Introduction to Latin squares
1.1. Latin square description and presentation
Latin squares are usually presented as arrays with indexed sites (cells), (i, j), occupied by numbers. These may be replaced by symbols or other entities (values). Throughout this article both numbers 1, 2, 3, 4, (5)… and coloured boards will be used to fill and highlight the sites of Ls’s. Symmetry and structure of a Ls show up particularly vivid in colour presentation. Sites are grounded upon the sites-lattice: a sub-lattice of the Bravais-lattice which is defined by the periodicities of the entire LS structure* (par.1. 2).
Ls-coordinates are x,y starting top left (0,0), site coordinates are (x=2i-1)/2, y=(2j-1)/2, with i and j 1...n. Intersite distances are 1 (adjacent), V2 (diagonal) , 2, V5 (knight’s move in chess), etc. Edge length (LS-order) is n. Areas are in corresponding units.
Figure 1.1.1. Various representations of the same Latin square.
(3) is the corresponding Latin-patterns composition diagram (par. 1.2).
Fig. 1.1.1 illustrates various ways in which a Latin square with a specific geometric structure can be presented. (Ls1) is in standard numeric format, snf*:first row in sequential order: 1 2 3 4; (Ls2) in our standard colour format scf or simply sf, with first row colours red(r) yellow(y) green (g) blue(b) (and purple (p) in order 5 Ls’s). The nomenclature-correspondence 1 -> red, 2 -> yellow, etc. (first rows of (Ls1) and (Ls2), is maintained throughout both squares. We have to establish a correspondence of numbers and colours to conclude that(Ls1) and (Ls2) are equal.
The equality holds if we produce “another” Latin square, (1gf), in general format, by permutation of the values of (1); here according to 1 2 3 4 → 2 3 1 4. The correspondence here is a permutation. Ls’s are really different * if such a correspondence does not exist. (Ls1) (Ls2) and (1gf), although symbolically and numerically different-looking , are equal*.
We will present Ls’s in the standard formats (sf) wherever possible. Converting to sf is a fast way to see whether two Ls’s are equal.
A LS will represent n! equal LS’s obtainable by permutation. This will bring the numbers of different* Ls’s of order 4 and 5: 576 and 161280, down already to (no more) than 24 and 1344 (ref.1).
Latin patterns. A meta-description of a Latin square is shown in Fig. 1.1.1 (3). It is obtained by connecting sites with equal symbol in (Ls1), (1gf) or (Ls2). The specificity (number, colour) of site entities can be entirely disregarded in (3); it is replaced by the coherence of the (hitherto) equally occupied sites. All networks defined in this manner (here two diagonals and two squares), already termed a Latin patterns (Lp’s) in the summary – are intermediate constituent of the structure of a Ls. Note that the Latin pattern concept implies the order n of a Ls twice: there are n Lp’s in a Latin square, distinct or equal, and each Lp has n members (sites). A Latin patterns diagram: Fig. 1.1.1 (3), the superposition of its constituting Lp’s, is now representing the Ls. It is independent of (= the same for) any specific representation or nomenclature of the LS.
A Lp composition formula can also be formed for each Ls, by naming and counting equal constituting Lp’s per unit cell (Chapter 3, formula 3.1.1), somewhat similar to assigning a chemical formula to a compound. Lp-diagram and Lp-formula will prove highly useful when classifying Latin squares’s in terms of structural building (par. 2.5 and 3.1).
Latin patterns often have a marked appearance (see Figs. 3.1.1 and 3.6.1 , reminiscent of structural components in crystals: layers, macromolecules, fibres, networks etc.
* Terminology which may be specific for this article is explained in the glossary (Appendix I)
1.2. Shape and crystal symmetries of Latin squares
You may wonder what can be said about the shape of a (Latin) square. In fact, all Latin “squares”, although nearly always (par.1.7) drawn as squares, lack the (conventional) point group symmetry properties for square symmetry: a fourfold rotation axis, or: diagonal and horizontal / vertical mirror lines. The first and last requirement contradict the Latin square condition for even and odd sized Ls’s. Instead, site/value arrangements within Ls’s, including those of Fig. 1.1.1, conform with a rectangular , oblique or trigonal axis (par.1.7) system for lack of (higher) internal symmetry.
The oblique axes system of many LS’s would readily show up if we would materialize them as a close-packed structure of discs and replace different colours by unequal disc-diameters or different attractions/repulsions.
Of course one is free to write and draw Ls’s as is suitable for well-ordered presentation and easy establishing (in)equalities between them, as illustrated in par. 1.4. However, when attributing a planar symmetry system and a planar group to a Latin square (Chapters 2 and 3), similar to a crystallographic space group, we will respect the symmetry originating from its specific spatial arrangement of values.
