CRYSTAL STRUCTURE AND SYMMETRY
We live in a universe of patterns
Geometry and symmetry of Latin squares are explored in this article. It is shown that collections of Latin squares of fixed order can be partitioned in a number of specific structure types according to different geometries. The full sets of 576 Latin squares of order 4 and 161280 of order 5 can be stepwise reduced to no more than 5 and 18 Latin squares respectively with different geometric structures.
Approaches to perform the reduction processes include: extending the notion of a Latin square to a cyclic or periodic structure, determining symmetry relations between different Latin squares, and utilizing that the numbers of different geometric patterns formed by same-symbol sites in Latin squares increase remarkably slowly with order; 2, 4, 10 and 28 for order 4, 5, 6 and 7 respectively. The numbers of ways in which these “Latin patterns” can be combined to form Latin squares with different structures are shown to be rather limited as well.
Aim and scope of this study
Latin squares (LS’s) are intriguing objects in general – applied – and recreational mathematics. Popular familiarity has currently been enhanced by the well-known Sudoku puzzles. The author’s interest is in Latin squares (LS’s) as geometric objects, and in such questions as: are the geometries of different Ls’s different as well or can they be related in terms of common symmetry and/or similar structure; and if so: can a minimal set of LS’s be assigned for each order comprising the geometries of all others; what symmetry types (planar groups) in terms of (2D) crystallographic symmetry classification are adopted by Latin squares?
Questions which are in fact so closely related to the everyday endeavour of a crystallographer that the 2010 title of this article “Geometric features of… ” has been changed in “Crystal structure and symmetry of…”.
Finally: are familiar combinatorial distinctions within LS sets, such as reduced-form appearance, division in isotopy classes, (self)orthogonality, etc. related to classifying their geometric structure?
Higher order LS’s, such as the order 9 Sudoku, are not the most obvious starting point for studying matters indicated above. After an introductory Chapter (1) we will start out with order 4 LS’s and see what features are encountered (Chapter 2). Next we will move to the (odd-dimensioned) LS’s of order 5, and add modest extensions toward order 6 – and 7 LS’s (Chapter 3). Special subjects including orthogonality are in Chapter 4. General approaches and insights gained are summarized in Chapter 5.
All in all, the author hopes that the contents of this article will be of interest for crystallographers who might be tempted to see how their skills can be targeted towards a non-material world, as well as for combinatorial mathematicians who might discover that despite the enormous literature on Latin squares there are some unperceived morphological aspects which are worth to be looked into.
Above all: enjoy the Art and Science of Latin squares!