In mathematics , a triangle group is a group that can be represented geometrically using successive reflections relative to the sides of the triangle . A triangle can be a regular Euclidean triangle, a triangle on a sphere, or a hyperbolic triangle . Any group of a triangle is a symmetry group of the parquet of congruent triangles in two-dimensional space , on a sphere or on the Lobachevsky plane (see also the article on the hyperbolic plane ).
Content
Definition
Let l , m , n be integers greater than or equal to 2. The triangle group Δ ( l , m , n ) is the group of motions of the Euclidean space, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by reflections relative to the sides of the triangle with angles π / l , π / m and π / n (measured in radians ). The product of reflections with respect to two adjacent sides is rotation by an angle equal to twice the angle between these sides, 2π / l , 2π / m and 2π / n . Thus, if the reflections are denoted by the letters a , b, and c , and the angles between the sides in a cyclic order, as indicated above, the following relations hold:
There is a theorem that all other relations between a, b, c are a consequence of these relations and that Δ ( l, m, n ) is a motions of the corresponding space. This triangle group is
The abstract group with this task is a Coxeter group with three generators.
Classification
If any natural numbers l , m , n > 1 are given, exactly one of the classical two-dimensional geometries (Euclidean, spherical or hyperbolic) admits a triangle with angles (π / l, π / m, π / n) and the space is paved with reflections of this triangle . The sum of the angles of a triangle determines the type of geometry according to the Gauss-Bonnet formula : the space is Euclidean if the sum of the angles is exactly π, spherical if it exceeds π and hyperbolic if it is strictly less than π. Moreover, any two triangles with given angles are congruent. Each group of a triangle defines a tiling, which is usually painted in two colors, so that any two adjacent mosaic elements have different colors.
In terms of the numbers l , m , n > 1, the following possibilities exist.
Euclidean plane
The triangle group is an infinite symmetry group of a parquet (or mosaic) of the Euclidean plane by triangles whose angles add up to π (or 180 °). Up to permutations, the triple ( l , m , n ) is one of the triples (2,3,6), (2,4,4), (3,3,3). The corresponding groups of triangles are representatives of the .
| (2,3,6) | (2,4,4) | (3,3,3) |
|---|---|---|
| Square parquet "Tetrakis" | Triangular parquet | |
| More detailed charts with marked vertices. It is shown how reflections act. | ||
Scope
A triangle group is a finite symmetry group of parquet on a unit sphere of spherical triangles, or Möbius triangles , the sum of the angles of which add up to a number greater than π. Up to permutation, triples ( l , m , n ) have the form (2,3,3), (2,3,4), (2,3,5) or (2,2, n ), n > 1. The spherical groups of triangles can be compared with the symmetry groups of regular polyhedra in three-dimensional Euclidean space: Δ (2,3,3) corresponds to a tetrahedron , Δ (2,3,4) corresponds to both a cube and an octahedron (they have the same symmetry group ), Δ (2,3,5) corresponds to both the dodecahedron and the icosahedron . The groups Δ (2,2, n ), n > 1, of dihedral symmetry can be considered as symmetry groups of the family of dihedrons , which are formed by two identical regular n- gons connected together, or, dually, an osohedron , which is formed by the union of n gons .
A spherical parquet corresponding to a regular polyhedron is obtained by the barycentric division of the polyhedron and the projection of the obtained points and lines onto the described sphere. There are four faces for a tetrahedron, and each face is an equilateral triangle, which is divided into 6 smaller parts by medians intersecting in the center. The resulting mosaic has 4 × 6 = 24 spherical triangles (this is a spherical tetrakishexahedron ).
These groups are finite, which corresponds to the compactness of the sphere - the disk areas on the sphere grow in terms of the radius, but ultimately cover the entire sphere.
Triangular tilings are given below:
| (2,2,2) | (2,2,3) | (2,2,4) | (2,2,5) | (2,2,6) | (2,2, n) |
|---|---|---|---|---|---|
| (2,3,3) | (2,3,4) | (2,3,5) | |||
Spherical parquet floors corresponding to the octahedron and icosahedron, as well as to dihedral spherical mosaics with even n , are centrally symmetrical . Therefore, each of these packages defines a parquet of the real projective plane, an . Their symmetry group is the factor group of the spherical group of triangles by central symmetry (- I ), which is the central element of order 2. Since the projective plane is a model of elliptic geometry , such groups are called elliptic triangle groups [1] .
Hyperbolic plane
The triangle group is an infinite symmetry group of parquet on a hyperbolic plane of hyperbolic triangles, the sum of the angles of which is less than π. All triples not listed above represent hardwood floors on a hyperbolic plane. For example, a triple (2,3,7) gives a triangle group (2,3,7) . There are infinitely many such groups. Below are the parquet floors associated with some small values.
| Examples of right triangles (2 pq) | ||||
|---|---|---|---|---|
| (2 3 7) | (2 3 8) | (2 3 9) | (2 3 ∞) | |
| (2 4 5) | (2 4 6) | (2 4 7) | (2 4 8) | (2 4 ∞) |
| (2 5 5) | (2 5 6) | (2 5 7) | (2 6 6) | (2 ∞ ∞) |
| Examples of general triangles (pqr) | ||||
| (3 3 4) | (3 3 5) | (3 3 6) | (3 3 7) | (3 3 ∞) |
| (3 4 4) | (3 6 6) | (3 ∞ ∞) | (6 6 6) | (∞ ∞ ∞) |
The hyperbolic groups of triangles are examples of and are generalized in the Gromov theory of hyperbolic groups .
