Chapter 2

Order 4 Latin squares, reducing them from  576 to 5 in number

2.1. Enumeration of the 24 4×4 Latin squares

An algorithm was set up to generate Ls’s in standard numeric format. The procedure, sequential enumeration (Appendix II), is applicable in principle to Ls’s of lower order and ensures that no Ls is overlooked or created twice. When programmed and run, 24 4×4 (and 1344 5×5) Latin squares emerge, as expected; for order 4 in the following order and notation*:

Table 2.1.1. All 24 different order 4 Latin squares

Table 2.1.1. All 24 symbolically different order 4 Latin squares, in standard numeric format.

LS’s 01, 02, 05, 07 exhibit so-called reduced form (first row and column equal, first symbol 1 in italics). They are also symmetric (equal to their transpose, main diagonal is a mirror line).
LS01 and 05 are double-symmetric: main and back diagonals are mirror lines.
LS’s 14 and 21 are diagonal: main – and back diagonals are transversals (carry all symbols once).
S: these twelve Ls’s are Sudoku’s with 4 2×2 sub-squares, in agreement with ref. 8.
Seven have an extra Sudoku type sub-square in their centre (Ss).

The order 4 LS’s were grouped in Table 2.1.1 in a 4×6 array. The same set, in the same order, is presented in Fig. 2.1.1 in colour format similar to Latin square (2) of Fig. 1.1.1 (this LS is in fact LS09 of Table and Fig. 2.1.1). A similar tableau has been presented by Bill Lombard (ref. 7).


Figure 2.1.1. All 24 Latin squares of order 4 in standard colour format. Symmetry properties of squares LS01, 02, 03, 04, 05, 07, 09, 11, 18, 24 are indicated in the text. 

Symmetry. The Latin square condition forbids ordinary horizontal and vertical mirror lines and fourfold axes for any order (par.1.3). But this set is nevertheless “full of symmetry”. We note: one diagonal mirror line (m) in squares LS02 (main (par. 1.2)), 03 (back), 05 (main) and 18 (back); two perpendicular (mm) in squares LS01 and 07, and inversion centres (i) in LS01, 07, 09 and 11. There is also a horizontal glide line (g) (reflection + translation over half of the edge length) in LS06 , and horizontal and vertical glide lines (gg) in LS11 and in many others.

Colour symmetry of LS09 (square (2) of Fig. (1 1 1) was discussed in Chapter 1. Additional examples are the horizontal and vertical “mirror” lines (written m) in LS01 with a two-pair colour switch (red-blue and green-yellow), and the main diagonal of LS03: a one-pair (blue-green) colour mirror line. There is a fourfold colour-axis (4) with  colour shifts: red-blue -green-yellow -red-etc. in LS24. Are there more? Find them!

The  symmetry notations for all squares, in terms of Bravais lattices and 2D symmetry groups (wallpaper groups), will be given at the end of this chapter (List and Fig. 2.4.1), showing that no order 4 Ls is “symmetry-free”.

2.2. Point symmetry relations between Latin squares

A mayor goal of this study is to represent all Ls’s of a fixed order (n), by a minimal set of Latin squares, which we will term a Geometric Representation. It will be shown that there is a systematic way to reduce the order 4 set towards such a “irreducible” representation of five LS’s.
A similar result will be achieved – in an alternative manner – for the order 5 set (Chapter 3).

We showed in par. 1.4 that symbolically different Latin squares (4) and (5), Fig. 1.4.1, were  point symmetry related. LS05 and LS18, related by reflection, offer another example.  These LS’s are brought into coindence by a point group operation (reflection), par.1 .4) followed by converting to sf *.The point group operation here is a symmetry operation of the dihedral group for a square, 4mm, par 1.4. We indicate this relation with parentheses: (LS05, 18). A systematic way to reveal all point symmetry relations is:

  • apply all elements of the dihedral group to each member of the LS 4×4 set; this will result in 8 squares per member and in a total of 192 LS’s,
  • re-label the resulting LS’s in sf,
  • identify the final result for each member as either identical with its original or with another member . In the latter case we term this other member, and its original LS, each other’s point group related counterparts.
The procedure produces the original in six cases: LS01, 04, 07, 09, 11, 24. Three symmetry related pairs are produced within the series LS02, 03, 05, 14, 18, 21, splitting the series in three different pairs (twofolds) : (LS02,03) (LS05,18) (Ls14,21). Likewise, the series LS
06, 08, 10, 12, 13 15, 16, 17, 19, 20, 22, 23 splits in three symmetry related fourfolds: (LS06,08,10,17), (Ls12,13,15,19), (LS16,20,22,23).

