Combination of 1981 glide reflections in $\mathbb{E}^2$ still a glide reflection?

Question

I was wondering if the combination of 1981 glide reflections over different lines is still a glide reflection over a line in $\mathbb{E}^2$ (so every glide reflection can be over a different line). Or may there be only limited cases in which this is true?

A glide reflection is a combination of a translation along a line $S$ and a reflection over a line $S$. We can see it as: $$S_b: \mathbb{E}^2 \to \mathbb{E}^2: x \mapsto S_b(x) = x + 2\overrightarrow{x\pi_s(x)} + b $$ Here is $\pi_s(x)$ the intersectionpoint of the orthogonal projection on the mirrorline $S$ and $S$ itself. $b$ has the direction alongside $S$.

Answer 1

Recall that every isometry is one of the following:

• the identity map,
• a rotation,
• a translation,
• a reflection,
• a glide reflection.

(if we allow degenerate cases, those classes overlap, but it doesn't matter)

Let me make the following ad-hoc definition: an isometry is orientation-preserving if it is the identity, a rotation or a translation, and it is orientation-reversing if it is a reflection or a glide reflection. We then have the following three results, which are relatively easy to verify case-by-case:

• A composition of two orientation-preserving isometries is orientation-preserving,
• A composition of two orientation-reverving isometries is orientation-preserving,
• A composition of an orientation-preserving isometry and an orientation-reverving isometry is orientation-reversing.

It is easy to deduce from those facts that a composition of an odd number of orientation-reversing is orientation-reversing (and of an even number is orientation-preserving). Therefore, a composition of 1981 glide reflections is still a glide reflection (or a pure reflection).

---

Now, let me just say that if you are willing to accept some linear algebra, then you can make all of the above less ad-hoc. Every isometry is an affine transformation, which means that it can be written as a composition of a translation and some invertible linear transformation. Let me denote the latter by $T(A)$ for an isometry $A$. A less ad-hoc definition is as follows: we call $A$ orientation-preserving if $\det T(A)>0$ and orientation-reversing if $\det T(A)<0$.

It's not hard to check that $T(A\circ B)=T(A)\circ T(B)$, therefore $$\det T(A\circ B)=\det(T(A)\circ T(B))=\det T(A)\cdot\det T(B).$$ From this it's very easy to deduce an odd number of orientation-reversing maps compose to an orientation-reversing map, and now we are done once we check that glide reflections are orientation-reversing (straightforward) and vice versa (less obvious, requires classification of isometries).