Order-4 hexagonal tiling | |
---|---|
Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | 6^{4} |
Schläfli symbol | {6,4} |
Wythoff symbol | 4 | 6 2 |
Coxeter diagram | |
Symmetry group | [6,4], (*642) |
Dual | Order-6 square tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.
Contents
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *222222 with 6 order-2 mirror intersections. In Coxeter notation can be represented as [6^{*},4], removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry. Adding all 6 bisectors leads to full *642 symmetry.
*222222 |
*443 |
*3222 |
*642 |
Uniform colorings
There are 7 distinct uniform colorings for the order-4 hexagonal tiling. They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions and Coxeter diagrams while three of them are undercolorings.
1 color | 2 colors | 3 and 2 colors | 4, 3 and 2 colors | ||||
---|---|---|---|---|---|---|---|
Uniform Coloring |
(1111) |
(1212) |
(1213) |
(1113) |
(1234) |
(1123) |
(1122) |
Symmetry | [6,4] (*642) |
[6,6] (*662) = |
[(6,6,3)] = [6,6,1^{+}] (*663) = |
[1^{+},6,6,1^{+}] (*3333) = = | |||
Symbol | {6,4} | r{6,6} = {6,4}^{1}/_{2} | r(6,3,6) = r{6,6}^{1}/_{2} | r{6,6}^{1}/_{4} | |||
Coxeter diagram |
= | = | = = |
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.
*n62 symmetry mutation of regular tilings: {6,n} | ||||||||
---|---|---|---|---|---|---|---|---|
Spherical | Euclidean | Hyperbolic tilings | ||||||
{6,2} |
{6,3} |
{6,4} |
{6,5} |
{6,6} |
{6,7} |
{6,8} |
... | {6,∞} |
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.
*n42 symmetry mutation of regular tilings: {n,4} | |||||||
---|---|---|---|---|---|---|---|
Spherical | Euclidean | Hyperbolic tilings | |||||
2^{4} | 3^{4} | 4^{4} | 5^{4} | 6^{4} | 7^{4} | 8^{4} | ...∞^{4} |
Symmetry mutation of quasiregular tilings: 6.n.6.n | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry *6n2 [n,6] |
Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||||||
*632 [3,6] |
*642 [4,6] |
*652 [5,6] |
*662 [6,6] |
*762 [7,6] |
*862 [8,6]... |
*∞62 [∞,6] |
[iπ/λ,6] | ||||
Quasiregular figures configuration |
6.3.6.3 |
6.4.6.4 |
6.5.6.5 |
6.6.6.6 |
6.7.6.7 |
6.8.6.8 |
6.∞.6.∞ |
6.∞.6.∞ | |||
Dual figures | |||||||||||
Rhombic figures configuration |
V6.3.6.3 |
V6.4.6.4 |
V6.5.6.5 |
V6.6.6.6 |
V6.7.6.7 |
V6.8.6.8 |
V6.∞.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) | |||||||||||
= = = |
= |
= = = |
= |
= = = |
= |
||||||
{6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4} | |||||
Uniform duals | |||||||||||
V6^{4} | 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) | |||||
= |
= |
= |
= |
= |
= |
||||||
h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} |
Uniform hexahexagonal tilings | ||||||
---|---|---|---|---|---|---|
Symmetry: [6,6], (*662) | ||||||
= = |
= = |
= = |
= = |
= = |
= = |
= = |
{6,6} = h{4,6} |
t{6,6} = h_{2}{4,6} |
r{6,6} {6,4} |
t{6,6} = h_{2}{4,6} |
{6,6} = h{4,6} |
rr{6,6} r{6,4} |
tr{6,6} t{6,4} |
Uniform duals | ||||||
V6^{6} | V6.12.12 | V6.6.6.6 | V6.12.12 | V6^{6} | V4.6.4.6 | V4.12.12 |
Alternations | ||||||
[1^{+},6,6] (*663) |
[6^{+},6] (6*3) |
[6,1^{+},6] (*3232) |
[6,6^{+}] (6*3) |
[6,6,1^{+}] (*663) |
[(6,6,2^{+})] (2*33) |
[6,6]^{+} (662) |
= | = | = | ||||
h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
Similar H2 tilings in *3232 symmetry | ||||||||
---|---|---|---|---|---|---|---|---|
Coxeter diagrams |
||||||||
Vertex figure |
6^{6} | (3.4.3.4)^{2} | 3.4.6.6.4 | 6.4.6.4 | ||||
Image | ||||||||
Dual |
Uniform tilings in symmetry *3222 | ||||
---|---|---|---|---|
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 |
4^{6} |
See also
Wikimedia Commons has media related to Order-4 hexagonal tiling. |
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.