Triangle groups, Fuchsian groups, and the wild symmetries of hyperbolic space
The symmetries of a hyperbolic tiling form a group. Reflect one triangle again and again and it fills the disk, and the group grows exponentially as it does.
Choose different (p, q, r) triples. Small values give big triangles; large values pack many tiny triangles into the disk.
The symmetries of a hyperbolic tiling form a Fuchsian group — a discrete subgroup of the isometry group of the hyperbolic plane. The simplest family is the triangle groups Δ(p, q, r), generated by reflections in the sides of a hyperbolic triangle with angles π/p, π/q, π/r. Whenever 1/p + 1/q + 1/r < 1 the triangle is hyperbolic, and the group tiles the disk with infinitely many reflected copies.
Choose parameters p, q, r satisfying 1/p + 1/q + 1/r < 1 to generate the Coxeter triangle group tiling. The fundamental triangle is reflected across its edges, producing a kaleidoscopic pattern that fills the entire Poincare disk. Triangles are colored in an alternating pattern so the reflection symmetry is visible.
Raise the depth slider to grow the tree of the free group F2. Colors show distance from the center, and the hyperbolic plane keeps room for every branch.
The free group on two generators, F2, has a Cayley graph that is a regular 4-valent tree. In Euclidean space this tree is impossibly tangled, but the hyperbolic plane has enough room to embed it with all edges as geodesic arcs. Vertices are colored by word length from the identity, revealing the exponential growth of the group.
Cayley graphs embed — The 4-valent tree of F2 fits naturally in hyperbolic space, where exponential area growth matches exponential branching
Increase the depth slider to add copies of the octagon. Each copy is one group element, and together they tile the disk with no gaps.
A regular octagon in the Poincare disk serves as the fundamental domain for a genus-2 surface group. Applying group elements — reflections and translations — generates copies that tile the disk without overlap or gaps, with alternating colors showing the group structure.
Raise the max distance and compare the growth curves. The hyperbolic group explodes exponentially while the flat grid grows slowly.
Left: Cayley graph of F2 with vertices colored by word distance. Right: the number of group elements at each distance n. The bars grow exponentially (~3n), matching the pink reference curve. The grey line shows polynomial growth for comparison (e.g., Z2).
In a finitely generated group, the word metric counts the minimum number of generators needed to reach an element from the identity. For hyperbolic groups the number of elements at distance n grows exponentially, in stark contrast to the polynomial growth of Euclidean lattices. The side graph compares these growth rates.
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