The plane tiling theorem states that the weight of a motif must equal 2. This small value makes it possible to consider every possible wallpaper symmetry: there are only seventeen wallpaper groups.
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Have you ever noticed? In friezes, tilings and other periodic patterns that repeat or exhibit symmetries, the artist's range of possibilities is actually quite limited. A fundamental theorem provides a complete enumeration of these patterns.
The decorative patterns of the Alhambra in Granada, a fortress built in the 14th century, encompass all the symmetry groups of plane tilings that can be repeated using two nonparallel translations. But this finding is recent, incomplete and somewhat controversial...
A tiling covers the plane, extending infinitely in two dimensions; a frieze extends in just one direction. The quintessential decorative motif, it is the decorative border you can put above your wallpaper. There can be only seven different types.
In geometry, what could be more natural than to seek, for each object, the transformations that leave it unchanged? This means dissecting those transformations and identifying the object's symmetries—and, surprise, groups emerge quite naturally...
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