Truncated tetraapeirogonal tiling

Truncated tetraapeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.8.
Schläfli symboltr{,4} or
Wythoff symbol2 4 |
Coxeter diagram or
Symmetry group[,4], (*42)
DualOrder 4-infinite kisrhombille
PropertiesVertex-transitive

In geometry, the truncated tetrapeirogonal 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}.

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

Wikimedia Commons has media related to Uniform tiling 4-8-i.

References

This article is issued from Wikipedia - version of the 4/3/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.