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1. Wallpaper Groups

   Linked via "orthogonal transformation"

   Mathematical Definition and Structure
   A wallpaper group $G$ is a subgroup of the affine group$\text{Aff}(\mathbb{E}^2)$ that preserves a lattice within the plane. Formally, if $g \in G$, then $g(\mathbf{v}) = A\mathbf{v} + \mathbf{t}$, where $A$ is a linear orthogonal transformation (a rotation, reflection, or glide reflection component) and $\mathbf{t}$ is a translation vector.
   The essential structural requirement is that $G$ must contain a translation subgroup …