Truncated tetraapeirogonal tiling

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}.

Related polyhedra and tilings

Symmetry

The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. 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 fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1⁺,∞,1⁺,4,1⁺] (∞2∞2) is the commutator subgroup of [∞,4].

A larger subgroup is constructed as [∞,4*], index 8, as [∞,4⁺], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞⁴), and another [∞*,4], index ∞ as [∞⁺,4], (∞*2) with gyration points removed as (*2^∞). And their direct subgroups [∞,4*]⁺, [∞*,4]⁺, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2^∞).

See also

• Tilings of regular polygons
• List of uniform planar tilings

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)

External links

• Hyperbolic and Spherical Tiling Gallery