A wallpaper pattern Overlaid patterns

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Baelde

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Several analogous patterns are overlaid in this "wallpaper", four tiling patterns.  An article calls "Pythagorean tiling" such a tiling covering entirely the Euclidean plane by squares of two different sizes, where any tile, by any edge, abuts against exactly one tile of another size.  Such tilings have necessarily a property of "wallpaper":  they are periodic.

Out of two small tiles with violet edges, a tile has an empty surface in bright blue, the other tile shows a central square bearing a tiny circle.  Leaving aside the central ornaments and the surface colors, the tilings of which the lines are violet or brown remain unchanged under the same group of translations.  This group is generated by two translations through equal distances in perpendicular directions, forming a 45 degrees angle with an edge of the image.

See other images for more informations:
• "Academ Translations depicted on a wallpaper"
• "A pattern Two explanatory grids"
See also the article "Wallpaper group", about a classification of groups of isometries that leave unchanged a wallpaper.
 

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