Truncated order-6 square tiling

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

Truncated order-6 square tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration8.8.6
Schläfli symbolt{4,6}
Wythoff symbol2 6 | 4
Coxeter diagram
Symmetry group[6,4], (*642)
[(3,3,4)], (*334)
DualOrder-4 hexakis hexagonal tiling
PropertiesVertex-transitive

Uniform colorings

The half symmetry [1+,6,4] = [(4,4,3)] can be shown with alternating two colors of octagons, with as Coxeter diagram .

Symmetry
Truncated order-6 square tiling with *443 symmetry mirror lines

The dual tiling represents the fundamental domains of the *443 orbifold symmetry. There are two reflective subgroup kaleidoscopic constructed from [(4,4,3)] by removing one or two of three mirrors. In these images fundamental domains are alternately colored black and cyan, and mirrors exist on the boundaries between colors.

A larger subgroup is constructed [(4,4,3*)], index 6, as (3*22) with gyration points removed, becomes (*222222).

The symmetry can be doubled as 642 symmetry by adding a mirror bisecting the fundamental domain.

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular order-4 hexagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

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)

=

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{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
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)

=
=
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h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}

It can also be generated from the (4 4 3) hyperbolic tilings:

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}
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
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
*n42 symmetry mutation of truncated tilings: n.8.8
Symmetry
*n42
[n,4]		 Spherical Euclidean Compact hyperbolic Paracompact
*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. 2.8.8 3.8.8 4.8.8 5.8.8 6.8.8 7.8.8 8.8.8 ∞.8.8
n-kis
figures
Config. V2.8.8 V3.8.8 V4.8.8 V5.8.8 V6.8.8 V7.8.8 V8.8.8 V∞.8.8
*n32 symmetry mutation of omnitruncated tilings: 6.8.2n
Sym.
*n43
[(n,4,3)]		 Spherical Compact hyperbolic Paraco.
*243
[4,3]		 *343
[(3,4,3)]		 *443
[(4,4,3)]		 *543
[(5,4,3)]		 *643
[(6,4,3)]		 *743
[(7,4,3)]		 *843
[(8,4,3)]		 *∞43
[(∞,4,3)]
Figures
Config. 4.8.6 6.8.6 8.8.6 10.8.6 12.8.6 14.8.6 16.8.6 ∞.8.6
Duals
Config. V4.8.6 V6.8.6 V8.8.6 V10.8.6 V12.8.6 V14.8.6 V16.8.6 V6.8.∞

See also
Wikimedia Commons has media related to Uniform tiling 6-8-8.

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.
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