infinite-order apeirogonal tiling

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|Uii_0}}

The infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.

Symmetry

This tiling represents the fundamental domains of *∞{{sup|∞}} symmetry.

This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.

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|120px
symmetry:
[(∞,∞,∞)] {{CDD|labelinfin|branch|split2-ii|node}}

|120px
t{{sub|0}}{(∞,∞,∞)}
{{CDD|labelinfin|branch_01rd|split2-ii|node}}

|120px
t{{sub|1}}{(∞,∞,∞)}
{{CDD|labelinfin|branch|split2-ii|node_1}}

|120px
t{{sub|2}}{(∞,∞,∞)}
{{CDD|labelinfin|branch_10ru|split2-ii|node}}

The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain.

: a{∞,∞} or {{CDD|node_h3|infin|node|infin|node}} = {{CDD|labelinfin|branch_01rd|split2-ii|node}} ∪ {{CDD|labelinfin|branch_10ru|split2-ii|node}}

John Horton Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, {{ISBN|978-1-56881-220-5}} (Chapter 19, The Hyperbolic Archimedean Tessellations)
{{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}

{{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
{{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
[http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery] {{Webarchive|url=https://web.archive.org/web/20130324095520/http://bork.hampshire.edu/~bernie/hyper/ |date=2013-03-24 }}
[http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
[http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]