FR2865054A1 - Decorative pattern composition determining process for use in e.g. textile industry, involves forming magic square comprising several numbered boxes, and determining mathematical relations between numbers - Google Patents

Abstract

The process involves forming a magic square comprising several numbered boxes, in which numbers are arranged according to a predetermined rule. Mathematical relations between the numbers represented in the boxes are determined from the magic square. An arrangement of units with symbolic values is extracted. A composition of decorative patterns is determined based on a transposition of the arrangement.

Description

The present invention relates to a method and a device for

  automated composition of possibly holographic symbolic patterns expressing a mathematical relationship. It uses in particular, but not exclusively, the properties of magic squares well known to mathematicians to define ranges of elements

remarkable properties.

  These ranges in question have among other qualities: - symmetrical in terms of geometry (aesthetic), harmonious in terms of arithmetic, - understandable in terms of (logical) reason, interpretable for what is is imagination (analogy).

  The set of images obtained thus defines a graphical chart with an understandable aesthetic that makes it possible to observe hologramorphism notably through the symbolism of Hermetic Philosophy. Given that the main interest of the proposed application is aesthetic, it is not here the exegesis, possible from other links with Judeo-Christian texts, especially through the Zohar with the Song of Songs.

  2865054 2 In general, we know that there are many methods to achieve a magic square, namely: the method of Bachet for all orders where n is odd.

  - Ibn Qunfudh's method for all orders where n is 2 times an even natural integer (a multiple of 4).

  Bachet's method (Claude Gaspar Bachet de Méziriac: 1581 - 1638) is a process of forming normal magic squares of odd orders using a serrated square whose grid is first enlarged by adjacent squares in pyramids on the four sides. The numbers are arranged in ascending order on each of the diagonals. Then, the numbers outside the square are brought into a cell of the same line or column according to the principle of the cylinder.

  Thanks to this method, non-normal odd-order magic squares can be constructed by choosing for example a sequence of n2 numbers and writing them in the order according to the algorithm proposed by Bachet.

  The method assigned to Ibn Qunfudh applies for all orders where n is a multiple of 4 (twice an even number). It implements an algorithm using a grid of nxn boxes, n being a multiple of 4 and includes a first step of marking each box extending over each of the diagonals of the square, a second step of counting the boxes starting from the box at the top right and proceeding from right to left along each line down line to the last line and assigning to each box marked the corresponding number (the unmarked boxes being counted but remaining empty ), a third step of counting the squares starting from the leftmost box of the last box and going from left to right and line in line to the box of the first line on the right. Each unmarked cell is then assigned the digit corresponding to that second count. We thus obtain a magic square in which all the lines, the columns and the diagonals have the same sum.

  The object of the invention is more particularly to extract, from these magic squares, geometrical shapes of arrangements of elements with symbolic values because of their remarkable properties. It applies in particular to the automatic composition of patterns in industrial production lines for example of fabric, wallpaper or tiles, or even of making tables or panels: Indeed, given the algorithmic approach used, this method allows a total automation of the composition process, for example from thematic parameters, as well as the production of materials from transpositions of the described arrangements; indeed, the mathematical properties of its forms can in particular allow creations as to the actual composition of the materials.

  More precisely, this method comprises the following steps: the programming of the evolutionary algorithm and the composition algorithms that it generates to advantageously automate the realization of higher order ranges of magic squares, the determination of mathematical relations between the numbers in the squares, for the tracing within the square of elements that express the said observable relations, these tracing elements constituting at least one pattern, - the determination of symbols associated with this pattern.

  The invention is not limited to magic squares obtained from Bachet's method and Ibn Qunfudh's method. It also applies to all remaining orders, ie all orders where n is twice an odd natural number.

  For this purpose, the invention proposes in particular: a cyclic symmetric harmonization algorithm applied to four element arrangement forms, defining a particular order 6 from which are gradually created on the scale of orders where n is twice an odd natural integer, the higher orders; the use of orders derived from this first algorithm and their symmetrical inverse as tiling elements to create higher order ranges by composition algorithms, transcribing the grouping rhythms into new paving units used. These rhythms of regroupings in new units of pavings create ranges of higher orders expressing the said rhythms which according to the programming can be null, with random progression or constant.

