Alternated order-4 hexagonal tiling
Appearance
Alternated order-4 hexagonal tiling | |
---|---|
![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | (3.4)4 |
Schläfli symbol | h{6,4} or (3,4,4) |
Wythoff symbol | 4 | 3 4 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [(4,4,3)], (*443) |
Dual | Order-4-4-3_t0 dual tiling |
Properties | Vertex-transitive |
In geometry, the alternated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}.
Uniform constructions
[edit]There are four uniform constructions, with some of lower ones which can be seen with two colors of triangles:
*443 | 3333 | *3232 | 3*22 |
---|---|---|---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() | ||
(4,4,3) = h{6,4} | hr{6,6} = h{6,4}1⁄2 |
Related polyhedra and tiling
[edit]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} |
Uniform hexahexagonal tilings | ||||||
---|---|---|---|---|---|---|
Symmetry: [6,6], (*662) | ||||||
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() = ![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
{6,6} = h{4,6} |
t{6,6} = h2{4,6} |
r{6,6} {6,4} |
t{6,6} = h2{4,6} |
{6,6} = h{4,6} |
rr{6,6} r{6,4} |
tr{6,6} t{6,4} |
Uniform duals | ||||||
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
V66 | V6.12.12 | V6.6.6.6 | V6.12.12 | V66 | 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} |
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 |
Similar H2 tilings in *3232 symmetry | ||||||||
---|---|---|---|---|---|---|---|---|
Coxeter diagrams |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() | ||||
![]() ![]() ![]() ![]() |
![]() ![]() ![]() |
![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
![]() ![]() ![]() |
![]() ![]() ![]() |
![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() | |
![]() ![]() ![]() |
![]() ![]() ![]() |
![]() ![]() ![]() |
![]() ![]() ![]() | |||||
Vertex figure |
66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
Image | ![]() |
![]() |
![]() |
![]() | ||||
Dual | ![]() |
![]() |
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.
See also
[edit]![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
Wikimedia Commons has media related to Uniform tiling 3-4-3-4-3-4-3-4.
External links
[edit]- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery Archived 2013-03-24 at the Wayback Machine
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch