Triangle group
A triangle group is a Coxeter group of rank three in mathematics, generated by three reflections a, b, and c corresponding to the sides of a triangle, with the standard presentation \langle a, b, c \mid a^2 = b^2 = c^2 = (ab)^p = (bc)^q = (ca)^r = 1 \rangle, where the integers p, q, r \geq 2 (or \infty) represent the orders of the products of adjacent reflections, related to the dihedral angles \pi/p, \pi/q, and \pi/r of the fundamental triangle.[1] These groups act as discrete reflection groups on the spherical, Euclidean, or hyperbolic plane, tiling the space with congruent copies of the fundamental triangle via the orbit of reflections, and their geometric realization is classified by the sum \frac{1}{p} + \frac{1}{q} + \frac{1}{r}: greater than 1 yields a finite spherical group acting on S^2 (e.g., \Delta(2,3,3), \Delta(2,3,4), \Delta(2,3,5)); equal to 1 produces an infinite Euclidean (affine) group acting on E^2 with translational symmetries (e.g., \Delta(2,3,6), \Delta(2,4,4), \Delta(3,3,3)); and less than 1 results in an infinite hyperbolic group acting discretely on H^2 (e.g., \Delta(2,3,7), \Delta(3,3,4)).[1][2] Triangle groups play a central role in the study of tilings and tessellations, as their actions produce regular triangulations of the underlying space, and they encompass important examples like the modular group \Gamma = \Delta(2,3,\infty) acting on the hyperbolic plane.[2] The even-length words in the generators form an index-two subgroup known as the von Dyck group or triangle rotation group D(p,q,r), which is generated by rotations of orders p, q, and r around the triangle's vertices and is orientation-preserving.[1] Finite triangle groups correspond precisely to the symmetry groups of the Platonic solids (tetrahedral, octahedral, icosahedral), linking them to polyhedral geometry and representation theory.[1] In broader contexts, triangle groups appear in arithmetic geometry, such as in the study of Fuchsian groups and modular surfaces, and in algebraic combinatorics through their connections to Coxeter graphs and Hecke algebras.[2] Their classification via the Gauss-Bonnet theorem ties the angle sum directly to the curvature of the ambient space, providing a foundational example of how abstract group presentations encode geometric structures.[1]Fundamentals
Definition
A triangle group is a discrete group generated by three reflections in the lines (or great circles) forming the sides of a triangle in a space of constant curvature, where the triangle has interior angles \pi/p, \pi/q, and \pi/r with p, q, r integers greater than or equal to 2.[3][4] These reflections act as orientation-reversing isometries of the underlying space.[3] The triangle serves as a fundamental domain for the action of the triangle group on the sphere S^2, the Euclidean plane \mathbb{E}^2, or the hyperbolic plane \mathbb{H}^2, depending on whether the sum of the angles exceeds, equals, or is less than \pi, respectively.[4][3] The group action tiles the space by repeated reflections across the triangle's sides, producing a tessellation where copies of the fundamental triangle cover the space without overlap except on boundaries.[4] Basic examples include the (2,3,3) triangle group, generated by reflections in a spherical triangle with angles \pi/2, \pi/3, and \pi/3, which realizes the symmetry group of the tetrahedron, and the (2,4,4) triangle group, generated by reflections in a Euclidean triangle with angles \pi/2, \pi/4, and \pi/4 (a right-angled isosceles triangle), which tiles the plane with squares.[3][5] The full triangle group consists of orientation-reversing isometries, but it contains an index-2 subgroup generated by the products of pairs of reflections, which consists of orientation-preserving transformations such as rotations.[3]Generators and Relations
Triangle groups are abstractly defined via a presentation involving three generators corresponding to reflections. These generators, denoted s, t, and u, represent reflections across the sides of a fundamental triangle and satisfy the relations s^2 = t^2 = u^2 = 1, as each reflection is an involution of order two.[6] The pairwise products of these generators obey additional relations (st)^p = (tu)^q = (us)^r = 1, where p, q, r are integers greater than or equal to 2 that classify the group type based on the geometry of the triangle. The complete presentation of the triangle group is thus \langle s, t, u \mid s^2 = t^2 = u^2 = (st)^p = (tu)^q = (us)^r = 1 \rangle. These parameters p, q, r directly correspond to the angles \pi/p, \pi/q, and \pi/r at the vertices of the triangle, with the relation (st)^p = 1 arising because the composition of reflections s and t yields a rotation by twice the angle between their lines of reflection, which closes after p applications when the angle is \pi/p.[6][3] In the geometric realization, these relations are tied to the triangle's angles through the law of cosines in the ambient space—Euclidean, spherical, or hyperbolic—which relates the angles to the side lengths and determines the existence of the triangle for given p, q, r. For instance, in hyperbolic geometry, the hyperbolic law of cosines \cosh c = \frac{\cos \gamma + \cos \alpha \cos \beta}{\sin \alpha \sin \beta} (where \alpha = \pi/p, \beta = \pi/q, \gamma = \pi/r) allows computation of side lengths, confirming the discrete group action when the angle sum is less than \pi. Similar formulas apply in spherical and Euclidean cases to verify the configuration.[6][1] A degenerate case occurs when one parameter, say r = \infty, corresponding to a zero angle, yielding the infinite dihedral group generated by two reflections across parallel lines or a straight angle.[6]Geometric Classifications
