Abstract
Mathematicians have established that there are exactly seventeen plane symmetry groups, known as “wallpaper
patterns.” Multiple mathematical fiber arts papers describe subsets of those patterns that are suitable for textile design
with grid-based media such as knitting and cross-stitch. This paper extends the existing library of these patterns to
include orientation-based variations and four-fold rotations. In particular we enumerate all plane lattice symmetries
that can be generated from square tiles using lattice translations and combinations of other lattice symmetries.
This work has application beyond mathematics as we also provide a useful patterning tool for knitting and fiber
arts designers; we introduce Symmetry Generator software created using Python/Processing and OpenSCAD which
provides a method for exporting plane lattice symmetry designs as hand knitting patterns or as physical punch cards
for use in knitting machines.
