Lecture 8: Friday, September 6, 2013

Over the next couple of lectures, we will generalize cyclic groups Z_n and dihedral groups D_n. Given positive integers m, n, and k, formally define the abstract group
D(m,n,k) = \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \ \biggl| \ {\gamma_0}^m = {\gamma_1}^n = {\gamma_\infty}^k = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 \right \rangle.
Such a group is called a Triangle Group, although it is also known as a von Dyck Group after the German mathematician Walther Franz Anton von Dyck (1856 — 1934). Today, we focus on why these are called “triangle groups.”

Basic Properties

It is easy to see that D(m, n, k) = D(k, m, n) = D(k, n, m), that is, the ordering of m, n, and k does not matter. Indeed, using the substitutions \gamma_0 = s, \gamma_1 = r, and \gamma_{\infty} = ( s \, r)^{-1}, we have the following diagram:

\begin{matrix}  {\begin{aligned} u & = (s \, r)^{-1} \\ v & = s \\ u \, v & = r^{-1} \end{aligned}} & {\begin{aligned} u^k & = 1 \\ v^m & = 1 \\ (u \, v)^n & = 1 \end{aligned}} & D(k,m,n) = \left \langle u, v \ \biggl| \ u^k = v^m = (u \, v)^n  = 1 \right \rangle \\  & & \uparrow \\  {\begin{aligned} s & = \gamma_0 \\ r & = \gamma_1 \\ s \, r & = {\gamma_\infty}^{-1} \end{aligned}} & {\begin{aligned} s^m & = 1 \\ r^n & = 1 \\ (s \, r)^k & = 1 \end{aligned}} & D(m,n,k) = \left \langle r, s \ \biggl| \ s^m = r^n = (s \, r)^k  =1 \right \rangle \\  & & \downarrow \\  {\begin{aligned} z & = s \, r \\ w & = r^{-1} \\ z \, w & = s \end{aligned}} & {\begin{aligned} z^k & = 1 \\ w^n & = 1 \\ (z \, w)^m & = 1 \end{aligned}} & D(k,n,m) = \left \langle z, w \ \biggl| \ z^k = w^n = (z \, w)^m  =1 \right \rangle  \end{matrix}

Triangles in the Affine Real Plane

We give a geometric interpretation of this group using matrices. Given positive integers m, n, and k which satisfy the identity \dfrac {1}{m} + \dfrac {1}{n} + \dfrac {1}{k} = 1, define the quantities
\begin{aligned}  x_P & = 0 & & & y_P & = 0 & & & & & A & = \dfrac {\pi}{m} \\[10pt]  x_Q & = \cos A \, \sin B + \sin A \, \cos B & & & y_Q & = 0 & & & \text{in terms of} & & B & = \dfrac {\pi}{n} \\[10pt]  x_R & = \cos A \, \sin B & & & y_R & = \sin A \, \sin B & & & & & C & = \dfrac {\pi}{k}.  \end{aligned}

Denoting \mathbb A^2(\mathbb R) as the affine real plane, consider the affine points P = (x_P, y_P), Q = (x_Q,y_Q) and R = (x_R, y_R) as well as the transformations \gamma_0, \, \gamma_1, \, \gamma_\infty: \mathbb A^2(\mathbb R) \to \mathbb A^2(\mathbb R) defined by


\begin{aligned}  \gamma_0: \qquad \left[ \begin{matrix} x \\[5pt] y \end{matrix} \right] & \qquad \mapsto \qquad \left [ \begin{matrix} \cos \, 2 A & -\sin \, 2 A \\[5pt] \sin \, 2 A & \cos \, 2 A \end{matrix} \right ] \left[ \begin{matrix} x - x_P \\[5pt] y - y_P \end{matrix} \right] + \left[ \begin{matrix} x_P \\[5pt] y_P \end{matrix} \right] \\[10pt]  \gamma_1: \qquad \left[ \begin{matrix} x \\[5pt] y \end{matrix} \right] & \qquad \mapsto \qquad \left [ \begin{matrix} \cos \, 2 B & -\sin \, 2 B \\[5pt] \sin \, 2 B & \cos \, 2 B \end{matrix} \right ] \left[ \begin{matrix} x - x_Q \\[5pt] y - y_Q \end{matrix} \right] + \left[ \begin{matrix} x_Q \\[5pt] y_Q \end{matrix} \right]  \\[10pt]  \gamma_\infty: \qquad \left[ \begin{matrix} x \\[5pt] y \end{matrix} \right] & \qquad \mapsto \qquad \left [ \begin{matrix} \cos \, 2 C & -\sin \, 2 C \\[5pt] \sin \, 2 C & \cos \, 2 C \end{matrix} \right ] \left[ \begin{matrix} x - x_R \\[5pt] y - y_R \end{matrix} \right] + \left[ \begin{matrix} x_R \\[5pt] y_R \end{matrix} \right]    \end{aligned}

Proposition.
Continue notation as above.


