**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 any 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:

##### Table 2.1.1. All 24 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. ).

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. Note the fourfold colour axis in the last Latin square (LS24) with clockwise colour shift: red → blue, blue → green, green → yellow , yellow → red, for all boards.

*Symmetry*. We saw that the Latin square condition forbids ordinary horizontal and vertical mirror lines and fourfold axes for any order (par.1.1). 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 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 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 in LS24. Are there more? Find them!

The symmetry notation for all squares, in terms of the 17 two-dimensional 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, termed 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. 1.1 that different Latin squares (4) and (5), Fig. 1.4.1, are point symmetry related. LS05 and LS18, related by reflection, offer another example. These LS’s, although numerically different, are brought into complete coincidence by a point group operation followed by a permutation of the site symbols, thus sharing the same structure type,. The point group operation here is a symmetry operation of the dihedral group for a square, *4mm* (par. 1.1). 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 standard format,
- 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 LS series LS06, 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 ”manyfolds”: 3 fourfolds, 3 twofolds and 6 singles, and thus in 12 different structure types. The manyfold 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 manyfolds , 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. Collecting all 4×4 Latin squares in 12 manifolds of symmetry related structure types

**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..*

We saw however in par. 1.2 (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.* These cyclically – or translation related counterparts are 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) operations of the cyclically shifting(csh) to all order 4 LS’s and relating the resulting 4×4 squares with the originals (par.1.2) after transforming them to standard format. The relationships then obtained and grouped in translation manyfolds, 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: 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 Collecting all 4×4 Latin squares in 7 manyfolds of translation related structure types

Table 2.3.1, is our reduction result for LS’s of order 4 if we count all cyclically shifted – or translation-related members as one.

LS’s, combined in a (translational) manyfold, such as LS01 and LS24, are different (shifted) “cuttings” from the same Latin-periodic structure. Its fourfold colour axis (Fig. 2.1.1 ) is centrally located in LS24, but off-centre in LS01 which may somewhat hamper its identification.

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 always 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 (par 1.3) that LS01 is centred; Its primitive cell is a rhombus; twice as small. Par. 2.3 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 is that all order 4 Latin squares share their geometric patterns with one of five independent Latin structure types represented by LS01, LS07, LS11, 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) and (12 13 15 19) |

##### List 2.4.1 Latin squares collected in manyfolds with common structure type (left) and multiplicities(right). Point group – and translation relationships indicated with ( ) and [ ]. Preferred representatives (par. 2.3) in bold.

row: | |||||
---|---|---|---|---|---|

1 | LS01 | LS07 | LS11 | LS05 | LS06 |

2 | 1 | 2 | 3 | 4 | 5 |

3 | |||||

4 | 2V2x2V2; 8 | 4×4; 16 | 4×4; 16 | 2V2xV2 | 4×4; 16 |

5 | rhombic | rect. | rect. | rect. | rect. |

6 | square | rect. | square | rect. | rect. |

7 | cmm* | pmm | pgg | pm | pg |

8 | c4m' | pmm | p4m' | pm | pg |

9 | 4ST (α) | 2ST.2CC | 4CC(α) | 4ST(β) | 4CC(β) |

10 | (1) | (2) | (1) | (2) | (2) |

##### Figure 2.4.1 Geometric representation representing all order 4 Latin squares, with 5 basic squares, 2D crystallographic data, Latin pattern compositions (Chapter 3) and isotopy classes. **row:**

** 1 preferred choice of independent Latin squares for representing the order 4 LS set.**

** 2 corresponding structure- type numbers (list 2.4.1)\**

** 4 translation periodicity of primitive cell; cell areas (par. 1.1)**

** 5 Bravais lattice type disregarding colour symmetry**

** 6 Bravais lattice type acknowledging colour symmetry**

** 7 conventional (planar) group in IUC-notation (ref.5))**

** 8 colour planar group**

** 9 Latin pattern compositions (par. 1.2 and 3.4).**

** 10 allocation in the isotopy-classes, (1) and (2), par. 2.5**

* Structure LS01 repeats itself halfway the body diagonals (see Fig. 1.2.2, LS05 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, Figs. 2.4.1 and 2.5.1*.

● Each column of Fig. 2.4.1 refers to one specific structure type shared by k Latin squares (k: full multiplicity: ( ) and [ ], List 2.4.1. The structure type figures are given as LS’s *as such* – for space-saving – but represent the corresponding *Latin structures* (Fig. 1.1.2) enclosing all symmetry related members.

● All order 4 Latin squares are shown to relate to one or more counterparts by point – and/or translation symmetry (Tables 2.2.1, 2.3.1 and Fig. 2.5.1).

##### 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 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 LS 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).

● The space groups given in the tables are presently based upon ordinary symmetry (disregarding colour symmetry). The corresponding Bravais lattices are rectangular or oblique (including incidentally rectangular). However, structure types 1, 2, 3, and 5 have the required colour symmetry meeting the condition for a a square Bravais lattice and a corresponding space group.

● 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 24Ss2 07 02 03 09 (16Ss20Ss22Ss23Ss) 3 11S 4 (14Ss21Ss) 4 (05 18) 5 (06 08 10 17) (12S13S15S19S)