  Embodiments of the invention will be described hereinafter by way of nonlimiting examples with reference to the accompanying drawings in which: Figure la is a magic square of order 6; this square is the basis of all the algorithms presented; FIGS. 1b to 1c illustrate a mode of filling the magic square of FIG. 1a. FIG. 1f represents plots resulting from the observation of the sums of two digits equal to n2 + 1; Figure 1g shows the four hexagrams extracted from the self-filling forms once merged and grouped into a composite set according to these four remarkable subsets; Figure 2a is a representation illustrating a first basic tiling mode 6; FIGS. 2b to 2f show the forms of self-filling of a square of order 30 obtained by the method illustrated in FIG. 2a; Figure 2g shows a base composite base base 6 obtained by the method illustrated in Figure 2a; Figure 3a shows a magic square of order 18 obtained by the method illustrated in Figure 2a; Figures 3b to 3e show self-filling shapes of the square of Figure 3a; Figure 3f is a representation showing the distribution of pairs of numbers whose sum is equal to n2 + 1; Figure 3g is a composition of order 210; Figures 3i to 3k show three forms of self-filling of the square shown in Figure 3g; Figure 3h is the simultaneous representation of Figures 3i and 3j; FIGS. 31 to 3n are the forms of self-filling according to a base of order 18; FIG. 30 being the simultaneous representation of FIGS. 3m and 3n; FIG. 4a illustrates a second base tiling method 6 according to the observation of FIG. 1g which can lead to consider the base 6 according to the 4 sets of remarkable arrangements of nine basic elements described in FIGS. 15c to 15f; The composite algorithm illustrated in FIG. 4a is also evolutive, and generates concentric bases that can all serve as a basic unit; Figs. 4b to 4e show self-filling forms of order 18 obtained by the method illustrated in Fig. 4a, Fig. 4f being a combination of Figs. 4e and 4d; Figure 4g shows the magic square of order 18 obtained by the method illustrated in Figure 4a; Figure 5a shows a magic square of order 30 obtained by the method illustrated in Figure 4a; Figs. 5b to 5e are the self-filling shapes of the square of Fig. 5a, Fig. 5f showing the combination of self-filling shapes of Figs. 5d and 5e; Figs. 5g to 51 illustrate the intermediate stage corresponding to order 12 (Fig. 5h being a magic square of order 12 and Figs. 5i to 51 showing the four forms of self-filling, while Fig. 5g shows the fusion of Figures 5k and 5j); FIGS. 6a to 6f show different plots obtained from a composition of order 210 obtained according to the method of FIG. 2a on a base 30, which is obtained according to the method of FIG. 4a; FIG. 6g shows a composition of order 162 (3x (3x18)) obtained according to the mode of symmetry of FIG. 2a applied to an order 54 itself obtained from the order 18 of FIG. 4g; Figs. 6h to 6k illustrate the four forms of self-filling of the square of Fig. 6g while Fig. 61 shows the fusion of the forms of Figs. 6i and 6j; FIGS. 6m to 6o show a composition of order 324 (3x [3x (3x12)]) composed in a constant progression of the unit base assemblies from order 12 of FIGS. 5g to 51, this composition is thus obtained from FIGS. 6q to 6t (order 108) which are obtained from FIGS. 6u to 6x (order 36), FIGS. 6u to 6x being for their part composed from an order 12; Figure 6p shows the fusion of Figures 6m and 6n; FIG. 7a is a magic square of order 10 obtained according to the progressively increasing algorithm; FIGS. 7b to 7e show the self-filling shapes of the square of FIG. 7a, FIG. 7f corresponding to the combination of the shapes of FIGS. 7a and 7e; Figures 7h and 7i show shapes whose combination which is illustrated in Figure 7g shows the distribution of the sums of two numbers, equal to n2 + 1; FIG. 7j illustrates an algorithm for arranging the multiplications of the base 10 and its symmetrical inverse transferring the properties of the base to the new set; FIG. 8a is a magic square of order 30 obtained by the method illustrated in FIG. 7j; Figures 8b to 8e are the self-filling shapes of the square of Figure 8a, Figure 8f showing the combination of the shapes of Figures 8d and 8e; Figure 8g shows the magic square of Figure 8a, when filled; Fig. 9a shows a composition corresponding to an order 210; Figures 9b to 9e are the self-filling shapes of the square of Figure 9a; Figure 9f shows the fusion of Figures 9d and 9e; Figure 10a is a magic square of order 14; Figs. 10b to 10e are the self-filling shapes of the square shown in Fig. 10a, Fig. 10f being the combination of the forms of Figs. 10d and 10c; Figure 10g is another superposition of the shapes of Figures 10d and 10e; Figure 10h illustrates the process of duplicating the base 14 to obtain the square shown in Figure 11a; Figure 10i shows Figure 10a without the numbers; Figure 10j shows the distribution of pairs of numbers equal to N2 + 1; FIG. 1a illustrates the evolutionary paving mode usable from a 14th order square so as to transfer the properties of the base to the new set; FIGS. 1b-1b are the self-filling shapes of the square of FIG. 11a, FIG. 11f corresponding to the combination of the shapes of FIGS. 11d and 11e; FIG. 12a illustrates a composition of order 210 obtained from a base 42, itself obtained from a base 14; Figures 12b to 12e are the self-filling shapes of the square of Figure 12a; Figure 13a is a magic square of order 54 obtained according to the evolutionary method (thirteenth step); Figures 13b to 13e show the self-filling shapes of the square of Figure 13a; Figure 13f shows the fusion of the shapes of Figures 13d and 13e; Figure 13g shows another form of superposition of the faunas of Figures 13d and 13e; FIG. 13h shows a disposition obtained by observing sums equal to n + 1 of two digits defining a straight line that does not pass through the center; Figure 14a shows a square of order 134 (33rd step); Figs. 14b to 14e are self-filling shapes of the square of Fig. 14a; Figure 14f is the superposition of Figures 14d and 14e; Figure 14g shows the symmetrically opposite points of the shapes of Figures 14d and 14e; Figure 15a shows a magic square of order 18 corresponding to Figure 3a; Fig. 15b is an expression of the magic square of Fig. 15a obtained from details shown in Figs. 15c to 15f; Figures 15g to 15j are plots expressing the self-filling forms of Figure 15b which is a way of expressing the arrangement of Figure 15a, Figure 15a being a duplicate of Figure 3a showing Figures 3b to 3rd; Figure 16a illustrates a magic square obtained by duplicating Figure 17a; FIGS. 16b to 16e and 17b to 17e are patterns obtained by multiplication expressing the harmony contained in the section of the four hexagrams, while FIGS. 16f to 16i and 17f to 17i illustrate the harmony obtained according to the section. four forms of self-filling; Figures 18 to 20 show plots obtained from the method of Ibn Qunfudh (Figure 18), the method of Bachet (Figure 19) and the previously described methods (Figure 20), which can be used in a second step to obtain decorative motifs.

