Self-tiling tile set

A self-tiling tile set, or setiset, of order n is a set of n shapes or pieces, usually planar, each of which can be tiled with smaller replicas of the complete set of n shapes. 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. Figure 1 shows an example for n = 4 using distinctly shaped decominoes. The concept can be extended to include pieces of higher dimension. The name setisets was coined by Lee Sallows in 2012, 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.

From the above definition it follows that a setiset composed of n identical pieces is the same thing as a 'self-replicating tile' or rep-tile, of which setisets are therefore a generalization. Setisets using n distinct shapes, such as Figure 1, are called perfect. Figure 2 shows an example for n = 4 which is imperfect because two of the component shapes are the same.

The shapes employed in a setiset need not be connected regions. Disjoint pieces composed of two or more separated islands are also permitted. 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 fewest number of pieces in a setiset is two. Figure 4 encapsulates an infinite family of order 2 setisets each composed of two triangles, P and Q. As shown, the latter can be hinged together to produce a compound triangle that has the same shape as P or Q, depending upon whether the hinge is fully open or fully closed. This unusual specimen thus provides an example of a hinged dissection.

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. Clearly, the twin actions of forming still larger and larger copies (known as inflation), or still smaller and smaller dissections (deflation), can be repeated indefinitely. In this way, setisets can produce non-periodic tilings. However, none of the non-periodic tilings thus far discovered qualify as aperiodic, because the prototiles can always be rearranged so as to yield a periodic tiling. Figure 5 shows the first two stages of inflation of an order 4 set leading to a non-periodic tiling.

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