Combining results with Latin square literature and general conclusion
5.1. Tabulating order- dependent numbers and structural parameters
We have collected significant parameters for Ls’s sets of orders up to 7 in Table 4.2.1.
|1.||Total Ls-number in general format||2||12||576||161280||812851200||61479419904000|
|2.||Total number in standard format||1||2||24||1344||1128960||12198297600|
|3.||Default multiplicity per structure type||32||72||128||200||288||392|
|4.||Average multiplicity per structure type||1||2||5||75||200||300|
|5.||Number of Latin patterns||1||1||2||4||10||27|
|6.||Number of composition families||1||1||3||14||?||?|
|7.||Number of structure types||1||1||5||18||100||2000|
|8.||Number of structures within a family||1||1||1 or 2||1 or 2||?||?|
|9.||PG ( par. 3.5 )||12||192 *||145164**||?|
|10.||TG ( par. 3.5)||7||60||?||?|
|11.||Number of isotopy classes||1||1||2||2||22||564|
|12.||Number of orthogonal pairs (st. format)||-||1||6||216||0||123292800|
*first determined by Barink
**determined by Joriki
numbers in bold determined by the author
author’s purely order-of-size guesses in italics
Table 5.1. Numbers of Latin squares and related quantities with increasing order
Ad r0w 3
Default multiplicity is the multiplicity in the absence of symmetry like we have in order 5 structure types 4 and 11. The maximum (default) multiplicity in standard format of a Latin square structure within an order n set is 8. n² (8. n² . n! in general format). Multiplicities in the lower order domain are far lower because of symmetry. Symmetry is expected to be (far) less dominant in higher order LS’s so the relative (actual versus default) multiplicity) might rapidly rise towards 1 for higher n.
Ad r0w 4
Set- multiplicity divided by the number of structure types. Very low at orders up to 5 but likely to rise towards default multiplicity at high orders due to the expected low number of symmetrical Ls’s
Ad r0w 5
Remarkably slow rise. Increasing by a factor 5 from n 4 to 6 where as the standard format set- multiplicity rises by a factor of 50000.
Ad row 7
The most intriguing number missing is the number of independent structure types for n equal 6. Averaging various extrapolations the author settles for something of the order of 100…..
5.2 General conclusion
We have studied the geometry of Latin squares in order to reach a classification of LS’s in structure types, rather similar to identifying building trends within a class of related crystal structures. Our approach included:
● Generalizing the Latin square construct as such into a cyclic or on-going structure by either putting it on the torus or repeating it wallpaper-wise in a Latin field; at the same time extending the “same structure” criterion of two Ls’s to isomorphism of the Latin periodic structures
● subjecting same-order sets of Latin squares to a rigorous symmetry analysis, including colour symmetry, in order to establish the (number of) independent structure types:
● proposing the Latin pattern concept: the geometric pattern defined by the positions of sites with identical symbol, adopting Latin patterns as the major constituents of a Ls, and setting up a Latin-patterns composition formula for describing a Ls, similar to the chemical formula of a compound. The Latin patterns number is a mayor structure parameter, at the same time controlling the set growth towardsr higher order
● showing that the LS sets of order up to 5 can be partitioned in a highly limited number of families with members sharing the same Latin pattern composition formula
● establishing a powerful vehicle in the lower order domain: LS’s with the same composition family share the same- or no more than two different structures-types.
● showing that our division of LS’s in (sub)families(= in structure types) is a subdivsion of the isotopy classification.
The decomposition of a Ls in Latin patterns, the very slow rise of Latin pattern numbers with n, the low numbers of independent structure types within same-order sets: 5 and 18 for order 4 and 5, the subdivision of the isotopy classes in structure type families, showing the analogy of the structure types of a full fixed-order set with the habitus of a crystal, a proposition to relate p-colour symmetry with order n Latin squares, would seem novel contributions to the knowledge of Latin squares. Our geometric analyses correlate well with often quoted examples of orthogonality. The same holds for obtaining the numbers of Ls’s in reduced form (par. 3.3) and for the correlation of short range order and structure type.
