The original nine-pointed star polygon is shown at the upper right
corner. The shapes into which it has been dissected tile a plane
Used with permission from David Eppstein's Geometry Junkyard - citation
Another type of tiling of the plane, though little-known, is known
as dissection tiling. Like simple tiling, dissection tiling uses
polygons to tile the plane, without gaps of overlaps. But unlike
simple tiling, dissection tiling has an added restriction.
To form a dissection tiling, begin with a polygon. Partition or
break this polygon into a collection of smaller polygonal pieces,
then use these pieces to tile the plane infinitely--use them to
create a standard tiling. The frequency of each smaller polygonal
piece, that is, how often it occurs in the dissection tiling,
must be in proportion to its area. Otherwise, one could take the
original polygon, cut out a small square from the middle, discard
the remainder, and just use the square to tile the plane.