Many of us made “snowflakes” as a kid, by folding paper in half horizontally and then vertically, then cutting a design through all the sheets. (Nevermind that snowflakes should have 6-fold symmetry instead of 4-fold.) Just as folding paper witnesses mirror reflection, rolling paper into a cylinder witnesses translational symmetry, rolling paper into a cone witnesses rotational symmetry, and rolling with a twist (like with Möbius band) witnesses glide reflection symmetry. By combining these elements, all 17 Wallpaper Symmetry groups can be realized.

Below are some activities inspired by the first third of The Symmetries of Things.

Portion of Running Bond brickwork.

Folded along a pair of parallel mirror lines

…then folded along a different pair of parallel mirror lines

Cut a slit to the 2-fold rotational point and rolled into a cone. The region of the previous step covers the cone doubly.

Punching a hole through all the layers. (Compass point used in lieu of proper hole punch.)

Additional injuries made by scissors.

Open up, and get copies of the injuries that respect the brickwork symmetry–and only those! (Ignore the slit needed to roll into a cone.)

Injuries highlighted by hand.

For more, see this handout with suggested activities, grids, reference for the 17 Wallpaper and 7 Frieze patterns, and mathematical development, or this collection of one sample of each of the 17 Wallpaper symmetry types, labeled by orbifold signature from Wikipedia. The same collection with markings for rotation points, reflection lines, and glide reflections.

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TIL of a website offering online free graph paper. You can get a square or triangular grid (among others) that are the basis for all Wallpaper Symmetry.

http://incompetech.com/graphpaper/