We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational center. As fundamental domain we have the quadrilateral.
For an asymmetric quadrilateral this tiling belongs to wallpaper group group p2. To produce a coloring which does, as many as seven colors may be needed.Ĭopies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centers at the midpoints of all sides, and translational symmetry with as minimal set of translation vectors a pair according to the diagonals of the quadrilateral, or equivalently, one of these and the sum or difference of the two. Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. The four color theorem states that for every tessellation of the plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length. When discussing a tiling that is displayed in colors, to avoid ambiguity one needs to specify whether the colors are part of the tiling or just part of its illustration. Tessellations and color File:Torus-with-seven-colours.pngīy infinitely repeating this region, one obtains a partitioning of the plane such that if each parallelogram is given a solid color, it requires at least seven colors to prevent same colored parallelograms from touching. Of the three regular tilings two are in the category p6m and one is in p4m. All seventeen of these patterns are known to exist in the Alhambra palace in Granada, Spain. Tilings with translational symmetry can be categorized by wallpaper group, of which 17 exist. 6 Number of sides of a polygon versus number of sides at a vertex.
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