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.