Wallpaper Symmetries

There are exactly 17 symmetry patterns for tiling the 2d plane.

They form the Wallpaper Group, named because most people encounter them in wallpaper.

Any periodic tiling matches one of these symmetry groups, so the group can classify appearances in art like pottery, glasswork, rugs, and more.

A periodic tiling of the infinite 2d grid means the the pattern has repetition if you apply certain transformations.

The five fundamental transformations are: identity, translation, rotation, reflection, and glide reflection.

A glide reflection is a reflection over a specific axis followed by a non-zero translation along that axis.

These transformations are fundamental because they are complete and do not overlap. Any isometric transformation of the 2d grid can be made using exactly one of them.

A drawing demo is available here.

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Bonus question: Wikipedia show that this herringbone pattern is PG, meaning the minimal symmetry has two glide reflections and no rotations or reflections. The challenge is to find the symmetry and replicate it using the tool.