Spherical Triangle Groups
Spherical triangle groups arise as finite Coxeter groups generated by reflections across the sides of a spherical triangle with vertex angles \pi/p, \pi/q, and \pi/r, where p, q, r are integers greater than or equal to 2 satisfying $1/p + 1/q + 1/r > 1. This inequality corresponds to the positive curvature of the sphere, ensuring the group's action tiles the sphere discretely with a finite number of triangular fundamental domains, resulting in a finite group order. The classification of such groups includes the dihedral groups \Delta(2,2,n) of order $4n for n \geq 2, and the three exceptional polyhedral groups \Delta(2,3,3), \Delta(2,3,4), and \Delta(2,3,5). The case \Delta(2,4,4) achieves equality in the inequality and represents the Euclidean limit, so it is excluded from the spherical classification.[1] The exceptional spherical triangle groups \Delta(2,3,3), \Delta(2,3,4), and \Delta(2,3,5) serve as the full symmetry groups (including reflections) of the Platonic solids, acting on the circumscribed sphere. Specifically, \Delta(2,3,3) is the symmetry group of the tetrahedron, with order 24; \Delta(2,3,4) is the symmetry group of the octahedron (or dual cube), with order 48; and \Delta(2,3,5) is the symmetry group of the icosahedron (or dual dodecahedron), with order 120. For these groups, the order is given by the formula |\Delta(p,q,r)| = 4 / (1/p + 1/q + 1/r - 1), which derives from the spherical excess of the fundamental triangle: the excess \pi(1/p + 1/q + 1/r - 1) determines the area of each triangular domain, and the sphere's total area $4\pi implies the number of domains is the reciprocal times 4, yielding the group order via the reflection action.[7][8] The orientation-preserving subgroups of these exceptional groups are the rotation groups of the Platonic solids, isomorphic to the alternating group A_4 (order 12) for the tetrahedron, the symmetric group S_4 (order 24) for the octahedron, and the alternating group A_5 (order 60) for the icosahedron. These rotation groups lift to central extensions in \mathrm{SU}(2), known as the binary polyhedral groups: the binary tetrahedral group of order 24, the binary octahedral group of order 48, and the binary icosahedral group of order 120, which play a key role in representations of 3-dimensional symmetries and quaternionic structures.[9]Euclidean Triangle Groups
Euclidean triangle groups are infinite discrete groups of isometries of the Euclidean plane \mathbb{E}^2 generated by reflections across the sides of a triangle with angles \pi/p, \pi/q, and \pi/r, where p, q, r are integers greater than or equal to 2 satisfying the condition \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 1.[10][11] This condition ensures that copies of the triangle tile the plane without gaps or overlaps, yielding a crystallographic action with translational symmetries.[10] Unlike finite spherical groups or infinite hyperbolic ones, these groups are affine and correspond to the symmetry groups of the three regular tilings of the plane.[11] Up to permutation, the integer solutions to the condition are the triples (2,3,6), (2,4,4), and (3,3,3).[10][11] These correspond to the triangular lattice (for (2,3,6)), the square lattice (for (2,4,4)), and the hexagonal lattice (for (3,3,3)).[10] Specifically:- The (2,3,6) group acts as the full symmetry group of the triangular tiling \{3,6\}, where tiles are equilateral triangles meeting six at each vertex, with rotation orders 2, 3, and 6 at the triangle's vertices.[10][12]
- The (2,4,4) group symmetries the square tiling \{4,4\}, with squares meeting four at each vertex and rotation orders 2, 4, and 4.[10][12]
- The (3,3,3) group symmetries the hexagonal tiling \{6,3\}, where regular hexagons meet three at each vertex, with all rotation orders 3, though it also relates to the dual triangular tiling in the hexagonal lattice.[10][12]
| Triple (p,q,r) | Tiling | Lattice | Wallpaper Group (Conway) | Angles (\pi/p, \pi/q, \pi/r) |
|---|---|---|---|---|
| (2,3,6) | Triangular \{3,6\} | Triangular | *632 | \pi/2, \pi/3, \pi/6 (90°, 60°, 30°) |
| (2,4,4) | Square \{4,4\} | Square | *442 | \pi/2, \pi/4, \pi/4 (90°, 45°, 45°) |
| (3,3,3) | Hexagonal \{6,3\} | Hexagonal | *333 | \pi/3, \pi/3, \pi/3 (60°, 60°, 60°) |