Seven Wallpaper Patterns. Or Frieze Patterns.

Question

The problem from "Notes on Geometry" by Rees:

Show that there are exactly seven distinct infinite subgroups $G \subset I(\mathbb{R}^2)$ such that $Z$ is an orbit of $G$, where $Z=\{(n,0):n \in Z \}$. For each such $G$, draw a subset $X(G) \subset \mathbb{R}^2$ such that $S(X(G))=G$. These subsets are called seven wallpaper patterns.

Here, $I(\mathbb{R}^2)$ means isometry group of $\mathbb{R}^2$. $S(X(G))$ means symmetry group of $X(G)$.

I've found seven distinct infinite subgroups.

• $G_1=<T_{(1,0)}>$ where $T$ is a translation.
• $G_2 = \{ G(x,(n,0))|n \in \mathbb{Z}\}$ where $x$ is the $x$-axis and $G$ stands for a glide.
• $G_3 = \{R_{l_{n/2}} | n \in \mathbb{Z} \}$ where $R$ stands for reflection, and $l_{n/2}$ are lines perpendicular to $x$-axis going through $(n/2,0)$.
• $G_4 = G_3 \cup {R_x} \cup \{R(n/2,\pi)|n\in \mathbb{Z}\}$ where $R_x$ is a reflection in $x$-axis, and $R(a,\alpha)$ stands for a rotation about $a$ for angle $\alpha$.
• $G_5 =\{R(n/2,\pi)|n\in \mathbb{Z}\}$
• $G_6 = G_5 \cup <T_{(1,0)}>$
• $G_7 = G_2 \cup <T_{(1,0)}>$

First of all, as I got these intuitively, I would like to know if these are the desired subgroups. And, more importantly, how could I find such subgroups algebraically?

Lastly, the second part is the one I'm having trouble with. So, $X(G)$ would mean the set of points where $G$ preserves all $X(G)$. But, for drawing, do you agree if it is sufficient to show for a few points only?

The problem is so vague to me, so I would like to get some help with this problem.

Thank you for your time!

Answer 1

There's not so much an algebraic way to find these subgroups, I would just say that there is a rigorously logical way to find them. Namely, do a complete case analysis, based on the classification of Euclidean isometries.

To start you off:

• Case 1: $G$ contains only translations. Use that to prove $G=G_1$.
• Case 2: $G$ contains only translations and glide reflections, including at least one glide reflection. Use that to prove $G=G_2$.
• Case 3: $G$ contains only translations and reflections across vertical lines, including at least one reflection across a vertical line. Use that to prove $G=G_3$.

and so on.

Regarding $X(G)$, whether you like more points or fewer points in your $X(G)$ set, it doesn't really matter. Frankly, what I would do is first to determine a fundamental domain for $G$. Then, inside that fundamental domain draw a funny, asymmetrical face. Finally, transport that funny, asymmetrical face to all other fundamental domains using the action of $G$.