| 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 |    or   
|
| 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}.
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 | ||
| 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 | 
|
| ||||||
