In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.
Tetrahexagonal tiling | |
---|---|
Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | (4.6)^{2} |
Schläfli symbol | r{6,4} or rr{6,6} r(4,4,3) t_{0,1,2,3}(∞,3,∞,3) |
Wythoff symbol | 2 | 6 4 |
Coxeter diagram | or or |
Symmetry group | [6,4], (*642) [6,6], (*662) [(4,4,3)], (*443) [(∞,3,∞,3)], (*3232) |
Dual | Order-6-4 quasiregular rhombic tiling |
Properties | Vertex-transitive edge-transitive |
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [6,4] kaleidoscope. Removing the last mirror, [6,4,1^{+}], gives [6,6], (*662). Removing the first mirror [1^{+},6,4], gives [(4,4,3)], (*443). Removing both mirror as [1^{+},6,4,1^{+}], leaving [(3,∞,3,∞)] (*3232).
Uniform Coloring |
||||
---|---|---|---|---|
Fundamental Domains |
||||
Schläfli | r{6,4} | r{4,6}^{1}⁄_{2} | r{6,4}^{1}⁄_{2} | r{6,4}^{1}⁄_{4} |
Symmetry | [6,4] (*642) |
[6,6] = [6,4,1^{+}] (*662) |
[(4,4,3)] = [1^{+},6,4] (*443) |
[(∞,3,∞,3)] = [1^{+},6,4,1^{+}] (*3232) or |
Symbol | r{6,4} | rr{6,6} | r(4,3,4) | t_{0,1,2,3}(∞,3,∞,3) |
Coxeter diagram |
= | = | = or |
The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) orbifold.
In the geometry of hyperbolic 3-space, the cube-octahedron honeycomb is a compact uniform honeycomb, constructed from cube, octahedron, and cuboctahedron cells, in a rhombicuboctahedron vertex figure. It has a single-ring Coxeter diagram, , and is named by its two regular cells.
A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in non-Euclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
CubohemioctahedronIn geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral.
It is given Wythoff symbol 4/3 4 | 3, although that is a double-covering of this figure.
A nonconvex polyhedron has intersecting faces which do not represent new edges or faces. In the picture vertices are marked by golden spheres, and edges by silver cylinders.
It is a hemipolyhedron with 4 hexagonal faces passing through the model center. The hexagons intersect each other and so only triangle portions of each are visible.
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 tetrahexagonal tilingIn geometry, the snub tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,4}.
Snub trihexagonal tilingIn geometry, the snub hexagonal tiling (or snub trihexagonal tiling) is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol of sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.
Conway calls it a snub hextille, constructed as a snub operation applied to a hexagonal tiling (hextille).
There are 3 regular and 8 semiregular tilings in the plane. This is the only one which does not have a reflection as a symmetry.
There is only one uniform coloring of a snub trihexagonal tiling. (Naming the colors by indices (3.3.3.3.6): 11213.)
Truncated pentahexagonal tilingIn geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one decagon, and one dodecagon on each vertex. It has Schläfli symbol of t0,1,2{6,5}. Its name is somewhat misleading: literal geometric truncation of pentahexagonal tiling produces rectangles instead of squares.
Truncated tetrahexagonal tilingIn geometry, the truncated tetrahexagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{6,4}.
*n42 symmetry mutations of quasiregular tilings: (4.n)^{2} | ||||||||
---|---|---|---|---|---|---|---|---|
Symmetry *4n2 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] |
[ni,4] | |
Figures | ||||||||
Config. | (4.3)^{2} | (4.4)^{2} | (4.5)^{2} | (4.6)2 | (4.7)^{2} | (4.8)^{2} | (4.∞)^{2} | (4.ni)^{2} |
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} |
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 |
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 |
This page is based on a Wikipedia article written by authors
(here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.