1.3. Conventional and colour symmetry
Let us consider the symmetry of the coloured (Ls2) once more. The only conventional symmetry element directly visible is an inversion centre (a twofold axis perpendicular to the paper) moving all sites towards sites with identical colour. Mirror lines through the centre are absent: horizontal and vertical reflections switch the colour of all sites; reflection across the diagonal lines turn red into red and blue into blue, but yellow and green switch colour.
We note though that the diagonal reflection induces a switch of all yellow and green sites . Similarly, the central horizontal/vertical lines, switch all red- blue and yellow-green pairs (known as perfect colouring).
In other words: these reflections perform permutations of colours of the Ls. So, in addition to its conventional symmetry elements, a Ls may carry colour mirror lines; more generally: elements of colour symmetry. We accept them as symmetry elements because we consider a Ls invariant under a overall switch of colour. It goes without saying that colouring can be replaced by other distinctions of site values.
In different words: An object is said to display colour symmetry if application of one or more of the isometric movements of the object followed by a permutation of its values – not being the unity permutation- can fully restore the original status. Or: if a full colour switch or a step in the colour sequence occurs in conjunction with the isometric movements of a square. For further reading we propose ref.8.
If an observer wants to test a possible element of crystal symmetry (mirror line, rotation axis etc.) in or on an object he must confirm first that the original and the image obtained by an isometric transformation (mapping) coincide fully as drawings (ref.2) , that is that their Latin patterns diagrams coincide. If he next does note distinct values, say colours, he will meet one of three possibilities: - each colour site will be mapped in the same colour of the original on the corresponding site: we have conventional symmetry, - all sites with common colour have adopted a single colour not being the original. We have a permutation of colours; we have colour symmetry, in this case a colour shift. If we proceed with the image in its new colour(s) we may return to the original colour or find still another colour etc. : we have a colour sequence. - the colours of the original and the mapping are not permutation related. The operation under consideration is not a symmetry operation.
Accepting colour symmetry elements in Ls- structures means firstly that we view Ls’s of or order n carrying colour symmetry as p-coloured symmetric objects, with p equaling n or lower. Secondly that the “additional” colour symmetry elements may raise the Bravais symmetry of the Ls’s, and thus their full space (planar) group symmetry as if they were not Ls-forbidden single colour conventional symmetry elements. For instance: if we establish a fourfold colour axis in a Ls under isometric movements we know that the entire Ls structure must be geometrically in accordance with square symmetry; in other w0rds: the Bravais lattice is square and the Ls adopts a square planar colour group. This will be illustrated in Chapters 2 and 3 for order 4 and 5 Ls’s.
1.4. Point symmetry relationships and structure types
We will use structure distinction or -correspondence of Ls’s as a vehicle to identify structure types, highly similar to what is practice in crystallographic classification. As an example we refer to two symbolically different Ls’s, (4) and (5), Fig. 1.4.1. left, and show that (5) is obtained by rotating (4) clockwise over 90 degrees (4rot) and next converting to standard format (sf).
(Ls4) and (5), although symbolically different as LS’s (par.1.1), are (point) symmetry related. They share the same structure type; they are dependent. All point symmetry relationships in any listing of same-order Ls’s can be established – and more structure types obtained – by testing them under the symmetry movements of he dihedral group for a square.
Figure. 1.4.1. (5) is obtained by rotating(4) over 90 degrees and bringing it back to sf; (6) is obtained by cyclic shifting of (1) and converting to standard format. Latin squares(4) (5) (1) (6) are LS02, 03, 09, 16 respectively of the full order 4 set (Fig. 2.2.1)
1.5. Cyclic (translation) symmetry operations
In Ls literature, operations on Ls’s include permutations of rows and columns. Such operations split a same-order set in so-called isotopy classes depending on whether Ls’s are mutually row-row (etc.) permutation-related or not. We will see (Chapters 2, 3) that our distinction in terms of structure types (above, par 1.4) is a subdivision of the isotopy classification. It follows that one may well “hop” from one structure type to the other with a row/row (etc.) switch. In other words: a Ls is generally not geometrically invariant under row/row etc. permutations.
A special case however – paramount for our entire approach – is cyclic permutation or – shifting, csh) of rows or columns This can be visualized by bending a Ls, for instance (1) of Figs. 1.1.1 and 1.1.2, into vertical cylindrical shape by connecting the end columns, reopen the cylinder elsewhere, obtain (1csh) and bring the result back in sf (6) . The resulting Ls (6) is (translation) symmetry related to the original (1). So, again, we have dependent, symmetry related, structure types. Translations can be repeated. The unit translation equals the period of the sites lattice.
