Tessellations with polygons

Next to the various tilings by regular polygons, tilings by other polygons have also been studied - wikipedia

Voronoi tiling]], in which the cells are always convex polygons - wikimedia

Any triangle or quadrilateral (even concave polygon) can be used as a prototile to form a monohedral tessellation, often in more than one way. Copies of an arbitrary quadrilateral can form a tessellation with translational symmetry and 2-fold rotational symmetry with centres at the midpoints of all sides. For an asymmetric quadrilateral this tiling belongs to Wallpaper group#Group p2. As fundamental domain we have the quadrilateral. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational centre. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.

If only one shape of tile is allowed, tilings exists with convex ''N''-gons for ''N'' equal to 3, 4, 5 and 6. For , see octagonal tiling.

For results on tiling the plane with polyominoes, see Polyomino tiling.

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# Tessellations with polygons