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Μάρκος Αυρήλιος

Παρασκευή, 26 Απριλίου 2013

Magic Squares

Quickie Introduction  
A curious arrangement of numbers includes what is referred to as a “magic square”.  The magic derives from the fact that numbers arranged in a square of equal sides all add to the same total, coming and going, up and down, and oft times even from an angle (diagonal).  For example:
6      1      8
7           3
2      9      4
Note that the total always adds to 15 (row, column or diagonal), the diagonals no longer necessarily add properly if either the row and/or columns are mixed, and the total of any three rows or columns is 45.  This is a magic square of rank 3.  
More information on this topic can be found at the Halexandria Forums.
One can also do a 4x4 magic square, e.g.  
  1    16    12      5
15      2    13      4
   8      9      6    11
  10      7      3    14  
Here the rows and columns add to 34, but in this particular case the diagonals do not.   
The 3x3 example above is considered Panmagic, Diabolical, Nasik, or Pandiagonal, while the 4x4 above is merely magic.  
It is also possible to start with zero, instead of one, so that a possible 5x5 magic square is:  
 21      2      8    14    15
13    19    20      1      7
  0      6    12    18    24
17    23      4      5    11
   9    10    16    22      3  
This particular version was taken from <http://www.grogono.com>, which is an excellent website on the subject.  Included is a very brief, traditional history, which notes that “all magic squares have at least eight variations: the square can be rotated into four positions and each of these rotations can be reflected - for a total of eight variations of any one unique design.  Most magic squares do not remain magic if one border is moved to the opposite edge - the change leaves the main diagonal no longer magic.  However, translocation - repeatedly moving one edge across to the opposite side or the top to the bottom - does not affect panmagic squares which have, therefore, additional variations.   
“In a 5 x 5 square this is equivalent to moving the starting square through all twenty-five positions - for a total of 25 x 8 = 200 variations.  For the order 7 square, each pan-magic square has 49 x 8 = 392 variations and for the size 11 square there are 121 x 8 = 968 variations.”  He also notes that, according to his book on Magic Squares and Cubes, William Andrews describes the construction of panmagic squares of order 5, and predicts that the total number of possible panmagic squares of order five will be 28,800.”  
The 6x6 magic square is particularly interesting.  Two examples include:  
  6    32      3    34    35      1                          32    29      4      1    24    21
  7    11    27    28      8    30                          30    31      2      3    22    23
24    14    16    15    23    19                          12      9    17    20    28    25
13    20    22    21    17    18                          10    11    18    19    26    27
25    29    10      9    26    12                          13    16    36    33      5      8
36       5    33      4     2     31                          14    15    34    35      6      7  
Here, every number between 1 and 36 is used.  All columns and rows (and the two diagonals) add to 111. The total of all rows (or all columns) is thus “666”!  Which also says that 1+2+3+...+36 = 666!  Note also that shifting the columns eliminates the fact that each of the diagonals add to 111.  
An intriguing aspect of the 6x6 magic square on the right is that if one looks at adjacent numbers, one obtains a pattern of the sequence of numbers in the 2x2 squares:  
X X X
X U X
U X U  
Such patterns are common in many magic squares, particularly when one uses formulas to derive the sequence of numbers.  
(9/22/9) Even more astounding in some respects is a three-dimensional magic... double tetrahedron (or star tetrahedron). See, for example, the forum thread on 09/09/09.
Magic?  
If Magic is nothing more than our current understanding of technology and science, then Magic Squares may indeed be magic.  There may be a hidden science of which we are blissfully unaware which magic squares describe, even when the mathematicians have exhaustively delved into the subject, thinking of it as no more than a mental exercise.  We may have become adept at manipulating magic squares and constructing them, but are they actually understood?  Probably not.  
By the same token, is it possible that there is really no profound hidden meaning to begin with?  Of course.  But as in Sacred Geometry, where we begin to see Philosophy as a mathematical discipline, then Magic Squares may have their own hidden values as well.  
One might notice, for example, that the total of the 6x6 magic square is 666, the so-called number of the beast.  It should be pointed out, however, that “666” shows up twice in the Bible, once in Revelation 13:18, but also earlier in King Solomon’s time [1 Kings 10:14]. In both cases, the phraseology is:  “six hundred threescore and six”; e.g.:  
Revelation 13:18: “Here is the wisdom.  Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is Six hundred threescore and six.”
 
