Tetrahexagonal tiling

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.

Tetrahexagonal tiling
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)
t0,1,2,3(∞,3,∞,3)
Wythoff symbol 2 | 6 4
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png or CDel node 1.pngCDel split1-64.pngCDel nodes.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png or CDel node.pngCDel split1-66.pngCDel nodes 11.png
CDel branch 11.pngCDel split2-44.pngCDel node.png
CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png
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

Constructions

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).

Four uniform constructions of 4.6.4.6
Uniform
Coloring
H2 tiling 246-2 H2 tiling 266-5 H2 tiling 344-5 3222-uniform tiling-verf4646
Fundamental
Domains
642 symmetry 000 642 symmetry 00a 642 symmetry a00 642 symmetry a0b
Schläfli r{6,4} r{4,6}​12 r{6,4}​12 r{6,4}​14
Symmetry [6,4]
(*642)
CDel node c3.pngCDel 6.pngCDel node c1.pngCDel 4.pngCDel node c2.png
[6,6] = [6,4,1+]
(*662)
CDel node c3.pngCDel split1-66.pngCDel nodeab c1.png
[(4,4,3)] = [1+,6,4]
(*443)
CDel branch c1.pngCDel split2-44.pngCDel node c2.png
[(∞,3,∞,3)] = [1+,6,4,1+]
(*3232)
CDel labelinfin.pngCDel branch c1.pngCDel 3ab.pngCDel branch c1.pngCDel labelinfin.png or CDel nodeab c1.pngCDel 3a3b-cross.pngCDel nodeab c1.png
Symbol r{6,4} rr{6,6} r(4,3,4) t0,1,2,3(∞,3,∞,3)
Coxeter
diagram
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node h0.png = CDel node.pngCDel split1-66.pngCDel nodes 11.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png = CDel branch 11.pngCDel split2-44.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node h0.png =
CDel labelinfin.pngCDel branch 11.pngCDel 3ab.pngCDel branch 11.pngCDel labelinfin.png or CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png

Symmetry

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.

Hyperbolic domains 3232
Ord64 qreg rhombic til
H2chess 246a
Order-6 hexagonal tiling and dual
Hyperbolic domains 3232
Ord64 qreg rhombic til
H2chess 246a
Order-6 hexagonal tiling and dual

See also

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.

External links

Cubic-octahedral honeycomb

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.

Cubohemioctahedron

In 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 tiling

In 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 tiling

In geometry, the snub tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,4}.

Snub trihexagonal tiling

In 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 tiling

In 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 tiling

In 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 Uniform tiling 432-t1 Uniform tiling 44-t1 H2 tiling 245-2 H2 tiling 246-2 H2 tiling 247-2 H2 tiling 248-2 H2 tiling 24i-2
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
Uniform tiling 63-t1
6.3.6.3
H2 tiling 246-2
6.4.6.4
H2 tiling 256-2
6.5.6.5
H2 tiling 266-2
6.6.6.6
H2 tiling 267-2
6.7.6.7
H2 tiling 268-2
6.8.6.8
H2 tiling 26i-2
6.∞.6.∞

6.∞.6.∞
Dual figures
Rhombic
figures
configuration
Rhombic star tiling
V6.3.6.3
H2chess 246a
V6.4.6.4
Order-6-5 quasiregular rhombic tiling
V6.5.6.5
H2 tiling 246-4
V6.6.6.6

