Self-tiling tile set

That is, the n shapes can be assembled in n different ways so as to create larger copies of themselves, where the increase in scale is the same in each case.

The name setisets was coined by Lee Sallows in 2012,[1][2] but the problem of finding such sets for n = 4 was asked decades previously by C. Dudley Langford, and examples for polyaboloes (discovered by Martin Gardner, Wade E. Philpott and others) and polyominoes (discovered by Maurice J. Povah) were previously published by Gardner.

Such pieces are described as disconnected, or weakly-connected (when islands join only at a point), as seen in the setiset shown in Figure 3.

The properties of setisets mean that their pieces form substitution tilings, or tessellations in which the prototiles can be dissected or combined so as to yield smaller or larger duplicates of themselves.

Figure 5 shows the first two stages of inflation of an order 4 set leading to a non-periodic tiling.

[5] Figure 6 shows a pair of mutually tiling sets of decominoes, in other words, a loop of length 2.

Alternatively, there exists a method whereby multiple copies of a rep-tile can be dissected in certain ways so as to yield shapes that create setisets.

Figure 1: A 'perfect' self-tiling tile set of order 4
Figure 2: A setiset with duplicated piece.
Figure 3: A setiset showing weakly-connected pieces.
Figure 4: An infinite family of order 2 setisets.
Figure 5: A setiset of order 4 using octominoes. Two stages of inflation are shown.
Figure 6: A loop of length 2 using decominoes.
Figure 7: A rep-tile-based setiset of order 4.
Figure 8: A rep-tile-based setiset of order 9.