Retrieving "Rhombus" from the archives

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1. Wallpaper Groups

   Linked via "rhombus"

   The group $p4$ requires a square fundamental domain because the $90^\circ$ rotation necessitates that the pattern repeats after a $90^\circ$ turn around a rotation center. This means that if the side length of the parallelogram defined by the translations is $L$, the side length of the square fundamental domain associated with $p4$ must be $L/\sqrt{2}$, implying that the underlying translation vectors are not the shortest possible repeats in …