Self-tiling tile set

From the definition above it follows that a set of self-tiling tiles, consisting of n identical figures, is a “fissile” tile , for which self-tiling tiles are a generalization ^[4] . Sets of n different shapes, such as in Figure 1, are called perfect . Figure 2 shows an example for n = 4 and it is not perfect , because the two tiles in the set have the same shape.

Shapes in sets do not have to be connected areas. Separated figures made up of two or more separate islets are also permitted. Such figures are considered disconnected or loosely connected (if the islands have one common point), as shown in Figure 3.

The smallest number of tiles in the set is 2. Figure 4 includes an infinite family of sets of order 2, each of which consists of two triangles P and Q. As shown in the figure, the triangles can be pivotally connected so that rotation around the hinge gives the same triangles P or Q (larger). These triangles give an example of articulation .

The properties of self-tiling tile sets mean that these tiles have the property of substitution , that is, they form a mosaic in which tiles can be cut or combined to produce a copy of themselves (smaller or larger). It is clear that repeating the process of combining tiles, you can get more and more copies (the process is called expansion) or smaller and smaller (compression), and these processes can continue indefinitely. In this way, self-tiling kits can form non-periodic mosaics. However, none of these non-periodic mosaics found is aperiodic , since protoplates can be combined to form a periodic mosaic. Figure 5 shows the first two stages of expanding a set of order 4, which leads to a non-periodic mosaic.

In addition to sets with self-tiling, which can be considered loops of length 1, there are longer loops or closed chains of sets of tiles in which each set piles the previous one ^[5] . Figure 6 shows a pair of mutually tiling sets of decamen tiles, in other words, a loop of length 2. Sellows and Schotel conducted an exhaustive search for sets of order 4 consisting of octamino . In addition to seven ordinary sets (with loops of length 1), they found a surprisingly large number of sets with loops of all lengths up to 14. The total number of loops found is about one and a half million. Further research in this direction has not been completed, but it seems true that other sets of tiles may contain loops ^[6] .

To date, two methods have been used to produce self-tiling tile sets. In the case when the set consists of figures of the type of polymino , in which the number of parts is fixed, a direct computer search is possible. It is easy to show that the number of tiles n must be a square ^[4] . Figures 1, 2, 3, 5, and 6 are examples found in this way.

Another way is to cut several copies of the "fissile" tile in some way, which leads to a self-sampling set. Figures 7 and 8 show the sets thus obtained. In them, each tile is a combination of two and three "dividing" tiles, respectively. In Figure 8 you can see how 9 tiles (top) together pave 3 “dividing” tiles (bottom), while these 9 tiles themselves are formed by combining the same three “dividing” tiles. Thus, each tile can be obtained by tiling each form with smaller tiles from the same set of 9 tiles ^[4] .

• Martin Gardner Chapter "Polyhexes and Polyaboloes" // Mathematical Magic Show. - Washington, DC: The Mathematical Association of Americf, 1989 .-- ISBN 0-88385-448-1 . (Reprint, 1977 books, Alfred A. Knopf)
• Lee Sallows. More On Self-Tiling Tile Sets // Mathematics Magazine. - 2014.- T. 87 , no. 2 .
• Lee Sallows. On Self-Tiling Tile Sets // Mathematics Magazine. - 2012.- T. 85 , no. 5 . - S. 323-333 .
• Geometric Hidden Gems by Jean-Paul Delahaye in Scilogs , April 07, 2013

• Self-Tiling Tile Sets website