  • The angles A, B, and C sum to 180^\circ, that is, A + B + C = \pi. Moreover, \cos^2 A + \cos^2 B + \cos^2 C + 2 \, \cos A \, \cos B \, \cos C = 1.

  • V = \{ P, \, Q, \, R \} \subseteq \mathbb A^2(\mathbb R) is a triangle with angles A, B, and C and area \dfrac {1}{2} \, \sin A \, \sin B \, \sin C.

  • The transformation \gamma_0 is a rotation around P by (2 \pi/m) radians counterclockwise, the transformation \gamma_1 is a rotation around Q by (2 \pi/n) radians counterclockwise, and the transformation \gamma_\infty is a rotation around R by (2 \pi/k) radians counterclockwise.

  • The compositions {\gamma_0}^m = {\gamma_1}^n = {\gamma_\infty}^k = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 are the identity transformation, while the composition

    \gamma_1 \circ \gamma_0: \qquad \left[ \begin{matrix} x \\[5pt] y \end{matrix} \right] \qquad \mapsto \qquad \left [ \begin{matrix} \cos \, 2 C & \sin \, 2 C \\[5pt] -\sin \, 2 C & \cos \, 2 C \end{matrix} \right ] \left[ \begin{matrix} x - x_R \\[5pt] y - y_R \end{matrix} \right] + \left[ \begin{matrix} x_R \\[5pt] -y_R \end{matrix} \right]
    has infinite order. In particular, D(m,n,k) is a finitely generated group which is not abelian and has infinitely many elements.

We leave the proof as an exercise for the reader.

There are only finitely many positive integers m, n, and k satisfying the conditions above, and up to symmetry they fit into the following table:






Group D(m, n, k)mnkTriangle V = \{ P, Q, R \}
Symmetries of the kisrhombille236 30^\circ - 60^\circ - 90^\circ
Symmetries of the kisquadrille24445^\circ - 45^\circ - 90^\circ
Symmetries of the deltille33360^\circ - 60^\circ - 60^\circ

These are examples of wallpaper groups. The images of the triangle V by transformations of the elements in D(m,n,k) form a tiling of the plane \mathbb A^2(\mathbb R). This explains why D(m,n,k) is called a “triangle group”.

Triangles on the Unit Sphere

Triangle groups D(m,n,k) are much more interesting when the integers m, n, and k are a bit larger. For example,

\begin{aligned}   Z_n & = \left \langle r \, \biggl| \, r^n = 1 \right \rangle & \simeq & \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \biggl| \, {\gamma_0}^1 = {\gamma_1}^n = {\gamma_\infty}^n = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 \right \rangle = D(1,n,n); \\[10pt]  D_n & = \left \langle r, \, s \, \biggl| \, s^2 = r^n = (s \, r)^2 = 1 \right \rangle & \simeq & \left \langle \gamma_0, \, \gamma_1, \, \gamma_\infty \, \biggl| \, {\gamma_0}^2 = {\gamma_1}^n = {\gamma_\infty}^2 = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 \right \rangle = D(2,n,2).  \end{aligned}

