Polypolygon Tilings

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Polypolygons

Polypolygons are figures assembled from other polygons. The most studied examples are figures made from triangles, squares and hegagons. Mathematician Solomon W. Golomb coined the term "polyomino" in 1954 for figures made from squares; by analogy, "polyiamonds" are figures made from triangles and "polyhexes" are figures made from hexagons. Although all sorts of other figures are possible, only these three types seem to lend themselves to a wide variety of interesting tiling problems. Polyominoes have been most thoroughly studied.

Number of Polypolygons
Order (Number of Units) Polyiamonds Polyominoes Polyhexes
1 1 1 1
2 1 1 1
3 1 2 3
4 3 5 7
5 4 12 22
6 12 35 83
7 24 108  
8 66 369  
9 160 1285  
10   4655  

Polyominoes

There are only one type of monomino and domino; both obviously tile the plane. The two trominoes and five tetrominoes all tile the plane as well.



Golomb registered the term "pentomino" as a trademark for puzzles involving this set of shapes. All 12 pentominoes tile the plane.



All 35 hexominoes Tile the Plane



Almost All Heptominoes Tile

There are 108 heptominoes. Only one requires reflection to tile (shown in yellow) and only four (in red) do not tile at all.

Polyiamonds

All Polyiamonds of Orders 1-6 Tile the Plane.



All Heptiamonds But One Tile

There are 24 heptiamonds. Only one, the V heptiamond (shown in red) does not tile the plane.





All Octiamonds Tile

Octiamonds that require reflection to tile are shown in dark green.

Polyhexes

All polyhexes through order 5 tile the plane.




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Created July 1, 1999, Last Update July 9, 1999