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}.

Truncated tetrahexagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.8.12
Schläfli symboltr{6,4} or
Wythoff symbol
Coxeter diagram or
Symmetry group[6,4], (*642)
DualOrder-4-6 kisrhombille tiling
PropertiesVertex-transitive

Dual tiling
The dual tiling is called an order-4-6 kisrhombille tiling, made as a complete bisection of the order-4 hexagonal tiling, here with triangles shown in alternating colors. This tiling represents the fundamental triangular domains of [6,4] (*642) symmetry.

From a Wythoff construction there are fourteen 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 7 forms with full [6,4] symmetry, and 7 with subsymmetry.

Symmetry
Truncated tetrahexagonal tiling with mirror lines in green, red, and blue:
Symmetry diagrams for small index subgroups of [6,4], shown in a hexagonal translational cell within a {6,6} tiling, with a fundamental domain in yellow.

The dual of the tiling represents the fundamental domains of (*642) orbifold symmetry. From [6,4] 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 alternately colored triangles show the location of gyration points. The [6+,4+], (32×) subgroup has narrow lines representing glide reflections. The subgroup index-8 group, [1+,6,1+,4,1+] (3232) is the commutator subgroup of [6,4].

Larger subgroup constructed as [6,4*], removing the gyration points of [6,4+], (3*22), index 6 becomes (*3333), and [6*,4], removing the gyration points of [6+,4], (2*33), index 12 as (*222222). Finally their direct subgroups [6,4*]+, [6*,4]+, subgroup indices 12 and 24 respectively, can be given in orbifold notation as (3333) and (222222).

Small index subgroups of [6,4]
Index 1 2 4
Diagram
Coxeter [6,4]
= 		[1+,6,4]
= 		[6,4,1+]
= 		[6,1+,4]
= 		[1+,6,4,1+]
= 		[6+,4+]
Orbifold *642 *443 *662 *3222 *3232 32×
Semidirect subgroups
Diagram
Coxeter [6,4+]
 [6+,4]
 [(6,4,2+)]
 [6,1+,4,1+]
= =
= = 		[1+,6,1+,4]
= =
= =
Orbifold 4*3 6*2 2*32 2*33 3*22
Direct subgroups
Index 2 4 8
Diagram
Coxeter [6,4]+
= 		[6,4+]+
= 		[6+,4]+
= 		[(6,4,2+)]+
= 		[6+,4+]+ = [1+,6,1+,4,1+]
= = =
Orbifold 642 443 662 3222 3232
Radical subgroups
Index 8 12 16 24
Diagram
Coxeter [6,4*]
= 		[6*,4]
 [6,4*]+
= 		[6*,4]+
Orbifold *3333 *222222 3333 222222

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.
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