Infinite-order apeirogonal tiling

<h2 id="symmetry">Symmetry</h2>
<p>This tiling represents the fundamental domains of *∞∞ symmetry.
</p>
<h2 id="uniform-colorings">Uniform colorings</h2>
<p>This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.
</p>
<table><tbody><tr><th><a href="/facts/Fundamental_domain/e68KDYWu">Domains</a></th><th>0</th><th>1</th><th>2</th></tr><tr><td>symmetry:[(∞,∞,∞)]  </td><td>t0{(∞,∞,∞)}</td><td>t1{(∞,∞,∞)}</td><td>t2{(∞,∞,∞)}</td></tr></tbody></table>
<h2 id="related-polyhedra-and-tiling">Related polyhedra and tiling</h2>
<p>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.
</p>

a{∞,∞} or  =  ∪ 
<table><tbody><tr><th colspan="7">Paracompact uniform tilings in [∞,∞] family <ul><li>v</li><li>t</li><li>e</li></ul></th></tr><tr><td>= = </td><td>= = </td><td>= = </td><td>= = </td><td>= = </td><td>= </td><td>= </td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>{∞,∞}</td><td><a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">t{∞,∞}</a></td><td><a href="/facts/Order-4_apeirogonal_tiling/dMECKVSx">r{∞,∞}</a></td><td><a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">2t{∞,∞}=t{∞,∞}</a></td><td>2r{∞,∞}={∞,∞}</td><td>rr{∞,∞}</td><td><a href="/facts/Truncated_order-4_apeirogonal_tiling/fqY73G6c">tr{∞,∞}</a></td></tr><tr><th colspan="7">Dual tilings</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>V∞∞</td><td>V∞.∞.∞</td><td>V(∞.∞)2</td><td>V∞.∞.∞</td><td>V∞∞</td><td>V4.∞.4.∞</td><td>V4.4.∞</td></tr><tr><th colspan="7">Alternations</th></tr><tr><td>[1+,∞,∞](*∞∞2)</td><td>[∞+,∞](∞*∞)</td><td>[∞,1+,∞](*∞∞∞∞)</td><td>[∞,∞+](∞*∞)</td><td>[∞,∞,1+](*∞∞2)</td><td>[(∞,∞,2+)](2*∞∞)</td><td>[∞,∞]+(2∞∞)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>h{∞,∞}</td><td>s{∞,∞}</td><td>hr{∞,∞}</td><td>s{∞,∞}</td><td>h2{∞,∞}</td><td>hrr{∞,∞}</td><td><a href="/facts/Snub_apeiroapeirogonal_tiling/y7XVR7QV">sr{∞,∞}</a></td></tr><tr><th colspan="7">Alternation duals</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>V(∞.∞)∞</td><td>V(3.∞)3</td><td>V(∞.4)4</td><td>V(3.∞)3</td><td>V∞∞</td><td>V(4.∞.4)2</td><td>V3.3.∞.3.∞</td></tr></tbody></table>
<table><tbody><tr><th colspan="7">Paracompact uniform tilings in [(∞,∞,∞)] family <ul><li>v</li><li>t</li><li>e</li></ul></th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(∞,∞,∞)h{∞,∞}</td><td><a href="/facts/Order-4_apeirogonal_tiling/dMECKVSx">r(∞,∞,∞)</a>h2{∞,∞}</td><td>(∞,∞,∞)h{∞,∞}</td><td><a href="/facts/Order-4_apeirogonal_tiling/dMECKVSx">r(∞,∞,∞)</a>h2{∞,∞}</td><td>(∞,∞,∞)h{∞,∞}</td><td><a href="/facts/Order-4_apeirogonal_tiling/dMECKVSx">r(∞,∞,∞)</a>r{∞,∞}</td><td><a href="/facts/Order-3_apeirogonal_tiling/UEkWcKPh">t(∞,∞,∞)</a>t{∞,∞}</td></tr><tr><th colspan="7">Dual tilings</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>V∞∞</td><td>V∞.∞.∞.∞</td><td>V∞∞</td><td>V∞.∞.∞.∞</td><td>V∞∞</td><td>V∞.∞.∞.∞</td><td>V∞.∞.∞</td></tr><tr><th colspan="7">Alternations</th></tr><tr><td>[(1+,∞,∞,∞)](*∞∞∞∞)</td><td>[∞+,∞,∞)](∞*∞)</td><td>[∞,1+,∞,∞)](*∞∞∞∞)</td><td>[∞,∞+,∞)](∞*∞)</td><td>[(∞,∞,∞,1+)](*∞∞∞∞)</td><td>[(∞,∞,∞+)](∞*∞)</td><td>[∞,∞,∞)]+(∞∞∞)</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th colspan="7">Alternation duals</th></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>V(∞.∞)∞</td><td>V(∞.4)4</td><td>V(∞.∞)∞</td><td>V(∞.4)4</td><td>V(∞.∞)∞</td><td>V(∞.4)4</td><td>V3.∞.3.∞.3.∞</td></tr></tbody></table>
<h2 id="see-also">See also</h2>

Wikimedia Commons has media related to Infinite-order apeirogonal tiling.

<ul><li><a href="/facts/Tilings_of_regular_polygons/tR5CQACy">Tilings of regular polygons</a></li>
<li><a href="/facts/List_of_uniform_planar_tilings/c1ALUDMm">List of uniform planar tilings</a></li>
<li><a href="/facts/List_of_regular_polytopes/Kng25Ymq">List of regular polytopes</a></li></ul>

<ul><li><a href="/facts/John_Horton_Conway/g3PA1zoK">John Horton Conway</a>, Heidi Burgiel, Chaim Goodman-Strauss, <i>The Symmetries of Things</i> 2008, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)</li>
<li>"Chapter 10: Regular honeycombs in hyperbolic space". <i>The Beauty of Geometry: Twelve Essays</i>. Dover Publications. 1999. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-40919-8. <a href="/facts/LCCN_(identifier)/5hb5Cw2V">LCCN</a> <a href="https://lccn.loc.gov/99035678">99035678</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/HyperbolicTiling.html">"Hyperbolic tiling"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/PoincareHyperbolicDisk.html">"Poincaré hyperbolic disk"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li>
<li><a href="http://bork.hampshire.edu/~bernie/hyper/">Hyperbolic and Spherical Tiling Gallery</a> <a href="https://web.archive.org/web/20130324095520/http://bork.hampshire.edu/~bernie/hyper/">Archived</a> 2013-03-24 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a></li>
<li><a href="http://geometrygames.org/KaleidoTile/index.html">KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings</a></li>
<li><a href="http://www.plunk.org/~hatch/HyperbolicTesselations">Hyperbolic Planar Tessellations, Don Hatch</a></li></ul>

Infinite-order apeirogonal tiling open-in-new

Infinite-order apeirogonal tiling