Like the frieze patterns, we can see the infinite translations within wallpaper patterns by focusing on a single piece that shifts over,

This time in multiple directions.

And again we can see how the entire pattern can shift with these translations. Because each shifting piece is followed by infinitely more pieces, the entire pattern is left unchanged.

These translations in different directions are symmetries of the wallpaper patterns that we can combine to see such translations in even more directions.

→ ✷ ↓ = ↘

Any 2 different directions of translation can be combined with each other, or with themselves, in any number of ways, again and again, to produce more,

→ ✷ → ✷ ↓ ✷ ↓ ✷ ↓ ...

Showing us how these 2 translations can be the generators for a group of translations that span across all directions of the wallpaper patterns.

And once again we can use color to alter our patterns, such as taking away some of these translations,

And doubling the shortest distance a pattern can translate vertically,

Or the many other ways that you will puzzle over.

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The simplest wallpaper patterns have translations as their only symmetries.

They get more interesting when we consider groups with more complex symmetries...

Challenge: Can you color the pattern so that the shortest possible translation is in a diagonal direction? Then add more color so that the shortest possible distance of translation is tripled.