We will use this to study triangles on the unit sphere S^2(\mathbb R) = \left \{ (u,v,w) \in \mathbb A^2(\mathbb R) \, \biggl| \, u^2 + v^2 + w^2 = 1 \right \}. Given positive integers m, n, and k which satisfy the inequalities \dfrac {1}{m} + \dfrac {1}{n} + \dfrac {1}{k} > 1 and m, \, n, \, k \geq 2, define the quantities
\begin{aligned}   x_P & = 0 & & & x_Q & = \dfrac {\sqrt{\delta}}{\sin A \, \sin B} & & & x_R & = \cos A \, \dfrac {\sqrt{\delta}}{\sin C \, \sin A} \\[5pt]  y_P & = 0 & & & y_Q & = 0 & & & y_R & = \sin A \, \dfrac {\sqrt{\delta}}{\sin C \, \sin A} \\[5pt]  z_P & = 1 & & & z_Q & = \dfrac {\cos C + \cos A \, \cos B}{\sin A \, \sin B} & & & z_R & = \dfrac {\cos B + \cos C \, \cos A}{\sin C \, \sin A} \end{aligned} \quad \text{where} \quad \begin{aligned}  A & = \dfrac {\pi}{m} \\[5pt]  B & = \dfrac {\pi}{n} \\[5pt]  C & = \dfrac {\pi}{k}   \end{aligned}

and \delta = 1 - \bigl( \cos^2 A + \cos^2 B + \cos^2 C + 2 \, \cos A \, \cos B \, \cos C \bigr) > 0. Denoting S^2(\mathbb R) as the unit sphere, consider the affine points

\begin{aligned}  P & = \left[ \begin{matrix} x_P \\[5pt] y_P \\[5pt] z_P \end{matrix} \right] = \gamma_P \left[ \begin{matrix} 0 \\[5pt] 0 \\[5pt] 1 \end{matrix} \right] \\[5pt]  Q & = \left[ \begin{matrix} x_Q \\[5pt] y_Q \\[5pt] z_Q \end{matrix} \right] = \gamma_Q \left[ \begin{matrix} 0 \\[5pt] 0 \\[5pt] 1 \end{matrix} \right] \\[5pt]  R & = \left[ \begin{matrix} x_R \\[5pt] y_R \\[5pt] z_R \end{matrix} \right] = \gamma_R \left[ \begin{matrix} 0 \\[5pt] 0 \\[5pt] 1 \end{matrix} \right]  \end{aligned} \quad \text{where} \quad \begin{aligned}   \gamma_P & = \left[ \begin{matrix} 1 & 0 & 0 \\[5pt] 0 & 1 & 0 \\[5pt] 0 & 0 & 1 \end{matrix} \right] \\[5pt]  \gamma_Q & = \left[ \begin{matrix} z_Q & 0 & x_Q \\[5pt] 0 & 1 & 0 \\[5pt] -x_Q & 0 & z_Q \end{matrix} \right] \\[5pt]  \gamma_R & = \left[ \begin{matrix} x_R \, z_R/\sqrt{x_R^2 + y_R^2} & - y_R/\sqrt{x_R^2 + y_R^2} & x_R \\[5pt] y_R \, z_R/\sqrt{x_R^2 + y_R^2} & x_R/\sqrt{x_R^2 + y_R^2} & y_R \\[5pt] -\sqrt{x_R^2 + y_R^2} & 0 & z_R \end{matrix} \right]  \end{aligned}
as well as the rotations \gamma_0, \, \gamma_1, \, \gamma_\infty: S^2(\mathbb R) \to S^2(\mathbb R) defined by

\begin{aligned}  \gamma_0: & \qquad \left[ \begin{matrix} x \\[5pt] y \\[5pt] z \end{matrix} \right] \qquad & \mapsto \qquad \left( \gamma_P \left[ \begin{matrix} \cos 2 A & -\sin 2 A & 0 \\[5pt] \sin 2 A & \cos 2 A & 0 \\[5pt] 0 & 0 & 1 \end{matrix} \right] {\gamma_P}^{-1} \right) \left[ \begin{matrix} x \\[5pt] y \\[5pt] z \end{matrix} \right] \\[5pt]  \gamma_1: & \qquad \left[ \begin{matrix} x \\[5pt] y \\[5pt] z \end{matrix} \right] \qquad & \mapsto \qquad\left( \gamma_Q \left[ \begin{matrix} \cos 2 B & -\sin 2 B & 0 \\[5pt] \sin 2 B & \cos 2 B & 0 \\[5pt] 0 & 0 & 1 \end{matrix} \right] {\gamma_Q}^{-1} \right) \left[ \begin{matrix} x \\[5pt] y \\[5pt] z \end{matrix} \right]\\[5pt]  \gamma_\infty: & \qquad \left[ \begin{matrix} x \\[5pt] y \\[5pt] z \end{matrix} \right] \qquad & \mapsto \qquad \left( \gamma_R \left[ \begin{matrix} \cos 2 C & -\sin 2 C & 0 \\[5pt] \sin 2 C & \cos 2 C & 0 \\[5pt] 0 & 0 & 1 \end{matrix} \right] {\gamma_R}^{-1} \right) \left[ \begin{matrix} x \\[5pt] y \\[5pt] z \end{matrix} \right]  \end{aligned}

Proposition.
Continue notation as above.