1.6. Torus and Latin field
Cyclic shifting can be generalized further by bending the cylinder (par.1.5) into a torus, (Fig. 1.6.1 , ref. 4 ). A torus allows for shifting a LS of order n in two directions: along and around the torus, thus exposing n²-1 translation-related representations of the original. Shapiro (ref. 3 1978) seems to have been the first to propose the torus as a natural object to extend the notion of a Latin square (ref. 3). A Latin square in toroidal representation has origin nor edges.
If we map the sites-lattice on the torus we find that the rows and columns are circles now, around and along the torus; the diagonals follow ellipses. There are two finite groups to consider: the translations group with n²-1 elements, and the point symmetry group which is the order 8 version of the dihedral group. Translation movements, following the sites lattice, can be numerous (n²-1), whereas the number of point symmetry operations is always 8.
We note (Fig. 1.6.1) that the point group – and translations group – movements have similarity on the torus; both groups are finite; the translations move sites (“cells”) and their values (colours) towards other sites; the point symmetry movements do the same but may leave certain site-values, or all values along certain lines, unchanged. This is why we consider the movements connecting translational-symmetry related LS’s: ((1) and (6) of Fig. 1.4.1), similar to those connecting point symmetry related (4) and (5).
Figure 1.6.1. Latin square LS01 (Table 2.1.1), in toroidal representation.
Since a Latin square on a torus is somewhat difficult to draw and overlook we may as well extend the structure of a Latin square wallpaper-wise, terming the resulting 2D-periodic array a Latin-periodic structure, or briefly: Latin field (Fig. 1.7. 1). An order 5 example is Fig. 3.1.2 . Latin patterns and Lp diagrams (par.1.1) can be likewise extended.
Note that extension of a Ls as such is convenient to fully describe its symmetry (conventional and/or colour -) of the structure. Looking once more at (2) we might conclude that conventional symmetry is limited to the inversion centre at the centre. However by adding a row and a column in all four directions we note a system of conventional mirror lines, diagonally following yellow-green-green-yellow sites.
1.7. Latin squares in an omniverse of related patterns
Once we generalize a Latin square “as such”, nxn, towards an ongoing unbounded Latin field we can compare the Latin square world with other similarly related families of patterns, 3D or 2D, such as crystal structures and wallpaper designs in particular, quilts , needle-work, tessellations, tilings, settings on (chess) boards (Ref. 4) and various architectic, decorative and artistic (Escher!, Ref. 2) creations. Ref. (5 ) on wallpaper designs is an excellent introduction to conventional planar symmetry groups in general
Subsequently,when in Ls-world, we can try to follow the symmetry-hierarchy of crystalline -and wallpaper structures (2D) with 7 (4) crystal systems (triclinic, monoclinic etc.) and 230 space groups (17 planar groups), and apply it in the Latin square environment.
However, the overall variety of symmetry systems and groups “available” for Latin squares is hampered by the Latin square conditions: lattice arrangement of sites, and: all values once and only once per period along two main axes. We have seen that square symmetry is impossible in order 4 and 5 unless we accept colour symmetry and so is trigonal unless we accept a Latin hexagon for order n=3 . Ref. (6) invites readers to design a Latin hexagon for n 6. This also illustrates the order-dependency of symmetry-variety . The possible Ls systems are reduced to oblique and rectangular with a subsequent reduction of the number of planar groups. However, colour symmetry may raise symmetry and diversity. Also, keep in mind that diverse structures may carry the same formal group symmetry (Chapter 3, par. 3.3).
Figure 1.7. 1. Order 4 Latin square (LS01, Table 2.1.1), extended towards an ongoing Latin periodic structure, briefly Latin field.
1.8. Translation movements and space (planar) group symmetry
The overall translational periodicity in a (Latin) structure is pinpointed by its Bravais-lattice. Some patterns repeat themselves completely within their Latin square boundaries as can best be seen in an ongoing structure, Fig. 1.7.1, a centred LS with repetition of the structure halfway the body diagonals.
Glide lines, combining reflection and translation halfway a full translation period, and all other symmetry operations of a Ls under consideration, constitute together its planar group (Chapter 2 and 3).
We summarize the various symmetry- and structure aspects (to be) raised in the present and following chapters:
1 : the various descriptions and representations of Ls’s, pinpointing Latin patterns as building blocks of Ls -structures;
2 ; the point group symmetry movements of any square (the dihedral group of order 8, 4mm);
3 ; the conventional point group and the symmetry movements of the colour group of a Ls;
4a; extending the Ls notion by putting a Ls unbounded on a torus and collect the translation symmetry movements in a finite group;
4b; alternatively: having the Ls 2D-repeated in a wallpaper-wise Latin field;
5a ; finding the point group- and/or translation symmetry relationships between Ls’s and:
5b; partitioning Ls’s of same order in structure types according to common symmetry;
6 ; determining the IUC designation (in terms of planar classes and groups) for Latin squares and – fields