Over the next couple of lectures, we will generalize cyclic groups and dihedral groups . Given positive integers , , and , formally define the abstract group

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 , that is, the ordering of , , and does not matter. Indeed, using the substitutions , , and , we have the following diagram:

### Triangles in the Affine Real Plane

We give a geometric interpretation of this group using matrices. Given positive integers , , and which satisfy the identity , define the quantities

Denoting as the affine real plane, consider the affine points , and as well as the transformations defined by

Proposition.

Continue notation as above.

- The angles , , and sum to , that is, . Moreover, .

- is a triangle with angles , , and and area .

- The transformation is a rotation around by radians counterclockwise, the transformation is a rotation around by radians counterclockwise, and the transformation is a rotation around by radians counterclockwise.

- The compositions are the identity transformation, while the composition

has infinite order. In particular, 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 , , and satisfying the conditions above, and up to symmetry they fit into the following table:

Group | Triangle | |||

Symmetries of the kisrhombille | 2 | 3 | 6 | |

Symmetries of the kisquadrille | 2 | 4 | 4 | |

Symmetries of the deltille | 3 | 3 | 3 |

These are examples of wallpaper groups. The images of the triangle by transformations of the elements in form a tiling of the plane . This explains why is called a “triangle group”.

### Triangles on the Unit Sphere

Triangle groups are much more interesting when the integers , , and are a bit larger. For example,

We will use this to study triangles on the unit sphere . Given positive integers , , and which satisfy the inequalities and , define the quantities

and . Denoting as the unit sphere, consider the affine points

as well as the rotations defined by

Proposition.

Continue notation as above.

- The angles , , and sum to more than , that is, .

- is a spherical triangle with angles , , and .

- The transformation is a rotation around by radians counterclockwise, the transformation is a rotation around by radians counterclockwise, and the transformation is a rotation around by radians counterclockwise.

- The compositions are the identity transformation.

*Sketch of Proof:* In order to verify that the triangle makes angles , , and on the sphere, we first compute the angles the vectors make with each other with respect to the origin . Denoting , , and as the angles between and , and , and and , respectively, we use inner products and cross products to verify that

Using the Spherical Law of Cosines, we see that

, , and . That is, is indeed a triangle with angle at vertex , angle at vertex , and angle at vertex .

The triangles such that their angles while , , and 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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