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+----- Thread: Uniform tilings in hyperbolic plane (/showthread.php?tid=1798)
Uniform tilings in hyperbolic plane - mklabgr - 09-02-2026
Summary
A uniform tiling of the hyperbolic plane is an edge-to-edge covering of hyperbolic space by regular polygons in which all vertices are equivalent under the symmetries of the tiling. This means that there is an isometry carrying any vertex to any other vertex. Uniform tilings may be regular, quasiregular, or semiregular, depending on whether their faces and edges are also equivalent. They are commonly described using a vertex configuration, which records the polygons meeting at each vertex. For example, $7.7.7$ means that three regular heptagons meet at every vertex; the same regular tiling can be written with the Schläfli symbol ${7,3}$.
A major difference from ordinary Euclidean geometry is that the hyperbolic plane admits infinitely many uniform tilings. Many of them can be systematically constructed using the Wythoff construction and hyperbolic reflection groups. The fundamental region is often a Schwarz triangle associated with three integers $(p,q,r)$. For a genuinely hyperbolic triangle, the parameters satisfy
$1p+1q+1r<1.\frac{1}{p}+\frac{1}{q}+\frac{1}{r}<1$.
Reflections in the sides of this triangle generate a hyperbolic triangle symmetry group. For each such symmetry family, different choices of active mirrors in the Wythoff construction generate several related uniform tilings, including regular, truncated, rectified, cantellated, omnitruncated, and snub forms.
The article classifies these tilings according to their fundamental domains. An important family uses right triangles $(p,q,2)$, which produces the regular tilings ${p,q}$ and their duals ${q,p}$. More general triangle and quadrilateral fundamental domains produce many additional families. The theory can also be extended by allowing parameters such as $p=\infty$, producing ideal tilings containing vertices at infinity or infinite-sided polygons called apeirogons. Altogether, uniform hyperbolic tilings illustrate how the negative curvature of hyperbolic geometry permits a far richer variety of symmetric tessellations than is possible in the Euclidean plane.
Key takeaways
- A uniform hyperbolic tiling consists of regular polygons with identical vertex arrangements.
- Vertex configurations such as $7.7.7$ describe which polygons meet at each vertex.
- Regular tilings are often represented by Schläfli symbols such as ${7,3}$.
- Hyperbolic geometry admits infinitely many uniform tilings.
- Their classification relies heavily on Wythoff constructions, Schwarz triangles, reflection groups, and Coxeter groups.
- The fundamental hyperbolic condition for a triangle group is
$1p+1q+1r<1.\frac1p+\frac1q+\frac1r<1$.
- Allowing infinite parameters leads to ideal vertices and apeirogons.
ARTICLE