Von Dick's groups
Denote by D ( l , m , n ) the subgroup with index 2 in Δ (l, m, n) generated by words of even length in the generators. Such subgroups are sometimes referred to as “regular” groups of triangles [2] or von Dick groups , named Walter von Dick . Spherical, Euclidean, and hyperbolic triangles correspond to elements of a group that preserves the orientation of the triangles. Projective (elliptic) triangles cannot be interpreted in this way, since the projective plane has no orientation, and there is no “orientation preservation” in it. Reflections, however, are locally orientated (and any manifold is locally orientable, since it is locally Euclidean). [3]
The groups D ( l , m , n ) are defined by the following task:
In terms of generators, this is x = ab, y = ca, yx = cb . Geometrically, the three elements x , y , xy correspond to rotations of 2π / l , 2π / m and 2π / n around three vertices of the triangle.
Note that D ( l , m , n ) ≅ D ( m , l , n ) ≅ D ( n , m , l ), so that D ( l , m , n ) does not depend on the order of the numbers l , m , n .
The von Dick hyperbolic group is the , a discrete group consisting of orientation preserving isometries of the hyperbolic plane.
Overlay Parquet
The groups of triangles preserve the parquet laying by triangles, namely, the fundamental region for the action (a triangle defined by reflection lines), called the Mobius triangle , and are given by a triple of integers ( l , m , n ) corresponding to triangles (2 l , 2 m , 2 n ) with a common peak. There are also parquet floors formed by overlapping triangles that correspond to Schwartz triangles with rational numbers ( l / a , m / b , n / c ), where the denominators are coprime to the numerators. This corresponds to the sides at an angle a π / l (resp.), Which corresponds to a rotation of of 2 a π / l (resp.), Which is of the order l and therefore identical to the element of the abstract group, but differs when presented in the form of reflections.
For example, the Schwartz triangle (2 3 3) gives on the sphere parquet 1, while the triangle (2 3/2 3) gives on the sphere parquet with density 3, but with the same abstract group. These overlay symmetries are not considered triangles.
History
The groups of triangles are dated at least by the representation of the as the rotation groups of the triangle (2,3,5) by Hamilton in 1856 in his article on icosians [4] .
Applications
| External video files | |
|---|---|
| Distorted modular parquet [5] - visualization of the mapping (2,3, ∞) → (2,3,7) by transforming the corresponding parquet. | |
Triangle groups arise in . The modular group generated by two elements, S and T , with the relations S² = (ST) ³ = 1 , is the rotation group of the triangle (2,3, ∞) and is mapped to all groups of triangles (2,3, n ) by adding the relation T n = 1. More generally, H q generated by two elements, S and T , with the relation S 2 = ( ST ) q = 1 (there is no relation separately for T ), is the rotation group of the triangle (2, q , ∞) and is mapped to all groups of triangles (2, q , n ) by adding the relation T n = 1. The modular group is the Hecke group H 3 . In the theory of , the Belyi function allows one to obtain a tiling of a Riemann surface corresponding to a certain group of a triangle.
All 26 sporadic groups are factor groups of triangle groups [6] , of which 12 are Hurwitz groups (factor group of group (2,3,7)).
See also
- Schwartz Triangle
- A Schwartz triangle map is a map of triangles to the upper half-plane .
- Geometric group theory
Notes
- ↑ ( Magnus 1974 )
- ↑ Gross & Tucker, 2001 .
- ↑ ( Magnus 1974 , p. 65)
- ↑ Hamilton, 1856 .
- ↑ Platonic tilings of Riemann surfaces: The Modular Group , Gerard Westendorp
- ↑ ( Wilson 2001 , Table 2, p. 7)
Literature
- Gross, Jonathan L. & Tucker, Thomas W. (2001), "6.2.8 Triangle Groups", Topological graph theory , Courier Dover Publications, p. 279–281 , ISBN 978-0-486-41741-7
- Magnus, Wilhelm (1974), "II. Discontinuous groups and triangle tessellations", Noneuclidean tesselations and their groups , Academic Press , p. 52-106 , ISBN 978-0-12-465450-1
- Wilson, RA (2001), " The Monster is a Hurwitz group ", Journal of Group Theory T. 4 (4): 367–374, doi : 10.1515 / jgth.2001.027 , < http: //web.mat.bham. ac.uk/RAWilson/pubs/MHurwitz.ps > . Retrieved December 24, 2017. Archived March 5, 2012 on Wayback Machine
- Sir William Rowan Hamilton Memorandum respecting a new System of Roots of Unity // Philosophical Magazine . - 1856. - T. 12 . - S. 446 .
Links
- Robert Dawson Some spherical parquet floors (A large number of interesting tilings of the sphere are shown, most of which are not parquet of a group of triangles.)
- Elizabeth r chen triangle groups (2010) Wallpaper on the display screen