This is part of a set of 17 things representing all wallpaper groups, designed for learning about symmetry.

"A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorative art, especially in textiles and tiles as well as wallpaper."

https://en.wikipedia.org/wiki/Wallpaper_group

This thing includes tiles for creating patterns with the appropriate symmetry.
The tiles have an embossed swirl pattern that is helpful for observing rotation, translation, reflections and glide planes.
Straight edges indicate mirror symmetry.

Files:
p31m-cutout.stl - the asymmetric unit.
p31m-cutout-swirl.stl - optional insert for the asymmetric unit.
p31m-cutout-m.stl - mirror image asymmetric unit.
p31m-cutout-swirl-m.stl - optional insert for the mirror image asymmetric unit.

The figure shows the symmetry elements for this group.

openSCAD files are available upon request.

Wallpaper group p6m has six tiles per unit cell.

The group p31m has three different rotation centres of order three (120°), of which two are each other's mirror image. It has reflections in three distinct directions. It has at least one rotation whose centre does not lie on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.

Orbifold signature: 3*3
Coxeter notation: [6,3+]
Lattice: hexagonal
Point group: D3

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