  • The angles A, B, and C sum to more than 180^\circ, that is, A + B + C > \pi.

  • V = \{ P, \, Q, \, R \} \subseteq S^2(\mathbb R) is a spherical triangle with angles A, B, and C.

  • The transformation \gamma_0 is a rotation around P by (2 \pi/m) radians counterclockwise, the transformation \gamma_1 is a rotation around Q by (2 \pi/n) radians counterclockwise, and the transformation \gamma_\infty is a rotation around R by (2 \pi/k) radians counterclockwise.

  • The compositions {\gamma_0}^m = {\gamma_1}^n = {\gamma_\infty}^k = \gamma_0 \, \gamma_1 \, \gamma_\infty = 1 are the identity transformation.

Sketch of Proof: In order to verify that the triangle V = \{ P, \, Q, \, R \} makes angles A, B, and C on the sphere, we first compute the angles the vectors make with each other with respect to the origin (0,0,0). Denoting a, b, and c as the angles between Q and R, R and P, and P and Q, respectively, we use inner products and cross products to verify that
\begin{aligned}  \cos a & = \dfrac {Q \cdot R}{\| Q \| \, \| R \|} & = \dfrac {\cos A + \cos B \, \cos C}{\sin B \, \sin C} \\[5pt]  \cos b & = \dfrac {R \cdot P}{\| R \| \, \| P \|} & =\dfrac {\cos B + \cos C \, \cos A}{\sin C \, \sin A} \\[5pt]  \cos c & = \dfrac {P \cdot Q}{\| P \| \, \| Q \|} & = \dfrac {\cos C + \cos A \, \cos B}{\sin A \, \sin B}  \end{aligned} \qquad \qquad \begin{aligned}  \sin a & = \dfrac {\| Q \times R \|}{\| Q \| \, \| R \|} & = \dfrac {\sqrt{\delta}}{\sin B \, \sin C} \\[5pt]  \sin b & = \dfrac {\| R \times P \|}{\| R \| \, \| P \|} & = \dfrac {\sqrt{\delta}}{\sin C \, \sin A} \\[5pt]  \sin c & = \dfrac {\| P \times Q \|}{\| P \| \, \| Q \|} & = \dfrac {\sqrt{\delta}}{\sin A \, \sin B}  \end{aligned}
Using the Spherical Law of Cosines, we see that
\cos A = \dfrac {\cos a - \cos b \, \cos c}{\sin b \, \sin c}, \cos B = \dfrac {\cos b - \cos c \, \cos a}{\sin c \, \sin a}, and \cos C = \dfrac {\cos c - \cos a \, \cos b}{\sin a \, \sin b}. That is, V is indeed a triangle with angle A at vertex P, angle B at vertex Q, and angle C at vertex R. \square

The triangles V = \{ P, \, Q, \, R \} such that their angles A + B + C > \pi while m = \pi/A, n = \pi/B, and k = \pi/C are integers are called Möbius triangles, named in honor of the German mathematician August Ferdinand Möbius (1790 — 1868).

In the next lecture, we’ll focus more of the geometry in this case. And we’ll show some pretty pictures!

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About Edray Herber Goins, Ph.D.

Edray Herber Goins grew up in South Los Angeles, California. A product of the Los Angeles Unified (LAUSD) public school system, Dr. Goins attended the California Institute of Technology, where he majored in mathematics and physics, and earned his doctorate in mathematics from Stanford University. Dr. Goins is currently an Associate Professor of Mathematics at Purdue University in West Lafayette, Indiana. He works in the field of number theory, as it pertains to the intersection of representation theory and algebraic geometry.
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2 Responses to Lecture 8: Friday, September 6, 2013

  1. Pingback: Lecture 9: Monday, September 9, 2013 | Lectures on Dessins d'Enfants

  2. Pingback: MA 59800 Course Syllabus | Lectures on Dessins d'Enfants

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