---- > [!definition] Definition. ([[plane crystallographic group]]) > Consider the following diagram, where each row corresponds to a [[short exact sequence]]: > > ```tikz > \usepackage{tikz} > \usepackage{amsmath} % Required for \text command in math mode > \usetikzlibrary{matrix, positioning, calc} > > \begin{document} > > \begin{tikzpicture} > \matrix (m) [matrix of math nodes, row sep=1em, column sep=4em] > { > T & M_2 & O_2 \\ > \Gamma \cap T & \Gamma \ \text{discrete} & \overline{\Gamma} \\ > & & \\ > (L,+) & & \\ > }; > \path[-stealth] > (m-1-1) edge (m-1-2) > (m-1-2) edge (m-1-3) > (m-2-1) edge (m-2-2) > (m-2-2) edge (m-2-3); > \node at ($(m-1-1)!0.5!(m-2-1)$) {$\cup$}; > \node at ($(m-1-2)!0.5!(m-2-2)$) {$\cup$}; > \node at ($(m-1-3)!0.5!(m-2-3)$) {$\cup$}; > \node[rotate=270] at ($(m-2-1.south)-(0,1em)$) {$\cong$}; > \end{tikzpicture} > > \end{document} > ``` > > In [[analyzing discrete subgroups of M2]] we show that $L$ is [[group isomorphism|isomorphic]] to one $\{ 0 \}, \mathbb{Z}, \mathbb{Z}^{2}$. In the case where $L \cong (\mathbb{Z}^{2}, +)$ is a [[lattice (groups)]], we call the group $\Gamma$ a **plane crystallographic group.** > > [[The crystallographic restriction|The crystallographic restriction]] allows one to (with work) classify the **plane crystallographic groups** into $17$ types, representative patterns for which are pictured below. > ![[CleanShot 2023-11-12 at 18.03.59.jpg|300]] ![[CleanShot 2023-11-29 at 19.10.53.jpg|300]] > > [!basicproperties] Classifications. > - [[plane crystallographic groups with a fourfold rotation in the point group]] > - [[plane crystallographic groups with point group isomorphic to D3]] ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```