Classification of discrete groups of rigid motions

This is a mathematical rediscovery project. We know the rigid motions of the plane can be classified as

• Translations
• Rotations
• Reflections
• Glide Reflections

As described in Chapter 1, a collection of these is a group if it is closed under composition and under taking inverses.

A group of transformations is discrete if there is a circular disk $D$ such that the image $T(D)$ of this disk under any transformation in the group, other than the identity, is disjoint from $D$.

The goal of this project is to describe all the possible discrete groups of rigid motions in a systematic way. An advanced reference is Chapter 5, Sections 1 through 4, of Michael Artin's Algebra [Artin, 1995]. But we will indicate the major steps in the classification, and the project will consist of filling in the details.

Here are some major steps. Let $G$ be a discrete group. What we want is a list of possibilities for the constructions of $G$.

• First we have to distinguish the orientation-preserving transformations (translations and rotations) from the orientation-reversing transformations (reflections and glide reflections). It's important to observe that the composition of two transformations of the same type is always orientation-preserving, while that of two transformations of different types is orientation-reversing. Based on that lemma, prove:

  The subcollection $G_0$ of all orientation-preserving transformations in $G$ is also a subgroup of $G$ (namely closed under inverses and composition).

  Then prove:

  Either $G=G_0$ or there is an orientation reversing transformation $T$ in $G$ such that $G$ consists of $G_0$ and $TG_0$ (meaning the composition of $T$ with every element of $G_0$.

• Now turn to the classification of the orientation-preserving groups $G_0$. These consist entirely of rotations or translations. We can do the same sort of reduction again by considering the subcollection $G_1$ of translations in $G_0$. Since the composition of any two translations is also a translation, $G_1$ is again a subgroup of $G_0$.

  Using complex numbers, each translation $T$ is of the form $T(z)=z+a$ for some complex constant $a$. Here $z=x+iy$ is a variable point in the plane. Thus, the collection $G_1$ of translations corresponds in a one-to-one way to the complex numbers $a$. Moreover, if $T_1(z)=z+a_1$ and $T_2(z)=z+a_2$, then $T_1\circ T_2(z)=z+(a_1+a_2)$, and $T_1^{-1}(z)=z-a_1$. Thus, $G_1$ corresponds to a collection of complex numbers $a$ which are closed under addition and subtraction. The condition of discreteness means there is a circular disk which does not contain any two points of the form $z$ and $z+a$ for some $a\neq 0$ in the collection. That means there is a distance $\epsilon>0$ such that $\vert a\vert \ge \epsilon$ for all $a\neq 0$ in the collection.

  Prove that any collection of complex numbers is one of

  1. $\{0\}$ (just the zero point)
  2. $\{m a\mid \text{for any integer $m$}\}$, for a fixed nonzero complex number $a$. (In other words all multiples of $a$.)
  3. $\{ma+nb \mid \text{for any integers $m,n$}\}$, for complex numbers $a$, $b$ which are not multiples of one another. (In other words, all integral linear combinations of $a$ and $b$.)
  The group is call rank 0, 1, or 2, respectively.

• Next we have to consider the rotations that can be added to a group of translations as described in the previous step. For the group to be discrete, prove the rotations must all be by a multiple of $2\pi/n$ radians.

  If there are no nonzero translations, the $n$ may be arbitrary, but all the rotations must be around the same point, which can be taken to be 0. This is called the cyclic group.

  If there are nonzero translations, then prove that $n$ has to be 1, 2, 3, 4, or 6. If the rank is 1, prove that $n=1$ or $2$.

  If the rank is 2, prove that $n=3$, $4$ or $6$ occurs only for special choices of the basic translations $a$ and $b$. That will complete the list of orientation-preserving discrete groups.

• Lastly, we have to decide when we can add a single reflection or glide reflection to one of the groups $G_0$ that we have listed. Here it is important to think about conjugacy. If $T$ is the orientation-reversing transformation that we add, and $S$ is any translation or rotation, then $T\circ S \circ T^{-1}$ is also translation or rotation in the group $G$. So the reflection or glide reflection that we add must conjugate our existing translations and rotations among themselves. That severely limits the choice of reflection or glide reflection that we can add. Once we analyze those choices, we'll have the complete list of wallpaper groups.