We already saw how infinitely repeating patterns can have rotations.

As we might expect, wallpaper patterns can have ½ turns, which we can see a single piece rotate around,

Or an entire pattern rotate around.

Since the pieces of our patterns repeat along translations, their rotation points must repeat along these same translations as well.

Wallpaper patterns can also have ¼ turns,

As well as ⅓ turns,

And ⅙ turns.

But there are no other types of rotations for the wallpaper patterns, and we can see why.

As each of our pieces rotates around a point, we can imagine it drawing a shape around itself as a boundary.

This bounding shape has the same rotations as the point it was drawn around, and it is centered on that rotation point.

When our starting piece shifts over in a translation, or is transformed by any other symmetry, all of the other pieces that share its rotation point must go with it,

And so its rotation point and bounding shape follow as well.

This collection of pieces, with their rotation point and bounding shape, keep moving with the infinite translations and symmetries of a wallpaper pattern,

So that we can see that bounding shape as part of an infinite grid of identical bounding shapes, providing structure for a pattern.

See, a square can make a grid for a pattern that has ¼ turns, with 4 other squares meeting perfectly at each of its sides. Each time it rotates by a ¼ turn, the surrounding squares rotate around it, each landing on an identical square, so that a pattern structured within this grid can be left unchanged.

Making perfect grids is possible with shapes that have the right number of rotations,

Such as the shapes with 2, 4, 3 and 6 rotations that can be drawn around our pieces as they make ½ turns, ¼ turns, ⅓ turns and ⅙ turns.

But making these grids will not work with shapes that have any other number of rotations.

Their angles cannot add up in a way to perfectly equal a full turn, so these other shapes cannot perfectly fit together in a grid.

For example, we can draw a shape with 5 rotations around a piece as its makes ⅕ turns,

But we cannot make a perfect grid by surrounding that shape with 5 copies of itself at each of its sides.

So we cannot have an infinitely repeating wallpaper pattern with ⅕ turns.

Challenge: Can you see the rotations in the following patterns? Can you imagine an underlying grid?

Challenge: Use color to remove the rotations in the patterns while maintaining their translations.