Truncated order-6 square tiling | |
---|---|
![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | 8.8.6 |
Schläfli symbol | t{4,6} |
Wythoff symbol | 2 6 | 4 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [6,4], (*642) [(3,3,4)], (*334) |
Dual | Order-4 hexakis hexagonal tiling |
Properties | Vertex-transitive |
In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.
Uniform colorings
[edit]![]() The half symmetry [1+,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons, with as Coxeter diagram ![]() ![]() ![]() |
Symmetry
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Truncated_order-6_square_tiling_with_mirrors.png/220px-Truncated_order-6_square_tiling_with_mirrors.png)
The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.
A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).
The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain.
Small index subgroups of [(4,4,3)] (*443) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Index | 1 | 2 | 6 | ||||||||
Diagram | ![]() |
![]() |
![]() |
![]() | |||||||
Coxeter (orbifold) |
[(4,4,3)] = ![]() ![]() ![]() (*443) |
[(4,1+,4,3)] = ![]() ![]() ![]() ![]() ![]() ![]() ![]() (*3232) |
[(4,4,3+)] = ![]() ![]() ![]() (3*22) |
[(4,4,3*)] = ![]() ![]() ![]() ![]() (*222222) | |||||||
Direct subgroups | |||||||||||
Index | 2 | 4 | 12 | ||||||||
Diagram | ![]() |
![]() |
![]() | ||||||||
Coxeter (orbifold) |
[(4,4,3)]+ = ![]() ![]() ![]() (443) |
[(4,4,3+)]+ = ![]() ![]() ![]() ![]() ![]() ![]() ![]() (3232) |
[(4,4,3*)]+ = ![]() ![]() ![]() ![]() (222222) |
Related polyhedra and tilings
[edit]From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
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 | |||||||||||
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() | |||||
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||
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) | |||||
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() | |||||
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||||
h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4} |
It can also be generated from the (4 4 3) hyperbolic tilings:
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} 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 | ||||||||||
![]() |
![]() |
![]() |
![]() |
|||||||
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 |
*n42 symmetry mutation of truncated tilings: n.8.8 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry *n42 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | |||||||
*242 [2,4] |
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] | ||||
Truncated figures |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||
Config. | 2.8.8 | 3.8.8 | 4.8.8 | 5.8.8 | 6.8.8 | 7.8.8 | 8.8.8 | ∞.8.8 | |||
n-kis figures |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | |||
Config. | V2.8.8 | V3.8.8 | V4.8.8 | V5.8.8 | V6.8.8 | V7.8.8 | V8.8.8 | V∞.8.8 |
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Sym. *n43 [(n,4,3)] |
Spherical | Compact hyperbolic | Paraco. | |||||||||
*243 [4,3] |
*343 [(3,4,3)] |
*443 [(4,4,3)] |
*543 [(5,4,3)] |
*643 [(6,4,3)] |
*743 [(7,4,3)] |
*843 [(8,4,3)] |
*∞43 [(∞,4,3)] | |||||
Figures | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | ||||
Config. | 4.8.6 | 6.8.6 | 8.8.6 | 10.8.6 | 12.8.6 | 14.8.6 | 16.8.6 | ∞.8.6 | ||||
Duals | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() | ||||
Config. | V4.8.6 | V6.8.6 | V8.8.6 | V10.8.6 | V12.8.6 | V14.8.6 | V16.8.6 | V6.8.∞ |
See also
[edit]References
[edit]- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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.