V6.7.6.7
H2chess 268a
V6.8.6.8
H2chess 26ia
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)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes.png
= CDel branch 11.pngCDel 3a3b-cross.pngCDel branch 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node 1.pngCDel split1-66.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png
= CDel node.pngCDel split1-66.pngCDel nodes 11.png
= CDel branch 11.pngCDel split2-44.pngCDel node.png
CDel 2.png
= CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes 11.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch 11.pngCDel split2-44.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
= CDel branch.pngCDel split2-44.pngCDel node 1.png
= CDel branch.pngCDel 2a2b-cross.pngCDel nodes 11.png
= CDel branchu 11.pngCDel 2.pngCDel branchu 11.pngCDel 2.pngCDel branchu 11.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel 2.png
CDel 2.png
= CDel branch 11.pngCDel 2a2b-cross.pngCDel nodes 11.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 246-1 H2 tiling 246-3 H2 tiling 246-2 H2 tiling 246-6 H2 tiling 246-4 H2 tiling 246-5 H2 tiling 246-7
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 246b H2chess 246f H2chess 246a H2chess 246e H2chess 246c H2chess 246d H2checkers 246
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)
CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
= CDel branch 10ru.pngCDel split2-44.pngCDel node.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.png
= CDel node h.pngCDel split1-66.pngCDel branch hh.pngCDel label2.png
CDel node.pngCDel 6.pngCDel node h1.pngCDel 4.pngCDel node.png
= CDel branch 10.pngCDel 2a2b-cross.pngCDel nodes 10.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel split2-44.pngCDel node h.png
CDel node.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-66.pngCDel nodes 10lu.png
CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.png
= CDel branch hh.pngCDel 2xa2xb-cross.pngCDel branch hh.pngCDel label2.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
Uniform tiling 443-t0 Uniform tiling 64-h02 Uniform tiling 64-h1 Uniform tiling 443-snub2 Uniform tiling 66-t0 Uniform tiling 3.4.4.4.4 Uniform tiling 64-snub
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)
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png = CDel nodes 10ru.pngCDel split2-66.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png = CDel nodes 10ru.pngCDel split2-66.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png = CDel nodes.pngCDel split2-66.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-66.pngCDel node 1.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png = CDel nodes 01rd.pngCDel split2-66.pngCDel node.png
= CDel node h1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node 1.png = CDel nodes 11.pngCDel split2-66.pngCDel node.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node.png
CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node 1.png =CDel nodes 11.pngCDel split2-66.pngCDel node 1.png
= CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node 1.png
H2 tiling 266-1 H2 tiling 266-3 H2 tiling 266-2 H2 tiling 266-6 H2 tiling 266-4 H2 tiling 266-5 H2 tiling 266-7
{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
CDel node f1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node f1.png
H2chess 266b H2chess 266f H2chess 266a H2chess 266e H2chess 266c H2chess 266d H2checkers 266
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)
CDel node h1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png = CDel branch 10ru.pngCDel split2-66.pngCDel node.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node h1.pngCDel 6.pngCDel node.png = CDel nodes 11.pngCDel 3a3b-cross.pngCDel nodes.png CDel node.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h1.png = CDel node.pngCDel split1-66.pngCDel branch 01ld.png CDel node h.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node h1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png CDel node.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h1.png CDel node h.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node h.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 6.pngCDel node h.png
Uniform tiling 66-h0 Uniform tiling verf 34343434 Uniform tiling 66-h0 Uniform tiling 64-h1 Uniform tiling 66-snub
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)
CDel branch 01rd.pngCDel split2-44.pngCDel node.png CDel branch 01rd.pngCDel split2-44.pngCDel node 1.png CDel branch.pngCDel split2-44.pngCDel node 1.png CDel branch 10ru.pngCDel split2-44.pngCDel node 1.png CDel branch 10ru.pngCDel split2-44.pngCDel node.png CDel branch 11.pngCDel split2-44.pngCDel node.png CDel branch 11.pngCDel split2-44.pngCDel node 1.png CDel branch hh.pngCDel split2-44.pngCDel node h.png CDel branch hh.pngCDel split2-44.pngCDel node.png CDel branch.pngCDel split2-44.pngCDel node h.png CDel branch 10ru.pngCDel split2-44.pngCDel node h.png
CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node h0.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node h0.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node.png CDel node h0.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png
Uniform tiling 443-t0 Uniform tiling 443-t01 Uniform tiling 443-t1 Uniform tiling 443-t12 Uniform tiling 443-t2 Uniform tiling 443-t02 Uniform tiling 443-t012 Uniform tiling 443-snub1 Uniform tiling 64-h1 Uniform tiling 66-t2 Uniform tiling verf 34664
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
Uniform tiling 66-t1 Ord64 qreg rhombic til Order4 hexakis hexagonal til Uniform tiling 66-t0
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
CDel node h0.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h0.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node h1.png CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 4.pngCDel node h0.png
CDel labelh.pngCDel node.pngCDel split1-66.pngCDel nodes 10lu.png CDel branch.pngCDel split2-44.pngCDel node h1.png CDel node h1.pngCDel split1-66.pngCDel nodes.png CDel branch 10ru.pngCDel split2-44.pngCDel node.pngCDel labelh.png CDel node h1.pngCDel split1-66.pngCDel nodes 10lu.png CDel branch 10ru.pngCDel split2-44.pngCDel node h1.png CDel labelh.pngCDel node.pngCDel split1-66.pngCDel nodes 11.png CDel branch 11.pngCDel split2-44.pngCDel node.pngCDel labelh.png
CDel branch 11.pngCDel 2a2b-cross.pngCDel branch.png CDel branch 10.pngCDel 2a2b-cross.pngCDel branch 10.png CDel branch 10.pngCDel 2a2b-cross.pngCDel branch 11.png CDel branch 11.pngCDel 2a2b-cross.pngCDel branch 11.png
Vertex
figure
66 (3.4.3.4)2 3.4.6.6.4 6.4.6.4
Image Uniform tiling verf 666666 Uniform tiling verf 34343434 Uniform tiling verf 34664 3222-uniform tiling-verf4646
Dual Uniform tiling verf 666666b H2chess 246a

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