Uniform tiling

In <a href="/facts/Geometry/wTQf5ffQ">geometry</a>, a uniform tiling is a <a href="/facts/Tessellation/7BWDywXC">tessellation</a> of the plane by <a href="/facts/Regular_polygon/ws89Nhpt">regular polygon</a> faces with the restriction of being <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a>.
Uniform tilings can exist in both the <a href="/facts/Euclidean_plane/tAUtRFcn">Euclidean plane</a> and <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic plane</a>. Uniform tilings are related to the finite <a href="/facts/Uniform_polyhedra/PHnprqRC">uniform polyhedra</a>; these can be considered uniform tilings of the <a href="/facts/Sphere/jywYLNM2">sphere</a>.
Most uniform tilings can be made from a <a href="/facts/Wythoff_construction/8yTYjp5e">Wythoff construction</a> starting with a <a href="/facts/Symmetry_group/CNqeseMp">symmetry group</a> and a singular generator point inside of the <a href="/facts/Fundamental_domain/e68KDYWu">fundamental domain</a>. A planar symmetry group has a polygonal fundamental domain and can be represented by its group notation: the sequence of the reflection orders of the fundamental domain vertices.
A fundamental domain triangle is denoted (p q r), where p, q, r are whole numbers > 1, i.e. ≥ 2; a fundamental domain right triangle is denoted (p q 2). The triangle may exist as a <a href="/facts/Spherical_triangle/3G3Fhk2h">spherical triangle</a>, a Euclidean plane triangle, or a hyperbolic plane triangle, depending on the values of p, q, and r.
There are several symbolic schemes for denoting these figures:

<ul><li>The modified <a href="/facts/Schl%25C3%25A4fli_symbol/jNQR77UU">Schläfli symbol</a> for a right triangle domain: (p q 2) → {p, q}.</li>
<li>The <a href="/facts/Coxeter-Dynkin_diagram/Ik6LmXJB">Coxeter-Dynkin diagram</a> is a triangular graph with p, q, r labeled on the edges. If r = 2, then the graph is linear, since diagram nodes with connectivity 2 are not connected to each other by a diagram branch (since domain mirrors meeting at 90 degrees generate no new mirrors).</li>
<li>The <a href="/facts/Wythoff_symbol/ECD0Wl5x">Wythoff symbol</a> takes the three integers and separates them by a vertical bar (|). If the generator point is off the mirror opposite to a domain vertex, then the reflection order of this domain vertex is given before the bar.</li>
<li>Finally, a uniform tiling can be described by its <a href="/facts/Vertex_configuration/wWaBS6t0">vertex configuration</a>: the (identical) sequence of polygons around each (equivalent) vertex.</li></ul>
All uniform tilings can be constructed from various operations applied to <a href="/facts/Regular_tiling/tR5CQACy">regular tilings</a>. These operations, as named by <a href="/facts/Norman_Johnson_(mathematician)/bQtL9w56">Norman Johnson</a>, are called <a href="/facts/Truncation_(geometry)/DYb589Je">truncation</a> (cutting vertices), <a href="/facts/Rectification_(geometry)/PkoUdq53">rectification</a> (cutting vertices until edges disappear), and <a href="/facts/Cantellation/kS24Tde3">cantellation</a> (cutting edges and vertices). <a href="/facts/Omnitruncation/hmbMrRm6">Omnitruncation</a> is an operation that combines truncation and cantellation. Snubbing is an operation of <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternate truncation</a> of the omnitruncated form. (See <a href="/facts/Uniform_polyhedron/PHnprqRC">Uniform polyhedron#Wythoff construction operators</a> for more details.)

Uniform tiling open-in-new

Uniform tiling