page_title: Self-tiling tile set
section_title: Methods of construction
hierarchical_section_title: Self-tiling tile set / Methods of construction
caption_attribution_description: English: A rep-tile-based setiset of order 9
context_page_description: 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 and polyominoes were previously published by Gardner.
context_section_description: To date, two methods have been used for producing setisets. In the case of sets composed of shapes such as polyominoes, which entail integral piece sizes, a brute force search by computer is possible, so long as n, the number of pieces involved, is not prohibitive. It is easily shown that n must then be a perfect square. Figures 1,2,3,5 and 6 are all examples found by this method.
  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. Figures 7 and 8 show setisets produced by this means, in which each piece is the union of 2 and 3 rep-tiles, respectively. In Figure 8 can be seen how the 9 pieces above together tile the 3 rep-tile shapes below, while each of the 9 pieces is itself formed by the union of 3 such rep-tile shapes. Hence each shape can be tiled with smaller duplicates of the entire set of 9.
caption: Figure 8:   A rep-tile-based setiset of order 9.
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