Table 2.2.1 summarizes the results, showing that the full set is rearranged in 12 ”manifolds”: 3 fourfolds, 3 twofolds and 6 singles, and thus in 12 different structure types. The manifold number is the (point symmetry) multiplicity of the particular structure type. We obtain a point symmetry representation of the order 4 set by taking (arbitrarily) one LS from all manifolds , i.e LS01, 02, 04, 05, 06, 07, 09, 11, 12, 14, 16, 24; thus including all independent structure types within the representation.

Table 2. 2. 1. All 4×4 Latin squares in 12 manifolds of symmetry related structures 

singles    :      (01 )        (04)        (07)        (09)        (11)        (24)
twofolds :        (02 03)                     (05 18)                  (14 21)
fourfolds :     (06 08 10 17)         (12 13 15 19)        (16 20 22 23)

Ordering a full Ls set in manyfolds with different multiplicities is in striking analogy with the ordering of the faces of a crystal. On a orthorhombic crystal we could have horizontal (0 0 1) faces, perpendicular to a vertical c-axis, and (1 1 0) faces at an oblique angle with (horizontal) a- and b-axes of the crystal. But these (110) faces would be fourfold: (1 1 0) (1 -1 0) (-1 1 0) and (-1 -1 0) on a well-developed crystal. A crystallographer considers these faces equivalent and will state that the (110) face has fourfold multiplicity, the (100) face twofold (top and bottom), etc.

2.3. Order 4 Latin squares “on the torus”; obtaining  translation-symmetry related structure types

Table 2.2.1, with a 12- structures representation, is our reduction result for LS’s of order 4 if we compare them “as such”, that is: as 4×4 Latin squares, counting all point symmetry related LS’s within one structure type as one..
We saw however in par. 1.4 (Fig. 1 4.1) that a Latin square (1) ( LS09 of Table 2.1.1) was converted in (6) (LS16) when the columns of (1) were shifted cyclically. Connections of cyclically (= translation related) counterparts will be written: [LS09, 16].
Again, the systematic way to find all translation relations, in addition to the point-symmetry equalities retrieved in par. 2.2), is: applying all (n² -1) cyclically shifting operations (csh) to all order 4 LS’s and relate the resulting 4×4 squares with the originals (par. 2.1) after transforming them to sf. The relationships then obtained and grouped in translation manifolds, are given in Table 2.3.1, showing that the entire set of 24 different Ls’s is present on the tori (or in the Latin structures) of no more than seven LS’s; for instance: LS01, 02, 04, 05, 06, 08, 18.

singles    :            [ 05]        [18 ]
twofolds :                              [01 24]
fourfolds :        [ 04 21 14 11]        [ 06 12 17 13]     [ 08 19 10 15]
eightfold :                        [02 03 22 23 20 09 16 07]

Table 2.3.1. All 4×4 Latin squares in seven manifolds of translation related structures. This is our reduction result for LS’s of order 4 if we count all cyclically shifted  (or: translation-related) members as one structure type. Ls’s, combined in a (translational) manifold, such as LS01 and LS24, are different (shifted) “cuttings” from the same  Latin-periodic structure (par.1.6). Its fourfold colour axis (Fig. 2.1.1 ) is centrally located in LS24, but off-centre in LS01 which may somewhat hamper its identification there.
A Latin-periodic structure and a Latin square resemble a crystal and its primitive cell. Within a crystal the primitive cell is the smallest parallelepiped in which the crystal structure is contained and from which the full crystal is obtained by repetition through translation. Its position within the periodic crystal structure is arbitrary. Of course, a certain cell position may be preferred, i.e. one in which the structure and symmetry are best presented. Analogously: a Latin- periodic structure has no specific origin. The geometric specificity of a Latin square structure is fully contained in any LS cut out of its Latin structure or torus. We may nevertheless require larger cuttings ( or the "full" Latin structure) to observe all symmetry elements. A specific LS is often preferred as representative (List 2.4.1; representative LS's in bold).
 Primitive cell and Latin square are not entirely in analogy . A Latin square is normally a nxn section of a Latin structure. A primitive cell in a Latin structure may have an area which is number of times smaller. We saw  that LS01 is centred; Its primitive cell is a rhombus; twice as small. Par. 2.4 includes another example (LS05).

