Terms and notions
(a b) [a b] { a b} (a,b) LS’s a and b are
(a b) point symmetry related ; [a b] translation symmetry related; {a b} point and translation symmetry related: (a b) form an orthogonal pair
(1) etc (1sf), (1csh) etc.; (a)
(1) etc. ; a specific LS ; (1sf) specific LS under or after a (symmetry movement
(a) is numbered isotopy class .
basic structure
A structure which has been adopted as a member of a geometric representation of a LS-set of fixed order.
colour-symmetry element
operation copying a LS geometrically and performing a switch of two or more of the site symbols throughout the entire structure
composition, composition formula, for a Latin square
specification of the types and numbers of the constituting Latin patterns of a Latin square.
conventional symmetry
symmetry disregarding colour symmetry
coordination square
LS matrix in which the site values have been replaced by their nnn (nearest neighbours number) at the site
counterpart (of a Latin square)
LS or LS’s which can be brought into geometric coincidence with a LS under consideration by one of the operations of the dihedral group and/or/or the translation group. Counterparts can be collected in manifolds. A LS with no counterpart is a single.
Latin patterns composition diagram (briefly: composition diagram)
Diagram showing a LS drawn by superposition of all n Latin patterns of that LS.. Composition diagrams may be extended as ongoing structures similar to Latin squares
Clint value
Clustering intensity value obtainned by summing nnn over the square and dividing by n^2
csh
cyclic shifting operation
dependent, see symmetry related
equal, ,(different) LS’s
LS’s A and B with different nomenclature systems, are equal, (different) if a (no) unambiguous correspondence can be established between these systems. If the nomenclature systems is the same then the unambiguous correspondence is a permutation. Both cases (different and same nomenclature) have been illustrated in par. 1.1.
Note that different LS’s can be different cuttings from the same Latin periodic structure. In the 5×5 domain we have 1344 different LS’s and 18 different (Latin-periodic) structures.
symmetry related, dependent (unrelated, independent) structures of LS’s
Structures of LS’s and Latin structures are symmetry-related, dependent,( symmetry-unrelated, independent) if the LS’s or the corresponding Latin -periodic structures can (cannot) be made to coincide by a combination of one of the symmetry operations of the dihedral group of order 8 and one of the group of n^2 translations defined on the sites lattice, including the “do nothing” .operations of both groups
extending a LS : adding one or more rows and/or columns to the LS’s, or entire copies of the LS’s, usually for better observation of symmetries 0r Latin patterns
family, subfamily
Collection of all LS’s – of the same order – sharing the same composition formula.
Families of orders up to 5 are either single- structured: all carrying the same structure type, or double- structured, with an alfa- and a beta structure type. The family can then be partitioned in a alfa- and a beta (sub)family.
Geometric Representation
If any LS of order n is symmetry related with exactly one member of a set of Latin squares then that set is a geometric representation of the entire order n set
glide line
symmetry operation: translation along the line followed halfway by reflection
LS, Latin square, Latin square as such
nxn bounded Latin square
Latin structure
Unbounded repetition of a Latin square, or tororoidal representation of a LS
LS structure
Collective term for both bounded and unbounded LS representations
LSi (1…i….24)
Latin square of order 4, numbered
Latin pattern
Line system of connections in a LS of sites occupied by a specific symbol. Latin patterns may be extended two-dimensionally similar to LS’s. Drawn connections may be limited for clarity to nearest – or nearest and next nearest neighbours in diagrams
Latin-periodic structure (of order n)
2D-periodic occupation of all sites of a lattice with periods n by symbols of n different kinds conforming with the Latin square condition.. All isometric transformations products of that distribution are considered identical.
Simpler: 2D- periodic extension of a Latin square on the (extended) sites-lattice (par. 1.2)
Latin structure
Shorthand for Latin-periodic structure; also representation of a LS on the torus
manyfold
LS combined with its counterparts
member
LS belonging to a same-order set
also: belonging to a family of LS’s (family-member)
also: belonging to a Geometric Representation (GR- member)
multiplicity
total number of point group- and/or translation-related counterparts
also: number of members in a same composition family
nearest neighbours / next nearest neighbours
identical symbols at positions (x, y) (x+1, y+1) etc. / (x, y) (x+1) (y+2), etc. (the knight ‘s move in chess)
nnn
number of nearest neighbours of a site
PR-representation
representation representing the full set if we count all point symmetry related squares as one (par. 2.2),
point group related LS’s:
LS’s coinciding under the action of an element of the symmetry group of a square ( the dihedral group of order 8) and a permutation
same (LS, LS-structure)
see different
set: full collection of LS’s of same order
single
site in a LS without same-value nearest neighbours
also: LS without point group – or translation symmetry related counterparts
SE sequential enumeration
see Appendix II
sf standard format
snf, standard numeric format of a LS
LS with first row 1 2 3 …..
structure
ordering of site values in a nxn or extended lattice in accordance with the Latin square condition. The structure can be adopted by a LS as such or by a Latin periodic structure
structure type
Latin structure of any member of a Geometric Representation , or a numbered LS isomorphous with a numbered member of a Geometric Representation,
TR-representation
Representing the full set if we count all l translation-related members as one (par. 2.3).