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Tetraoctagonal tiling

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Tetraoctagonal tiling
Tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.8)
Schläfli symbol r{8,4} or { 8 4 } {\displaystyle {\begin{Bmatrix}8\\4\end{Bmatrix}}}
rr{8,8}
rr(4,4,4)
t0,1,2,3(∞,4,∞,4)
Wythoff symbol 2 | 8 4
Coxeter diagram or
or

Symmetry group , (*842)
, (*882)
, (*444)
, (*4242)
Dual Order-8-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, , gives , (*882). Removing the mirror between the order 2 and 8 points, , gives , (*444). Removing both mirrors, , leaves a rectangular fundamental domain, , (*4242).

Four uniform constructions of 4.8.4.8
Name Tetra-octagonal tiling Rhombi-octaoctagonal tiling
Image
Symmetry
(*842)
 =
(*882)
= 		=
(*444)
= 		=
(*4242)
= or
Schläfli r{8,4} rr{8,8}
=r{8,4}/2 		r(4,4,4)
=r{4,8}/2 		t0,1,2,3(∞,4,∞,4)
=r{8,4}/4
Coxeter = = = or

Symmetry

The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.

Related polyhedra and tiling

*n42 symmetry mutations of quasiregular tilings: (4.n)
Symmetry
*4n2
 Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
 *442
 *542
 *642
 *742
 *842
... 		*∞42
  
Figures
Config. (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.∞) (4.ni)
Dimensional family of quasiregular polyhedra and tilings: (8.n)
Symmetry
*8n2
 Hyperbolic... Paracompact Noncompact
*832
 *842
 *852
 *862
 *872
 *882
... 		*∞82
  
Coxeter
Quasiregular
figures
configuration
3.8.3.8
4.8.4.8
8.5.8.5
8.6.8.6
8.7.8.7
8.8.8.8
8.∞.8.∞ 		 
8.∞.8.∞
Uniform octagonal/square tilings
, (*842)
(with (*882), (*444) , (*4222) index 2 subsymmetries)
(And (*4242) index 4 subsymmetry)

=

=
=
=
=
=

=

=

=
=


=
{8,4} t{8,4}
 r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
V8 V4.16.16 V(4.8) V8.8.8 V4 V4.4.4.8 V4.8.16
Alternations

(*444)
(8*2)
(*4222)
(4*4)
(*882)
(2*42)
(842)

=
=
=
=
=
=
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
V(4.4) V3.(3.8) V(4.4.4) V(3.4) V8 V4.4 V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: , (*882)
=
= 		=
= 		=
= 		=
= 		=
= 		=
= 		=
=
{8,8} t{8,8}
 r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
V8 V8.16.16 V8.8.8.8 V8.16.16 V8 V4.8.4.8 V4.16.16
Alternations

(*884)
(8*4)
(*4242)
(8*4)
(*884)
(2*44)
(882)
= = = =
= 		=
=
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
V(4.8) V3.4.3.8.3.8 V(4.4) V3.4.3.8.3.8 V(4.8) V4 V3.3.8.3.8
Uniform (4,4,4) tilings
Symmetry: , (*444)
(444)
(*4242)
(4*22)










t0(4,4,4)
h{8,4} 		t0,1(4,4,4)
h2{8,4} 		t1(4,4,4)
{4,8}/2 		t1,2(4,4,4)
h2{8,4} 		t2(4,4,4)
h{8,4} 		t0,2(4,4,4)
r{4,8}/2 		t0,1,2(4,4,4)
t{4,8}/2 		s(4,4,4)
s{4,8}/2 		h(4,4,4)
h{4,8}/2 		hr(4,4,4)
hr{4,8}/2
Uniform duals
V(4.4) V4.8.4.8 V(4.4) V4.8.4.8 V(4.4) V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V8 V(4,4)

See also

References

External links

Categories: