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List of loops

File Loops.txt provides a list of the 1,429,281 loops discovered by Henk Schotel using a Java program due to Frank Myer that implements an algorithm of Donald B. Johnson's to find all elementary cycles in a directed graph. ElementaryCycles.zip  can be downloaded from http://normalisiert.de. Loops.txt  is around 100 MB. A zipped version (9 MB) can be downloaded here.


The file contains 1,429,281 numbered rows, each showing a closed chain of quads. As it happens, the number of different quads involved in the octomino network is 62, for which reason the quads are labelled Q01, .., Q62. The notation Q01->Q02 is used to indicate that Q01 tiles Q02, or more precisely: the 4 octominoes in Q01 will tile each of the 4 bigocto's represented by Q02. The arrow at the righthand end of each chain indicates that the last quad tiles the first. For the piece numbers that identify the 4 octominoes represented by each quad click on 'The 62 quads'. The number preceding each chain is its length. Loop lengths run from 1 up to 14. Table 1 in "On Self-Tiling Tile Sets" shows the number of loops found for each length. By a loop we mean a closed cycle of quads in which each tiles its successor. These include bi-directional loops in which quads tile both their successor and predecessor. The latter are not identified as such in the file, but give rise to two separate entries, one for each direction. For example, among others, the 'Quintet' of quads featured in the article ´╗┐includes the loop of length 5: Q06->Q30->Q27->Q24->Q07->, which appears in row 3122 of the list along with Q06->Q07->Q24->Q27->Q30-> in row 2407.

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