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1. Wallpaper Groups

   Linked via "isometry groups"

   Wallpaper groups, also known as two-dimensional crystallographic groups, constitute the set of all possible isometry groups of the Euclidean plane ($\mathbb{E}^2$) that possess a discrete subgroup of translations $T$ such that the quotient space $\mathbb{E}^2/T$ is compact. These groups are fundamental in describing the symmetry inherent in patterns that repeat infinitely in two dimensions, such as those found in tiling, textile design…

2. Wallpaper Groups

   Linked via "isometry groups"

   Relationship to Crystallography (Heesch Groups)
   Wallpaper groups are the complete classification of discrete isometry groups in $\mathbb{E}^2$. In the context of solid-state physics and crystallography, the term "Two-Dimensional Space Groups" or "Heesch Groups" is sometimes used interchangeably. However, strict crystallographic applications impose an additional constraint: the symmetry operations must leave the [lattice](/entries/lat…