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However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a diagonal line, we do not get the same pattern back; what we do get is the original pattern shifted across by a certain distance. This is part of the reason that the wallpaper group of A and B is different from the wallpaper group of C.

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The types of transformations that are relevant here are called Euclidean plane isometries. For example:

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Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., a colored tiling is equivalent with one in black and white with the colors coded radially in a circularly symmetric "bar code" in the centre of mass of each tile.

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A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns that repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry — when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.

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Examples A and B have the same wallpaper group; it is called p4m. Example C has a different wallpaper group, called p4g. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.

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Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns that are very different in style, color, scale or orientation may belong to the same group.

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Wallpaper groups are two-dimensional symmetry groups, intermediate in complexity between the simpler frieze groups and the three-dimensional crystallographic groups (also called space groups).

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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. There are 17 possible distinct groups.