1 Kings 10:14:  “Now the weight of gold that came to Solomon in one year was six hundred threescore and six talents of gold.”  
The degree of magic in a magic square might also be hinted at from its history.  In another excellent website, <http://mathforum.org/alejandre/magic.square.html>, the author notes that, “Magic squares have been around for over 3,000 years. They are descendants of the oldest known number mystery, the legend of Lo Shu, found in China in a book entitled Yih King.”  
The story of Lo Shu is basically one of a huge flood in ancient China, whereby sacrifices to the river god -- to calm his anger -- seem ineffective.  Each time a turtle came out of the river and walked around the sacrifice, as if to suggest that the river god had not accepted the sacrifice.  Until a child noticed a curious figure on the turtle shell -- in effect, the 3x3 magic square shown above.  On this basis, the people realized the correct amount of sacrifice to make, and thus appeased the river god.  
Apparently, these ancient Chinese believed in the magic!  
In 1514, Albrecht Dürer created an engraving named Melancholia that included a magic square.  In the bottom row of his 4x4 magic square, he placed the numbers “15” and “14” side by side to reveal the date of his engraving.  
16      3      2    13 
  5    10    11      8 
  9      6      7    12 
   4    15    14      1  
Possibly the premier website on the subject, complete with lots of math and an excellent bibliography is <http://mathworld.wolfram.com/MagicSquare.html>.  This website notes, among many, many other aspects, that, “Various numerological properties have also been associated with magic squares.  Pivari associates the squares illustrated below with Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon, respectively.”
4    9    2                        4    14    15      1
3    5    7                        9      7      6    12
8    1    6                        5    11    10      8
                                     16      2      3    13
Saturn = 15/45             Jupiter = 34/136

11    24      7    20      3                    6    32      3    34    35      1
  4    12    25      8    16                    7    11    27    28      8    30
17      5    13    21      9                  19    14    16    15    23    24
10    18      1    14    22                  18    20    22    21    17    13
23      6    19      2    15                   25    29    10      9    26    12
                                                         36      5     33      4      2    31
    Mars = 65/325                                 Sun = 111/666

22    47    16    41    10    35      4                                 8    58    59      5      4    62    63      1
  5    23    48    17    42    11    29                              49    15    14    52    53    11    10    56
30      6    24    49    18    36    12                              41    23    22    44    45    19    18    48
13    31      7    25    43    19    37                              32    34    35    29    28    38    39    25
38    14    32      1    26    44    20                              40    26    27    37    36    30    31    33
21    39      8    33      2    27    45                              17    47    46    20    21    43    42    24
46    15    40      9    34      3    28                                9    55    54    12    13    51    50    16
                                                                                     64      2      3    61    60      6      7     57 
Venus = 175/1225                                                     Mercury =  260/2080
                                                                                        
37    78    29    70    21    62    13    54      5
  6    38    79    30    71    22    63    14    46
47      7    39    80    31    72    23    55    15
16    48      8    40    81    32    64    24    56
57    17    49      9    41    73    33    65    25
26    58    18    50      1    42    74    34    66
67    27    59    10    51      2    43    75    35
36    68    19    60    11    52      3    44    76
77    28    69    20    61    12    53      4    45       
the Moon = 369/3321
The patterns in these squares formed from adjacent numbers are quite interesting.  Note also, the sequence from Saturn to Jupiter and on to the Moon.  In each case, the period of the cycles of each with respect to the Earth is decreasing -- just as in the Days of the Week and their association with the planets.  
Pivari, F. <http://www.geocities.com/CapeCanaveral/Lab/3469/examples.html>, has described briefly, the connection between these Magic Squares and Numerology, basing much of his material on the original writings of Cornelius Agrippa (1486-1535).  Heinrich Cornelius Agrippa von Nettesheim was Counseller to Charles the Fifth, Emperor of Germany, and Iudge of the Prerogative Court, and was one of the more influential writers of renaissance esoterica.  His de occulta philosophia in three books was considered a systematic exposition of various kinds of magic.  Whether or not he had a clue about what he was writing is not immediately obvious, but one suspects there is some profound and practical stuff to be gleamed from his writings.  For anyone brave enough to traverse through sixteenth century German, or seventeenth century English, link to <http://www.esotericarchives.com/agrippa/agripp2b.htm>. 
The beginnings for example are:  
                “It is affirmed by Magicians, that there are certain tables of numbers distributed to the seven planets, which they call the sacred tables of the planets, endowed with many, and very great vertues of the Heavens, in as much as they represent that divine order of Celestiall numbers, impressed upon Celestials by the Idea's of the divine mind, by means of the soul of the world, and the sweet harmony of those Celestiall rayes, signifying according to the proportion of effigies, supercelestiall Intelligencies, which can no other way be expressed, then by the marks of numbers, and Characters.  For materiall numbers, and figures can do nothing in the mysteries of hid things, but representatively by formall numbers, and figures, as they are governed, and informed by intelligencies, and divine numerations, which unite the extreams of the matter, and spirit to the will of the elevated soul, receiving through great affection, by the Celestiall power of the operator, a power from God, applyed through the soul of the universe, and observations of Celestiall constellations, to a matter fit for a form, the mediums being disposed by the skill, and industry of Magicians; But let us hasten to explain the tables severally.”  
If that is completely clear to you, then link to Agrippa and study it all.  [And pay particular attention to pages 19 through 25 of Part 2 of Book II. Celestial Magic.  Some of this is reproduced below for convenience.]  
Meanwhile, we might note that in the 5x5, 7x7, and 9x9 magic squares, not only are the diagonal totals the same as the rows and columns, but the two diagonals at equal distances from the center diagonal add to the same total when divided by two.  Furthermore, in the right-slanting diagonals, each of the adjacent numbers have a difference of exactly 5, 7, or 9 -- depending on the magic square.  [However, one may have to complete the cycle to note this.  For example, in the 9x9 magic square, going from 78 to 6, involves a 78, 79, 80, 81, 1, 2, 3, 4, 5, 6 sequence -- just as ten days from July 25 is August 4.]  Adjacent numbers, and numbers with exact differences in sequence seem to be important.  
We can look into the numbers associated with these seven magic squares by noting that:  
Planet Row Reduced* Inverse Total Reduced* Inverse
  (x)   (1/x) (y)   (1/y)
             
