# Truncated order-4 octagonal tiling

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Truncated order-4 octagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.16.16
Schläfli symbol t{8,4}
tr{8,8} or ${\displaystyle t{\begin{Bmatrix}8\\8\end{Bmatrix}}}$
Wythoff symbol 8
2 8 8 |
Coxeter diagram
or
Symmetry group [8,4], (*842)
[8,8], (*882)
Dual Order-8 tetrakis square tiling
Properties Vertex-transitive

In geometry, the truncated order-4 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,4}. A secondary construction t0,1,2{8,8} is called a truncated octaoctagonal tiling with two colors of hexakaidecagons.

## Constructions

There are two uniform constructions of this tiling, first by the [8,4] kaleidoscope, and second by removing the last mirror, [8,4,1+], gives [8,8], (*882).

Two uniform constructions of 4.8.4.8
NameTetraoctagonalTruncated octaoctagonal
Image
Symmetry [8,4]
(*842)
[8,8] = [8,4,1+]
(*882)
=
Symbolt{8,4}tr{8,8}
Coxeter diagram

## Dual tiling

 The dual tiling, Order-8 tetrakis square tiling has face configuration V4.16.16, and represents the fundamental domains of the [8,8] symmetry group.

## Symmetry

The dual of the tiling represents the fundamental domains of (*882) orbifold symmetry. From [8,8] symmetry, there are 15 small index subgroup by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images unique mirrors are colored red, green, and blue, and alternatively colored triangles show the location of gyration points. The [8+,8+], (44×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1+,8,1+,8,1+] (4444) is the commutator subgroup of [8,8].

One larger subgroup is constructed as [8,8*], removing the gyration points of (8*4), index 16 becomes (*44444444), and its direct subgroup [8,8*]+, index 32, (44444444).

The [8,8] symmetry can be doubled by a mirror bisecting the fundamental domain, and creating *884 symmetry.

Small index subgroups of [8,8] (*882)
Index 124
Diagram
Coxeter [8,8]
[1+,8,8]
=
[8,8,1+]
=
[8,1+,8]
=
[1+,8,8,1+]
=
[8+,8+]
Orbifold *882 *884 *4242 *4444 44×
Semidirect subgroups
Diagram
Coxeter[8,8+]
[8+,8]
[(8,8,2+)]
[8,1+,8,1+]
= =
= =
[1+,8,1+,8]
= =
= =
Orbifold8*42*444*44
Direct subgroups
Index248
Diagram
Coxeter[8,8]+
[8,8+]+
=
[8+,8]+
=
[8,1+,8]+
=
[8+,8+]+ = [1+,8,1+,8,1+]
= = =
Orbifold88288442424444
Index1632
Diagram
Coxeter[8,8*]
[8*,8]
[8,8*]+
[8*,8]+
Orbifold*4444444444444444
*n42 symmetry mutation of truncated tilings: 4.2n.2n
Symmetry
*n42
[n,4]
Spherical Euclidean Compact hyperbolicParacomp.
*242
[2,4]
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*42
[,4]
Truncated
figures
Config. 4.4.4 4.6.6 4.8.8 4.10.10 4.12.12 4.14.14 4.16.16 4..
n-kis
figures
Config. V4.4.4 V4.6.6 V4.8.8 V4.10.10V4.12.12V4.14.14V4.16.16V4..
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [,4,] (*4222) index 2 subsymmetries)
(And [(,4,,4)] (*4242) index 4 subsymmetry)

=

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=

=
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=

=

=
=

=
{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
V84 V4.16.16V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(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)4V3.(3.8)2V(4.4.4)2V(3.4)3V88V4.44V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: [8,8], (*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
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8V4.16.16
Alternations
[1+,8,8]
(*884)
[8+,8]
(8*4)
[8,1+,8]
(*4242)
[8,8+]
(8*4)
[8,8,1+]
(*884)
[(8,8,2+)]
(2*44)
[8,8]+
(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)8V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8V(4.8)8 V46 V3.3.8.3.8

## Related Research Articles

In geometry, the truncated trioctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of tr{8,3}.

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}.

In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.

In geometry, the truncated order-6 square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

In geometry, the truncated tetrapentagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1,2{4,5} or tr{4,5}.

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

In geometry, the truncated order-8 triangular tiling is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex. It has Schläfli symbol of t{3,8}.

In geometry, the truncated order-6 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically constructed as a cantic order-6 square tiling, h2{4,6}

In geometry, the truncated tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of tr{4,7}.

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

In geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

In geometry, the truncated order-8 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t0,1{8,8}.

In geometry, the order-3 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

In geometry, the truncated order-4 apeirogonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{∞,4}.

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.

In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

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.

In geometry, the truncated hexaoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

In geometry, the truncated order-6 octagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}.

In geometry, the truncated order-8 hexagonal tiling is a semiregular tiling of the hyperbolic plane. It has Schläfli symbol of t{6,8}.

## 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.