  As previously mentioned, in addition to the conventional methods of making magic squares such as the Bachet method (odd orders) and the method of Ibn Qunfudh, the invention proposes a charter that applies more particularly to the orders n in which n is equal to twice an odd natural number other than 1.

  Figures la to illustrate the filling mode of a square of order 6 (Figure 17a) according to a method taking into account the harmony of four forms of self-filling, namely: the "cross" (Figure lb ) which defines in particular the two diagonals in all the squares of this category presented: the right diagonal starting with the number 1 and progressing by successive additions of n + 1, this diagonal contains the smallest and the largest number (a and Q) and defines the axis of symmetry, the left diagonal starting with the digit n and progressing n times by successive additions of n - 1, a harmony here named third (Figure 1 c) which consists of a form balancing the "cross"; this form self-filling the square from the inverse of the starting point of the cross; - 12 - two other forms (Figure 1d and Figure le) identical self-filling share the remaining space; These two forms (Figure 1d and Figure 1c) similar balance on the axis 5 defined by the diagonal comprising the first and the last digit to self-fill the square from the two ends remaining.

  The filling of the boxes of the magic square of order 6 is carried out as follows: The filling of the boxes constituting the cross is carried out according to arrows FL1 indicated in figure lb starting from the box in top on the right, of right on the left and online line up to the lowest line. Only the numbers of the boxes constituting the cross are preserved.

  The filling of the boxes constituting the third form is effected according to the arrows FL2 shown in Figure 1c starting from the box at the bottom left, from left to right and line in line to the highest line.

