Problem of the Month (October 2002)
Shapes that can be tiled with smaller congruent copies of themselves are called reptiles. Several small polyominoes are reptiles:
Shapes that can be tiled with smaller copies of themselves, not necessarily all the same size, are called irreptiles. Whereas reptiles have been wellstudied, irreptiles have not. Here are a few to whet your appetite:
What irreptiles can you find? For a given shape, we call its order the smallest number of copies needed to tile it. What are the orders of some irreptiles? Can you find a shape with order 2? Are there irreptiles with arbitrarily large orders? Are there irreptiles with all possible orders? Can you find a polyomino irreptile that cannot tile any rectangles?
ANSWERS
Jeremy Galvagni found some trapezoid irreptiles. This one has order 10:
Stewart Hinsley is an expert in fractile reptiles. See his page here.
Andrew Bayly found that right triangles have order 2, and all other triangles have order 4. He also found 2 of the 3 rectangles with order 3, and that all other rectangles have order 4. He came up with a sequence of trapezoids which he thought might have all large orders, but he didn't know how to prove this.
Stewart Hinsley gave a similar construction but couldn't prove that these trapezoids didn't have a smaller order. He did eventually prove that there are trapezoids of all odd orders. Here is his trapezoid of order 5:
Stewart Hinsley also says he can prove that there are rectangular reptiles corresponding to all unit quadratic Pisot numbers.
Andrew Bayly also came up with a sequence of polyomino examples to show that arbitrarily large orders exist. If we take a polyomino where 2 of them tile a square, and one of them has a very thin part, it takes a lot of copies to tile the thin part.
Both Andrew Bayly and Jeremy Galvagni found polyiamond examples of irreptiles using the same ideas:
Mike Reid found an infinite family of cyclic quadrilaterals with order 3:
Karl Scherer found a polygonal reptile with order 2, which he calls the "golden bee":
Ernesto Amezcua Found these three Lshaped reptiles of order 7. The left one has height/width ratio of √3, the center one has height/width ratio of √(5/2), and the right one has height/width ratio of √(5/3).
Polyominoes
Shape  Smallest Irreptile Packing  Source 
 4  Trivial 
 4  Trivial 
 4  Trivial 
 16  Karl Scherer 
 4  Trivial 
 10  Karl Scherer 
 40  Karl Scherer 
 8  Karl Scherer 
 9  Mike Reid 
 12  Rodolfo Kurchan 
 18  Rodolfo Kurchan 
 22  Mike Reid 
 30  George Sicherman 
 63  Livio Zucca 
 10  Karl Scherer 
 14  George Sicherman 
 68  Mike Reid 
 10  Mike Reid 
 10  Karl Scherer 

Shape  Smallest Irreptile Packing  Source 
 10  George Sicherman 
 10  George Sicherman 
 10  George Sicherman 
 10  Karl Scherer 
 40  George Sicherman 
 43  Mike Reid 
 12  Erich Friedman 
 12  Erich Friedman 
 12  Erich Friedman 
 12  Erich Friedman 
 12  Erich Friedman 
 12  Erich Friedman 
 14  George Sicherman 
 17  Mike Reid 
 9  George Sicherman 
 7  Mike Reid 
 62  Mike Reid 

Polyaboloes
Shape  Smallest Irreptile Packing  Source 
 3  Karl Scherer 
 8  George Sicherman 
 34  George Sicherman 
 8  Karl Scherer 
 5  Karl Scherer 
 14  George Sicherman 
 34  Karl Scherer 
 17  George Sicherman 
 8  Karl Scherer 
 16  George Sicherman 
 5  George Sicherman 
 16  George Sicherman 
 5  George Sicherman 
 8  George Sicherman 

Polyiamonds
Shape  Smallest Irreptile Packing  Source 
 10  George Sicherman 
 6  Karl Scherer 
 6  Karl Scherer 
 16  Karl Scherer 
 14  George Sicherman 
 10  Karl Scherer 
 10  Karl Scherer 
 20  George Sicherman 
 14  Karl Scherer 
 10  George Sicherman 
 5  Karl Scherer 
 65  George Sicherman 

Polydrafters
Shape  Smallest Irreptile Packing  Source 
 2  George Sicherman/td> 
 5  Karl Scherer 
 8  George Sicherman 
 8  George Sicherman 
 17  George Sicherman 
 35  George Sicherman 
 6  George Sicherman 
 10  George Sicherman 
 6  Karl Scherer 
 7  George Sicherman 
 12  George Sicherman 
 10  George Sicherman 

Polydoms
Shape  Smallest Irreptile Packing  Source 
 2  George Sicherman 
 6  Karl Scherer 
 8  George Sicherman 
 9  George Sicherman 
 9  Karl Scherer 
 15  George Sicherman 
 15  George Sicherman 
 6  George Sicherman 
 17  George Sicherman 
 17  George Sicherman 
 6  George Sicherman 
 6  George Sicherman 
 6  George Sicherman 
 6  George Sicherman 
 6  George Sicherman 
 6  George Sicherman 
 9  George Sicherman 
 13  George Sicherman 
 13  George Sicherman 
 8  Karl Scherer 

If you can extend any of these results, please
email me.
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