Tessellation

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A tessellation is a tiling of the plane with one or more shapes so as to leave no gaps and no shapes overlap.

When there exists a translation symmetry then the pattern must be one of 17 "wallpaper groups". http://www.clowder.net/hop/17walppr/17walppr.html

Of the regular polygons only the equilateral triangle, square and hexagon tessellate.

Many interesting patterns can be found with the inclusion of more that one regular polygons.

All triangles tessellate.

All quadrilaterals tessellate.

There are 14 different classifications of irregular pentagons which tesselllate. http://www.mathpuzzle.com/tilepent.html

There are 3 different classifications of irregular hexagons which tessellate. http://mathworld.wolfram.com/HexagonTiling.html

No convex shapes with 7 or more sides will tessellate.