  The filling of the cells constituting the first image is carried out according to the arrows FL3 indicated in FIG. 1, starting from the box located at the top left, from left to right and from line to line at the lowest line.

  The filling of the cells constituting the second image is effected according to the arrows FL4 indicated in FIG. 1d starting from the bottom right, from the right to the left and from the line in line to the highest line.

  We thus obtain a magic square expressing the symmetrical harmony of four forms of arrangement of elements grouped into a composite set, according to the logic of the four sets shown in Figure 1g.

  2865054 - 13 - In this magic square, the path connecting two elements whose sum is equal to nxn +1 (here 37) without passing through the center presents a symmetrical shape necessary for the construction of the squares of this category (figure 1f), this form is the fifth component or constraint of the progressively increasing algorithm, making it possible to determine the symmetrical harmony of the 4 forms of self-filling.

  The advantage of the method previously described therefore consists in that it is scalable, in that it can evolve according to an increasing progression in the scale of orders where n is 2 times an odd natural integer other than 1.

  This increasing algorithm will be explained below, with reference to FIGS. 7a to 7e, 10b to 10e, 13b to 13e, 14b to 14e which represent the evolutions of the self-filling modes obtained on orders 10, 14, 54 and 134 .

  The increasing algorithm is thus characterized by: the use as a basis of the order 6 of FIG. 1a for extracting the four basic forms of self-filling, of which the first three stages of evolution (ie FIGS. 10a) contain all the symmetry methods used; more precisely, FIG. 7a shows the second rung, ie a square of order 10 obtained by an enlargement effect of the four forms of self-filling represented in FIGS. 1b to 1e and which are indicated in FIGS. 7b. at 7th. These four forms are defined in particular by means of the plots connecting two elements whose sum is equal to n2 +1 (here 37) which is illustrated in FIG. 7g which, itself, results from the combination of the forms illustrated in FIG. 7h (symmetries specific to the category of order concerned) and 7i (sums passing through the center).

  2865054 - 14- For all the other orders that follow, not concerned with the cyclic variant of the third rung, the symmetrical harmonization algorithm between the four forms of self-filling is similar to that of the second rung, ie the order 10, ( Figure 7a), except the fourth rung is Order 18.

  The third rung is order 14 (Fig. 10a) contains the symmetrical harmonization process between the four forms to be cyclically reproduced every 10th from the third rung or every 40th from the 14th rung.

  Fig. 10a therefore shows a magic square of order 14 which includes both the square of order 6 and the square of order 10.

  Figures 10b, 10c, 10d, 10e show the evolution of the four forms of self-filling.

  The fourth step of progression is the order 18 contains a last variant on the forms that generate the first step of an additional rule of symmetry as to the arrangement of two similar forms according to the so-called third form, viewable thereafter between the midpoints of the sides of the formed assembly (see Figures 14c and 14f).

  Principle of the symmetry-increasing harmonization algorithm, starting from the first 4 rungs Figure 13a illustrates the 13th rung of the evolutionary method, the magic square of order 54 which is self-filled by the fill shapes illustrated on FIGS. 13b, 13c, 13d, 13e, FIG. 13g being the superposition of FIGS. 13d and 13e, the dark points being the symmetrically opposite points.

  This increasing algorithm is also illustrated with regard to the forms of FIGS. 14a to 14g, namely the order 134, the step 33 carried out according to the symmetry harmonization algorithm, consequently containing concentrically 33 squares therein. magical.

  Figure 14a shows a magic square of order 134.

  Figures 14b to 14e are changes in the self-filling shapes of the shapes shown in Figures 1b to 1c.

  Figure 14f is the superposition of Figures 14d and 14e.

  Figure 14g shows the symmetrically inverted points of shapes 14d and 14e.

  The first step of the algorithm consists in gradually projecting the evolutionary cross according to the nature of the order in the cycle, defining for the form at the stage of progression concerned either a central symmetry or an axial symmetry as here with the figure 14b.

  This first form makes it possible to deduce the number of components of its antipode (the so-called third form), it being understood that they are balanced in particular by having the same number of components.

