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Order-8 square tiling

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Order-8 square tiling
Order-8 square tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 4
Schläfli symbol {4,8}
Wythoff symbol 8 | 4 2
Coxeter diagram
Symmetry group , (*842)
Dual Order-4 octagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-8 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,8}.

Symmetry

This tiling represents a hyperbolic kaleidoscope of 4 mirrors meeting as edges of a square, with eight squares around every vertex. This symmetry by orbifold notation is called (*4444) with 4 order-4 mirror intersections. In Coxeter notation can be represented as , (*4444 orbifold) removing two of three mirrors (passing through the square center) in the symmetry. The *4444 symmetry can be doubled by bisecting the fundamental domain (square) by a mirror, creating *884 symmetry.

This bicolored square tiling shows the even/odd reflective fundamental square domains of this symmetry. This bicolored tiling has a wythoff construction (4,4,4), or {4}, :

Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4).

*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact

{4,3}

{4,4}

{4,5}

{4,6}

{4,7}

{4,8}...

{4,∞}
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 (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

Tessellation
Periodic


Aperiodic
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By vertex type
Spherical
Regular
Semi-
regular
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bolic
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