## 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.’”

More information on this topic can be found at the Halexandria Forums.

Forward to:

Appendix

According to <http://www.esotericarchives.com/agrippa/agripp2b.htm>, a website devoted to the works of Heinrich Cornelius Agrippa, “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...”  Allegedly these tables represent great virtue, divine order, and “sweet harmony” (among a host of other things).  Agrippa, in fact, goes to some lengths to describe the wonders and profundities of such tables, and ultimately attempts to explain the tables in some detail.

[Much of Agrippa’s work sounds very much like Astrology, and in his day and time, there was general acceptance of such occult arts.]

“The first of them is assigned to Saturn, and consists of a square of three, containing the particular numbers of nine, and in every line three every way, and through each Diameter making fifteen. Now the whole sum of numbers is fourty five. Over this are of Divine names set such names as fill up the numbers with an Intelligency to what is good, with a spirit to what is bad, and out of the same numbers is drawn the seal, or Character of Saturn, and of the spirits thereof, such as we shall beneath ascribe to its table. They say that this table being with a fortunate Saturn engraved on a plate of lead, doth help to bring forth, or birth, and to make a man safe, and powerfull, and to cause success of petitions with princes, and powers: but if it be done with an unfortunate Saturn, that it hinders buildings, plantings, and the like, and casts a man from honours, and dignities, and causes discords, and quarrellings, and disperses an Army.

“The second is called the table of Jupiter, which consists of a Quaternian drawn into it self [i.e. 4 times 4], containing sixteen particular numbers, and in every line, and Diameter four, making thirty four. Now the Sum of all is 136. And there are over it divine names with an Intelligence to good, with a spirit to bad, and out of it is drawn the Character of Jupiter, and the spirits thereof. They say that if it be impressed upon a Silver plate with Jupiter being powerfull, and ruling, it conduceth to gain, and riches, favor, and love, peace, and concord, and to appease enemies, to confirm honors, dignities, and counsels, and dissolve enchantments if it be engraven on a corall.

“The third table belongs to Mars, which is made of a square of five containing twenty five numbers, and of these in every side and Diameter five, which make sixty five, and the sum of all is 325. And there are over it Divine names... [and so forth]. These with Mars being fortunate, being engraven on an Iron plate, or sword, makes a man potent in war, and judgments, and petitions, and terrible to his enemies, and victorious against them; and if engraven upon the Stone Correola, it stops blood, and the mestrues [menses]; but if it be engraven with Mars being unfortunate, on a plate of red Brass, it hinders buildings, casts down the powerfull from dignities, honors, and riches, and causeth discord, strife, and hatred of men, and beasts, chaseth away Bees, Pigeons, and Fish, and hinders Mils, and renders them unfortunate that go forth to hunting, or fighting, and causeth barreness in men and women, and other Animals, and strikes a terror in all enemies, and compels them to submit.

“The fourth table is of the Sun, and is made of a square of six, and contains thirty six numbers, whereof six in every side, and Diameter, produce 111, and the sum of all is 666. There are over it divine names... [as before]. This being engraven on a Golden plate with the Sun being fortunate, renders him that wears it to be renowned, amiable, acceptable, potent in all his works, and equals a man to Kings, and Princes, elevating him to high fortunes, inabling to do whatsoever he pleaseth: but with an unfortunate Sun, it makes a tyrant, and a man to be proud, ambitious, unsatisfiable, and to have an ill ending.

“The fifth table is of Venus, consisting of a square of seven drawn into it self, viz. of fourty nine numbers, whereof seven on each side and Diameter make 175. and the sum of all is 1225. And there are over it divine names with an Intelligency to good, and spirit to evil; And there is drawn out of it the Character of Venus, and the spirits thereof. This being engraven on a Silver plate, Venus being fortunate, procureth concord, endeth strife, procureth the love of women, conduceth to conception, is good against barreness, causeth ability for generation, dissolves enchantments, and causeth peace between man, and woman, and maketh all kind of Animals and Cattle fruitful; and being put into a Dove-house, causeth an increase of Pigeons. It conduceth to the cure of all melancholy distempers, and causeth joyfulness; and being carryed about travellers make them fortunate. But if it be formed upon Brass with an unfortunate Venus, it causeth contrary things to all that hath bin above said.