  The so-called third form progresses by defining an axial symmetry at the progression steps where the cross defines a central symmetry and cyclically equilibrates with a central symmetry at the progression steps where this cross becomes an axial symmetry, this thanks in particular to the possible projection of the superimposition of the two other forms (FIG. 14g), thus making it possible to deduce in advance the elements to be integrated by this so-called third form to harmonize the two other forms, it being understood that the cells of the non-grid must be integrated in the so-called third form; covered by the first form and symmetrically opposed to the points defined by Figure 14g, when this is rearranged to form a unit of the type of Figure 14f.

  It being understood that these two figures 14 d and 14 e lose their own axial symmetry only at the progressions step concerned by an axial symmetry of the form named cross, FIGS. 10g, 13g, 14g, (with the exception of the fourth step).

  The number of remaining components is to be divided by two to predefine that of the other two forms. Its two other forms (figures 14d and 14e) being identical to the echelons concerned by the central symmetry of the first form known as the evolutionary cross, they do not become symmetrically inverse until the fourth echelon and the progression steps where this evolutionary cross becomes axial.

  By gradually transferring the modes of symmetry to the four forms of self-filling according to the cycle described, and respecting the interdependence of these forms by a last step which defines a symmetrical image by lines expressing the sum of two boxes, when this sum is equivalent to (nxn) +1, it becomes possible to automate the definition of the upper echelons.

  Advantageously, each order obtained by this evolutionary algorithm can serve as a basis for a so-called "composite" square realized in a very simple way by multiplying the central square (base) and its symmetrical inverse.

  Principle of the composite algorithm If we define the order 6 as the basis, we obtain two types of symmetry, according to the use of the base, to realize a higher order magic square: - 17 - the use of orders from this first algorithm and their symmetrical inverse as tiling elements to create higher order ranges by composition algorithms, transcribing the grouping rhythms into new paving units used. These rhythms of regroupings in new units of tilings create ranges of higher orders expressing the said rhythms which according to the programming can be null, to progression random or constant; either by symmetry of the base as a whole, resulting in progression 12/12 (Fig. 2a); either by symmetry of the hexagrams (squares of order 3) (figure 1g), which can lead to a symmetry evolving on the scale of progression of orders of magic squares of order n such that n is a multiple of six (figure 4a ); By way of example, FIG. 2a is a representation illustrating a base tiling algorithm 6 as a whole, ie a progression of 12 to 12. The multiplications of the base and its symmetrical inverse are arranged in order to recreate a magic square of higher order, this composite algorithm is evolutionary, as the dotted line suggests, it generates a concentric series of magic squares all of which can serve as their base units.

  In this example, the background consists of a base of order 6 duplicated and arranged according to symmetries materialized by the axes of symmetry A, B, C, D - A ', B', C ', D' respectively parallel to the sides of the base.

  Thus, the base is surrounded by eight duplicates whose outer sides delimit a first concentric square surround at the base, namely: four duplicates arranged symmetrically with respect to the axes of symmetry B, C-B ', D', four duplicates located in the angles of the first entourage; each of these duplicates is arranged in mirror with respect to the vertex of an angle of the base.

  This entourage defines with the base a square of order 18, see figure 3a.

  The eight duplicates are themselves duplicated in sixteen duplicates forming a second concentric square surround at the base.

  This second entourage defines with the first as well as with the base a square of order 30, see Figure 2g.

  To achieve a tiling in this mode it will be enough to create 2 basic copies namely: a base and its inverse; and to arrange them according to the logic indicated previously; (The described arrangement of these two elements may be suggested on the back of a pavement with directions marks).

  As previously indicated this process is evolutionary, it constitutes an algorithm generating higher order bases; The bases of superior orders obtained can all evolve on the same rhythm of flat progression, and generated variants.

  It is also possible to program the grouping frequencies in a tiling unit, the frequency being more or less regular depending on the desired effect, the aim here being to be able to propose a source of creation of two inverse basic meshes, which can advantageously be arrange to create a remarkable symmetry.

  This arrangement symmetry algorithm can vary on the diagonals according to the nature of the order, in fact for order 10 and order 14, the base can simply be translated (respectively FIGS. 7j and 10h).

  Advantageously, the method can thus provide ranges of balanced tiling units, the size of these units being definable according to the desired effect.