Saturn 15 6 0.06666666... 45 9 0.022222222...
Jupiter 34 7 0.02941176... 136 1 0.073529412...
Mars 65 2 0.01538462... 325 1 0.003076923...
Sun 111 3 0.00900900... 666 9 0.001501501...
Venus 175 4 0.00571428... 1225 1 0.000816326...
Mercury 260 8 0.00384615... 2080 1 0.000480769...
Moon 369 9 0.00271002... 3321 9 0.000301114...
*As per Numerology.  
Other possibilities  
<http://mathforum.org/alejandre/magic.star/msuzuki1.html> notes a Magic Star by Mutsumi Suzuki, i.e.:  
 
Other variations on magic squares include Bimagic Squares (where replacing each number by its square in a magic square results in another magic square) and Border Squares.  The latter represents the case where a magic square remains magic when its border is removed.  An example < http://www.geocities.com/CapeCanaveral/Lab/3469/examples.html> is a 3x3 within a 5x5 within a 7x7, and is:  
40      1      2      3    42    41    46
38    31    13    14    32    35    12
39    30    26    21    28    21    11
43    33    27    25    23    17      7
  6    16    22    29    24    34    44
  5    15    37    36    18    19    45
  4    49    48    47      8      9    10
Note that in the above example that opposing border numbers always add to 50 (e.g. 38 + 12, 2 + 48, 13 + 37).   This sum is the difference between the row totals of the 3x3 (75), and the 5x5 (125), and 7x7 (175).  
Variations on magic squares can also be constructed using letters (either in defining the square or as entries in it), such as the alphamagic square and the Templar magic square.   An alphamagic is “A magic square for which the number of letters in the word for each number generates another magic square. This definition depends, of course, on the language being used.  In English, for example,
 5    22    18                                                           4      9      8
28    15      2                      becomes                   11      7      3
12      8    25                                                           6      5    10  
where the magic square on the right corresponds to the number of letters in
five              twenty two          eighteen
twenty eight       fifteen                two          
  twelve               eight             twenty five  
The Templar Magic Square is a magic square-type arrangement of the words in the Latin sentence “Sator Arepo tenet opera rotas” (“the farmer Arepo keeps the world rolling”). This square has been found in excavations of ancient Pompeii.  
S      A      T      O      R
A      R      E      P      O
T      E      N      E      T
O      P      E      R      A
R      O      T      A      S  
Another version of this story, and referred to merely as the Sator Rotas magic square comes from <http://mathforum.org/alejandre/magic.square.html>, and notes that, “In Rome during the Middle Ages this square was inscribed on a variety of common, everyday objects such as utensils and drinking vessels.  It was also found above doorways. It was believed that the square had magical properties, and that making it visible would ward off evil spirits. The words on this square roughly translate to ‘The Creator (or Savior) holds the working of the spheres in his hands.’”   

 http://www.halexandria.org/dward090.htm

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