“The sixt table is of Mercury resulting from the square of eight drawn into it self, containing sixty four numbers, whereof eight on every side and by both Diameters make 260. and the sum of all 2080, [which is, curiously enough, the diameter of the Moon in miles!  Not a coincidence!] and over it are set divine names with an Intelligency to what is good, with a spirit to what is evil, and from it is drawn a Character of Mercury, and the spirits thereof; and if it be with Mercury being fortunate engraven upon Silver, or Tin, or yellow Brass, or be writ upon Virgin Parchment, it renders the bearer thereof gratefull, and fortunate to do what he pleaseth: it bringeth gain, and prevents poverty, conduceth to memory, understanding, and divination, and to the understanding of occult things by dreams: and if it be an unfortunate Mercury, doth all things contrary to these.

“The seventh table is of the Moon, of a square of nine multiplied into it self, having eighty one numbers, in every side and Diameter nine, producing 369. and the sum of all 3321. And there are over it divine names with an Intelligency to what is good, and a spirit to what is bad. And of it are drawn the Characters of the Moon, and of the spirits thereof. This fortunate Moon being engraven on Silver, renders the bearer thereof grateful, aimiable [amiable], pleasant, cheerfull, honored, removing all malice, and ill will. It causeth security in a ourney, increase of riches, and health of body, drives away enemies and other evil things from what place thou pleaseth; and if it be an unfortunate Moon engraven in a plate of Lead, where ever it shall be buried, it makes that place unfortunate, and the inhabitants thereabouts, as also Ships, Rivers, Fountains, Mills, and it makes every man unfortunate, against which it shall be directly done, making him fly from his Country, and that place of his abode where it shall be buried, and it hinders Physitians [physicians], and Orators, and all men whatsoever in their office, against whom it shall be made.

“Now how the seals, and Characters of the Stars, and spirits are drawn from these tables, the wise searcher, and he which shall understand the verifying of these tables, shall easily find out.”

Pivari, F. <http://www.geocities.com/CapeCanaveral/Lab/3469/examples.html>, has described briefly, the connection between these Magic Squares and Numerology, i.e.:

“Magic Square and Numerology: Cornelius Agrippa (1486-1535) joined the seven known planets with these seven Magic Square:

Saturn Magic Square: the magic constant is 15 and the sum of all the numbers of the square is 45. 15 and 45 are triangular numbers. The colours are: white for the numbers and black for the background.  The metal of Saturn is lead

Jupiter Magic Square: the magic constant is 34 the product of 2, the first female number (even are female, odd, without 1 are male) and 17 a male prime number.  The sum of all the numbers of the square 136.  136 is strangely bound to 244.  13 +33 + 63 = 244 and 23 +43 + 43 = 136.  The colours are: orange-bronze for the numbers and blue for the background.  The metal of Jupiter is tin.

Mars Magic Square: the magic constant is 65, the second number equal to the sum of 2 squares 1 + 82 and 42 + 72 and product of 5 and 13, two important numbers.  The sum of all the numbers of the square is 325, the smallest number sum of 2 squares in three different mode: 1 + 182, 62 + 172, 102 + 152.  The colours are: green for the numbers and red for the background.  The metal of Mars is iron

Sun Magic Square: the magic constant is 111, three one. The sum of all the numbers of the square is 666, the number of the Beast of th Apocalypse. The colours are: purple or magenta for the numbers and yellow for the background.  The metal of the Sun is gold.

Venus Magic Square: the magic constant is 175 equal 1 + 72 + 52. [?] The sum of all the numbers of the square is 1225 = 352 is also triangular. The colours are: lemon yellow for the numbers and dark green for the background. The metal of Venus is bronze.

Mercury Magic Square: the magic constant is 260, the sum of all the numbers of the square is 2080. The colours are: clear blue for the numbers and orange-bronze for the background. The metal of Mercury is mercury.

Moon Magic Square: the magic costant is 369, formed by 3x1, 3x2, 3x3. The sum of all the numbers of the square is 3321. The colours are opposite Sun colours: yellow for numbers and purple or magenta for the background. The metal of Moon is silver.

“Cornelius Agrippa (1486-1536) deduced the imperfection of the 4 elements (air, ground, fire, water) with the impossibility to make a Magic Square of 4 elements. This Magic Square was also a simbol of the original sin.”