  Figures 2b to 2e show the four forms of self-filling of the magic square of order 30 shown in Figure 2g obtained from the method described with reference to Figure 2a.

  FIG. 3a shows a composite base 6 magic square 6 obtained by the method previously described, while FIGS. 3b to 3e show filling shapes of this magic square.

  FIGS. 151 to 15f show the details of FIG. 1g from which it is possible to express the magic square of FIG. 3a or its duplicate 15a in the form of FIG. 15b from which it is possible to deduce the illustrated patterns. Figures 15g to 15j corresponding to the forms of self-filling of Figures 3b to 3e.

  Figure 3g is a composition of order 210 (7x [5x6]) obtained by the method illustrated in Figure 2a but applied to a base of order 30, base 6 is Figure 2g.

  FIGS. 31 to 3n illustrate a possible combination of order 270 5x [3 (3x6)], obtained by the method illustrated in FIG. 2a but applied to a base of order 18 to form an order 54, which multiplies, will form a composition order 270.

  FIG. 4a illustrates a second basic tiling algorithm 6 using the symmetry properties generated by taking into account as a basis the four subassemblies constituting FIG. 1g.

  In this example, the square of order 6 is surrounded by a first entourage of twelve duplicates, namely: eight duplicates which are symmetrical of the hexagonal shapes with respect to axes B, B ', D, D', four duplicates which are respectively the translations of the hexagonal shapes opposite to the angle of the base.

  These twelve duplicates are themselves duplicated in twenty duplicates forming a second concentric square surround at the base.

  The previously described composite method is easily transposable to create squares with orders equal to multiples of 6, ie 12, 18, 24, 30, 36, etc., for base unit.

  Figures 4b to 4e are the four self-filling forms of order 18.

  Figure 4f shows the combination of Figures 4c and 4d. Figure 4g shows the magic square of order 18 thus obtained.

  Figures 6a to 6g show different plots obtained from a composition of order 210, base 30, with progression from 6 to 6.

  FIGS. 6b to 6e show the self-filling shapes of the magic square of FIG. 6a, FIG. 6f being the superposition of FIGS. 6d and 6e.

  Figure 6g is, for its part, the superposition of Figures 6b and 6c.

  Moreover, as previously indicated, each square of the evolutionary method and its symmetrical inverse can serve as a basis for composite orders by being multiplied and arranged according to their nature. Thus, for example, the order 10, resulting from the extension of the 6, can serve as a basis for a progression according to the orders n in which n is an odd multiplied by 10.

  FIG. 7j shows a mode of arrangement of the multiplications of the base 10 making it possible to obtain a variant of the order 30 illustrated in FIG. 8a. This square of order 30 is obtained using the four forms of self-filling illustrated in FIGS. 8b to 8e.

  Figure 8f showing the combination of shapes 8d and 8e.

  Figure 8g shows the magic square when filled.

  FIG. 9a shows a composition of order 210 obtained by duplicating a base 30 derived from a base 10. FIGS. 9b to 9e are the four self-filling shapes of the square of FIG. 9a.

  Figure 1 illustrates the arrangement advantageously applicable from the square of order 14 with the four forms of self-filling l lb to Il e, Figure 11a showing the magic square obtained by this method.

  Figure 12a shows a possible order 210 composition obtained from the 14th order composite square, with Figures 12b through 12e being the four corresponding self-filling shapes. Of course, these forms of self-filling can be combined with each other.

  Thanks to the previously described evolutionary method, it becomes possible to create an infinity of magic squares making it possible to obtain a multiplicity of layout shapes of elements constituting ranges of units from the four basic forms of the order 6 These shapes can be exploited for the production of decorative motifs, in particular for textiles (weaving and / or printing), or for the programming of light panels. The advantage of this solution is that, by its very nature, it is particularly suitable for numerically controlled systems.

  Of course, the multiplication of base shapes obtained from magic squares in order to obtain a pattern does not necessarily lead to obtaining a magic square.

  In order to better highlight the arrangements of this category of order, the invention proposes a method of extracting traces to form images expressing the symmetrical properties of the magic squares obtained.

  The process is based on: the determination of mathematical relations between the numbers in the squares, for the tracing of elements within the square of elements that express the said observable relations, these tracing elements constituting at least one motif , the determination of symbols associated with this pattern.