The approach for obtaining most results could be termed semi-empirical; many of them could well have been obtained faster in a more analytic manner. But then the article would have been fairly incomprehensible for non-combinatorial readers
The approaches offered in this paper are extensible to Latin squares of larger magnitude, given appropriate computational means. Identifying Ls’s as combinations of Latin patterns, and placing them in composition families, is a straightforward means to ”format” the immense populations of higher order Latin squares, at the same time eliminating what is not really relevant: multiplication by permutation, different orientation and – boundaries, thus establishing their true geometric diversity.
Latin square presentations in which the site symbols are (coloured) objects (boards, circles, connected in nets and threads or highlighted as “singles”, were shown to be useful in assessing structure types, symmetries, multiplicities. They might also be inspiring for designs of tile fillings (tilings), mosaics, fabrics, quilts (ref. 15) and perhaps for recreational toys and educational games.
An example of a tile filling based on the couple-couple Latin pattern in 4×4 Latin squares is presented underneath. The order 5 LS-structures offer abundant inspiration for further aesthetic endeavours.
Figure 5.2.1. Tile filling, known as snub square tiling (ref. 13), based on the couple-couple (CC) Latin pattern (in black) in 4×4 Latin squares (par. 3.4, fig. 3.4.1) as an example of designs which can be taken from Latin squares and Latin patterns.
Creation of sets of Ls’s up to order 6, and all calculations, permutations, symmetry – , orthogonality – and other numeric or symbolic operations on LS structures required throughout this article were programmed and performed on a Hewlett Packard scientific(graphic) calculator (HP48 and later versions). Handy programs – for putting LS’s on standard numeric format, checking for symmetry relationship or orthogonality, can be made available on request. Some figures were prepared using a version of Processing, some others with a homemade graphic program based on Forth.
The author is indebted to Marc van der Zalm for creating the font cover header.
1. Cited for instance in Wolfram MathWorld http://mathworld.wolfram.com/LatinSquare.html 2. Douglas Dunham, University of Minnesota, Creating Repeating Patterns with Color Symmetry. Web Site: http://www.d.umn.edu/~ ddunham/ 3. Henry D. Shapiro, Generalized Latin squares on the torus, Discrete Mathematics, volume 24, issue 1, 1978, p. 63 -77 4. John J. Watkins, Across the Board, Chapter 5, The Torus and the Cylinder, 2007, Princeton University Press 5. Wikipedia, Wallpapergroup http://en.wikipedia.org/wiki/Wallpaper_group#Group_p16. 6. Latin Hexagon, math.sunysb.edu/...-squaresII-901/latinII3.html, page presently(2017) not accessible, but try: Latin Hexagon, origame live journal.com/40923.html 7. Bill Lombard (2009), try: Google Analytic: finitegeometry.org/sc/16/latin. 4x4.html 9. http://mathworld.wolfram.com/RooksProblem.html 10. H. W. Barink, webside: How many structurally different Latin squares order 5 do exist, https://latinsquares5x5, wordpress.com 11. Joriki, Mathematics http://www.stackexchange.com/questions,1381193 page presently (2017) not accessible. 12. Terry Ritter http://www.ciphersbyritter.com/GLOSSARY.HTM#OrthogonalLatinSquares 13. G.P. Graham & C.E. Roberts, 2006. Enumeration and isomorphic classification of self-orthogonal Latin squares, Journal of Combinatorial Mathematics and Combinatorial Computing, 59, pp. 101–1114 14. A.P. Burger, M.P. Kidd, J.H. van Vuuren, 2010, Enumeration of Self-orthogonal Latin squares,http://www.researchgate.net/publication/228942665 15. Wikipedia, Snub square Tiling, http://en.wikipedia.org/wiki/Snub_square_tiling