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 tilingIn 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 tilingIn 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 tilingIn 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 tilingIn 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 tilingIn 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} | |||||||||||
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Spherical | Euclidean | Compact hyperbolic | Paracompact | ||||||||
{4,3} |
{4,4} |
{4,5} |
{4,6} |
{4,7} |
{4,8}... |
{4,∞} |
Regular tilings {n,6} | ||||||||
---|---|---|---|---|---|---|---|---|
Spherical | Euclidean | Hyperbolic tilings | ||||||
{2,6} |
{3,6} |
{4,6} |
{5,6} |
{6,6} |
{7,6} |
{8,6} |
... | {∞,6} |
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) | |||||||||||
= = = |
= |
= = = |
= |
= = = |
= |
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{6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
Uniform duals | |||||||||||
V64 | V4.12.12 | V(4.6)^{2} | V6.8.8 | V4^{6} | 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) | |||||
= |
= |
= |
= |
= |
= |
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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) | |||||||
h{6,4} t_{0}(4,4,3) |
h_{2}{6,4} t_{0,1}(4,4,3) |
{4,6}1/2 t_{1}(4,4,3) |
h_{2}{6,4} t_{1,2}(4,4,3) |
h{6,4} t_{2}(4,4,3) |
r{6,4}^{1}/_{2} t_{0,2}(4,4,3) |
t{4,6}^{1}/_{2} t_{0,1,2}(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} h_{1}(4,3,4) |
Uniform duals | ||||||||||
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} | V6^{6} | V4.3.4.6.6 |
Uniform tilings in symmetry *3222 | ||||
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6^{4} |
6.6.4.4 |
(3.4.4)^{2} |
4.3.4.3.3.3 | |
6.6.4.4 |
6.4.4.4 |
3.4.4.4.4 | ||
(3.4.4)^{2} |
3.4.4.4.4 |
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