  Advantageously, the plots of each of the above elements comprise line segments passing through the center of each of the boxes whose numbers express a relation. Each segment expressing a relation may be drawn in a color or shape different from that of the other lines. Similarly, the geometrically shaped surfaces delimited by the traces may be represented in different colors so as to better highlight the particularities of the arrangements mentioned.

  Of course, the method according to the invention may comprise the superposition of plots, forms of observations and forms of self-filling, obtained from magic squares of different orders, by centering them on a point at respective scales so as to obtain images by transparency effect, which can 2865054 - 23 - in particular thanks to their consistency serve as graphic charts for multimedia applications (video games, holograms, illuminated panels, stationery, game cards to collect) and industrial (making materials) according to a method of surface arrangement composed of more or less complex binary units (symmetric axial base and symmetrical inverse) transposing their conductive qualities to the whole.

  Figure 18 contains plots obtained from Ibn Qunfudh's method: These plots can serve as a basis for obtaining decorative patterns according to the method previously described.

  Figure 19 shows traces obtained by the method of Bachet also used to obtain other types of decorative patterns.

  Figure 20 shows plots obtained by the previously described methods from squares where n is twice an odd number or a multiple of 6.

2865054 - 24 -

Claims (16)

claims   1. A method for the automated composition of patterns in an industrial production line of objects using these patterns, this method being exploitable in particular in the textile industry or in the display of light information, characterized in that it comprises: the realization of a magic square comprising a plurality of numbered boxes, whose numbers are arranged according to a predetermined law, the determination of mathematical relationships between the numbers in the boxes, the extraction, from this magic square, of arrangements of elements with symbolic values due to their remarkable properties, the composition of said patterns or the production of materials from the transposition of said arrangements.   2. Method according to claim 1, characterized in that the plots of each of said elements comprise line segments passing through the center of each of the boxes whose figures express a mathematical relationship.   3. Method according to one of claims 1 and 2, characterized in that each segment expressing a relationship can be traced in a color different from that of the other traces.   4. Method according to one of the preceding claims, characterized in that the surfaces of geometric shapes delimited by the traces are represented in different colors.   2865054 - 25 - 5. Method according to one of the preceding claims, characterized in that the boxes whose figures express the same relationship are represented in the same color.   6. Method according to one of the preceding claims, characterized in that it comprises a preliminary step of constitution of a background which defines the mode of construction of the square and its proportions.   7. Method according to claim 6, characterized in that the aforesaid background is obtained by means of a cyclic symmetric harmonization algorithm applied to four forms of arrangement of elements, defining a particular order 6 from which create progressively higher orders on the scale of orders where n is twice an odd natural integer.   8. Method according to claim 7, characterized in that, to create higher order ranges, it comprises a paving process using, as a paving element, orders from the above algorithm and their symmetrical inverses through algorithms. of 20 compositions transcribing the grouping rhythms into new paving units employed.   9. The method of claim 8, characterized in that the aforesaid grouping rhythms create ranges of higher orders expressing said rhythms which, according to the programming, may be zero, random or constant progression.   10. Method according to one of the preceding claims, characterized in that it comprises the substitution of all or part of the squares of the 30 magic squares of the pixel arrangement composed patterns, whose shape and dimensions are 2865054 -26- determined by the order of the magic square and the desired effect.   11. Method according to one of the preceding claims, characterized in that it comprises the superposition of plots obtained from magic squares of different orders, by centering them on a point to respective scale so as to obtain images by effect transparency.   12. Method according to one of the preceding claims, characterized in that the magic squares odd orders are made by the method of Bachet.   13. Method according to one of claims 1 to 11, characterized in that the squares whose orders are multiples of 4 are made by the method of Ibn Qunfudh.   14. Method according to one of the preceding claims, characterized in that it comprises the determination of self-filling forms of the aforesaid magic square and the use and / or association of these forms of self-filling to obtain the aforesaid reason.   15. The method of claim 14, characterized in that the elements constituting the forms of self-filling are represented by remarkable elementary forms.   16. Method according to one of the preceding claims, characterized in that the magic square is obtained by multiplying a basic magic square and / or its symmetrical inverse in a symmetrical arrangement with respect to the center of the square at its diagonals and or his mediators.