2.4. Combining translation – and point group symmetry

Let us finally see whether the number of minimally required independent structures can be further reduced by combining the translation – and point group symmetry reduction results (Tables 2.3.1 and 2.2.1). We note first that the “translation independent” singles LS05 and LS18 (Table 2.3.1) were shown to be point symmetry connected (par. 2.2). Furthermore, the translation fourfolds [LS06 12 17 13] and [08 19 10 15] (Table 2.3.1) have several cross-over point group connections: (LS06 08), (10, 17), (12, 15), (13, 19), best shown in Fig. 2.5.1, indicating that these fourfolds share the same Latin-periodic structure all together.

Our final result  for order 4 is that all Latin squares share their geometric pattern with no more than one of five independent Latin structure types, represented by LS01, LS07, S11, LS05, LS06 (List and Fig. 2.4.1); our ultimate Geometric Representation of order 4 LS’s.

Structure types Latin squares  numbers following Table 2.1.1 multiplicities (k) totalling 24
1 [01 24 ] 2
2 [07 09 (02 03) (16 20 22 23)] 8
3 [11 04 (14 21)] 4
4 (05 18) 2
5 [(06 17) (12 13)] [(08 10) (15 19)] 8
with (06 17 08 10)  (12 13 15 19)
List 2.4.1 Latin squares collected in manifolds with common structure type (left) and multiplicities(right). Point group – and translation relationships indicated with ( ) and [ ]. Preferred representatives (par. 2.3) in bold.
Figure 2.4.1 Geometric representation representing all order 4 Latin squares

1 Ls01 Ls07 Ls11 Ls05 Ls06
2 1 2 3 4 5
3 2V2x2V2; 8 4×4; 16 4×4; 16 2V2xV2 4×4; 16
4 rhombic rect. rect. rect. rect.
5 square square square rect. rect.
6 cmm pmm pgg pm pg
7 4' m'm' i4' mm' 4' m'm' m' m'
8 4ST (α) 2ST.2CC 4CC(α) 4ST(β) 4CC(β)
9 (1) (2) (1) (2) (2)

Figure 2.4.1 Geometric representation representing all order 4 Latin squares, with 5 preferred squares taken from each of the 5 manifolds (List 2.4.1), 2D crystallographic data, Latin pattern compositions (Chapter 3) and isotopy classes.
1 selected Ls-members for representing the order 4 LS set.
2 corresponding structure- type numbers (list 2.4.1)\
3 translation periodicity of primitive cell; cell areas (par. 1.1)
4 Bravais lattice type disregarding colour symmetry
5 Bravais lattice type acknowledging colour symmetry
6 conventional (planar) group in IUC-notation (ref.5))
7 colour planar group elements
8 Latin pattern compositions (par. 1.2 and 3.4).
9 allocation in the isotopy-classes, (1) and (2), par. 2.5
Structure LS01 repeats itself halfway the body diagonals (see Fig. 1.7.1), LS05 a quarter-way along the back diagonal, The other order 4 Latin squares correspond with the primitive cells of their Latin structures.

2.5. Remarks and observations related to List 2.4.1, Figures 2.4.1 and 2.5.1.

General. The structure type figures 2.4.1 are given as LS’s “as such” – for space saving – but they represent the corresponding “ongoing” Latin structures (like Fig. 1.6.2) enclosing all symmetry related members (List 2.4.1).

The  IUC space (planar) group notations (Fig.2.4.1, row 6) follow the conventional symmetry elements.  All corresponding Bravais lattices are rectangular (row 4). Note that all structure types  (1 to 5, row 8) are one to one related with the (conventional) planar groups (row 6) of order 4.

Presently there is apparently no standard nomenclature for groups of symmetry elements in 4- and 5- coloured planar objects. Thus we have restricted ourselves to establishing their colour symmetry elements as such (r0w 7). However, most of the  structure types  have the required colour symmetry t0 grant them higher Bravais lattice symmetries (row 5). Structure types 1, 2, 3 have  fourfold colour rotation axes, meeting the condition of  a square Bravais lattice.

Preferred Ls’s. We will comment on  structure and conventional/colour symmetry of each of  the preferred  Latin squares, adding a hyphen ( ‘ ) to  the (colour) symmetry symbols 4,  m and i. It might be helpful, dear reader, to prepare coloured Ls-field copies (similar to Fig.1.6.2. of each of them) exceeding the “as such” representations with one LS-c0py in all 8 directions and draw  the symmetry elements and the structural components mentioned above and underneath. All  counterparts (List 2.4.1, Fig. 2.1.1) will appear by shifting their “as such” boundaries across these Latin field copies.

