Tessellation
Tessellations with polygons

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

Any triangle or quadrilateral (even non-convex) 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 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 exist with convex N -gons for N equal to 3, 4, 5, and 6. For N = 5, see Pentagonal tiling, for N = 6, see Hexagonal tiling, for N = 7, see Heptagonal tiling and for N = 8, see octagonal tiling.

With non-convex polygons, there are far fewer limitations in the number of sides, even if only one shape is allowed.

Polyominoes are examples of tiles that are either convex of non-convex, for which various combinations, rotations, and reflections can be used to tile a plane. For results on tiling the plane with polyominoes, see Polyomino § Uses of polyominoes.
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