LS01 (type 1) has four pairwise perpendicular Strings (par.3.1)  two by tw0 parallel  The strings are  als0 diagonal mirror lines m: (red, green) main-directed  and (blue , yellow) back- directed. Ther are also inversion centres (tw0f0ld axes). The lattice is centered and the cell rhombohedric; twice as small as the LS as such. The conventional planar symmetry group is cmm2.

Similar to LS07 (below) the edges of the square are colour- switching mirror lines; with in between an additional set of horizontal and vertical  m’ lines crosssing the centers.  The resulting fourfold colour axis is best seen (centrally located) in counterpart LS 24  (Fig. 2.1.1 The values are  undergoing colour shifts r-b-g-y-r etc. under clockwise rotation of the axis.

LS11 (type 3) has horizonal and vertical glide lines (m’ in colour symmetry), conventional planar group pgg2. Centre and vertices are inversion centres.

Centre and vertices are inversion centres. The hor.and vert.lines crossing them are colour reflection lines. A fourfold axis is situated at  the LS11 position (1,1)  but this 4′ axis is centrally located in  counterpart LS04 and as such easily seen.    The sequence is r-b-y-g under clockwise rotation.

LS07 (type 2) is particularly symmetry-rich. The 2red/2blue diagonals are  mirror lines m implying conventional inversion centers i  at all centers and vertices; the conventional planar group therefore is pmm2.

As for the colour symmetry:  there are several sets of colour mirror lines m’: the edges of the square are redblue– and greenyellow switching mirror lines; in between we have a similar hor./vert. set of colour switching m’ lines crossing the centers.  Finally,  between the m-diagonals: additional diagonal m‘ lines which are green and yellow Strings (Chapter 3) running respectively in main- and back direction. We thus have a square Bravais lattice (par. 1.2) and fourfold colour axes at all centers and vertices with the special fourfold sequence a b a b  because the d geometry here results from a 4′ axis and an inversion centre i.  Because of the similarity with  p4mm we might tentatively term the colour planar group p4’mm’.





Figure 2.5.1 Overview of the order 4 set showing 24 point symmetry – and/or translation related Ls’s, 5 structure types, 3 families (introduced in par.3.2 and 3.6), and 2 isotopy classes.                                                                                                                                  (1), (2):                 isotopy classification within the order 4 set
1,2, 3, 4, 5:           structure type numbers, introduced in Table 2.4.1.Types with the  same  Latin pattern  composition (par. 3.4) are combined ( / \ ) in families (par.3.2).
LS01,……..LS24:    LS numbering ( Table 2.1.1).

The structure of LS05, type 4, is a packing of of four back-diagonal Strings (par.3.1) with colour mirror lines m’ in between and with one main-diagonal mirror line m; planar group pm ;  colour planar group pmm’ (tentatively).The rectangular axes run diagonally as well and define a primitive cell which is four times as small as the LS.

LS06, structure type 5, has  horizontal glide line through the centre and the vertices, The conventional planar group is pg . Since the the glide lines are also colour reflection lines m’ the group might also be written pm’ in analogy with pm.

Order 4 network. The overall symmetry-connections network within the order 4 set is rather complex (Fig. 2.5.1), with translation- and point group symmetry relationships interweaved (Fig. 2.5.1), although the cube-structure of type 5 is neat. The following order 4  couples and – quadruple are point group – and translation related, { }, : {LS02, 03}, {14, 21}, {06 17}, {08 10}, {12 13}, {15 19}, {6 20 22 23}.

There are two isotopy classes for order 4 (par.1.3). It is easily shown that all LS’s of structure types 1 and 3 form one class (1) and the other structure types 2,4, 5 class (2) (Fig. 2.5.1).

Sudoku-categories s, S, sS (Table 2. 1. 1) in bold) have been added to the LS numbering underneath for correlation with structure type classification, first column (Table 2 4 1). Point group related LS's (parentheses) will share the same Sudoku category because the symmetry operations of the dihedral group leave the Sudoku property intact.

1       01     24Ss
2       07  02 03 09 (16Ss 20Ss 22Ss 23Ss)
3       11S 4 (14Ss 21Ss)
4       (05 18)
5       (06 08 10 17) (12S 13S 15S 19S)