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To: alt.magick.tyagi,alt.divination,alt.magick,alt.metaphysics.sacredgeometry,rec.puzzles From: Seyfert1Subject: Magic Square 'Solutions' Date: Thu, 30 Oct 2003 05:28:55 GMT 50031029 viii om first installment, 0, 1, 2x2, 3x3 (+ 4x4 notes) as part of an examination of Planetary sequences for occult study, I've been reading a bit of Henry Cornelius Agrippa von Nettesheim, Three Books of Occult Philosophy, in which a sequence of magic squares keyed to Planets appears (in the Sequence of the Ancients, or 'Chaldean Sequence'). someone in the Sacredlandscapelist was coincidentally reading Agrippa and asking about the 'solution' to these squares ("once and for all"!). here is the result so far of our conversation: Martin: # ...I propose ...we try to solve the magic squares once # and for all. I think I know what you mean. I tried something similar with an 8x8 'Mercury' Square in order to configure it properly for Yijing divination using it and Chess. I found out that while there are several solutions (as with many magic squares), there is one most simple (transposition) solution. there are famous, named squares, however, with intriguing properties of identical sum throughout that make it difficult to do more than grade such solutions in rank as regards their features. 'solution once and for all' is, without further definition of what solution means, too ambiguous to produce. mathematics constitutes the field of endeavor which includes number arrangement (logic, puzzles, computers, and, for some, thaumaturgy or theurgy). orders (sizes) of magic square were provided assignment by occultists to planets, gods, and other quirky principles, its apparent fact of sequence mapped to the apparent Planets of the time. with the discovery of 3 more planets, and as I'm beginning a refashioning for mages of my ilk in the PISCEAN age what should have been done long ago, I agree that some kind of new Solution should be devised with specified parameters for the construction of novel magical devices and systems. the tendency to remain in the past, accept the wellworn constructions of the past, even though their knowledgebase is superceded by plain contradiction, is to set into Golden Age remembrance what should be oriented to the present and personal as a mapping onetoone correspondence. # I believe that they do represent a higher form of mathematics # that currently is not in use by our scientist. magic squares are formal structures, not formulae in a sense as to be recognizable by many modern scientists, and to call them a higher form of mathematics is to provide them a greater range of meaning than should include them. yes, the magic square is a device usable by magicians to influence the universe (they think), and typically magic is considered a form of science unrecognized by the witness (esp. for fans of ACClarke). I'm not sure what is supportable beyond this. # I also believe that we collectively have the ability to solve # the apparent complexities concerning these squares. very true, many are gifted in the area of mathematics that have the time and interest in lending a hand. references are also available, interestingly enough (many online). the best appear to be long treatises that haven't made it online yet. I'm very seriously considering transcribing some of Andrews' book if it hasn't yet been done. # Anybody who wants to partake or have suggestions/input is # more than welcome to join in and help out. agreed! :> # I have read the three books of occult philosophy by cornellius # agrippa, which translation? I've been looking at it carefully lately and noting its resonances, geometrics and attributions. the Llewellyn Tyson version has some helpful but sometimes ambiguous notes that are supplemented with generous Appendices. to read the thing straight through is different than using it as a reference (indexing, or via ToC for topics). his magic squares are perhaps in need of refinement (Tyson corrected many errors in translations at least, a few from the original manuscript, if I recall arightly). # and have gained some insights which I would like to share. thank you! more!! # ...the 3x3 square is probably useless and/or unlucky in that its # total result is equal to 666 and any attempt at solving it would # be futile. LOL! the 3x3 square, also known as the Lo Shu or Lo River diagram, also known as the Saturn Square, is widely known for its luck, power and the benefits it affords its magical users. its actual total is 136 (the 'Sun' square 6x6 adds to 666). the questions are to what ought the 3x3 be associated and whether there is more than one solution to it. 3x3 is certainly the lowest *order* of magic square, by the usual minimum definition (rows, columns, and major diagonals composed of a set of sequential numbers adding to the same total). as I have suggested elsewhere, fundamentals of magic square construction allow earlier orders by restricted qualification. more below. ============================================================= Magic Squares : Arcane Speculation on Mathematical Objects in Series by Seyfert1 (nagasiva)  ProtoSquares what follows is a beginning of analysis of magic squares for arcane reformation purposes and intrinsic morphological content. (order) ========= (1? 0?) 1 the smallest order (0?) of magic square? only in a sense like anything to the 0 power produces the result of 1. :> perhaps this could be attributed somehow (with the next order(s 0/1/2)) with a Without/Limitless/LVX trinity of preKetherian Qabalistic Nothingness. :> one might also construct the square out of letters, in which case we're probably talking about A or perhaps + (t/200? x/600?)  (2) 1 2 1 4 A  3 4 B  2 3 no possible solution obtains for a 2x2 square, but there is a configuration with the least differential in summation (sums in parens): (5)(5) (3) (7) (4) 1 3 A C = C  (6) 4 2 D B (Ulian Schemata) the extreme sums displaced to the diagonals. 2x2 pursued in letters might double upon itself like so: (4) A B 1 2 (3) = B A 2 1 (3) (2) (3)(3) simultaneously referencing paternal authority (ABBA), or (1600) A Z 1 800 (801) = Z A 800 1 (801) (2) (801)(801) symbolizing the Beginning and the End (Romanization of the Greek, popular Alpha et Omega). General  any letter / number / figure might be utilized within a square of this order to combine 2 (or 4) elements (e.g. Jupiter(6) and Saturn(7): 6 7 F G = 7 6 G F or the use of their sigils inside an amulet or some ceremonial diagram which calls for their qualities).  Proper Magic Squares (3x3 => 13x13) the first of the odd magic squares (Andrews divides them into families of odd and even because of methods of their derivation and construction) has *1* solution, and is the only magic square that has a single solution. the variation of this square is not its numerical arrange ment so much as its *orientation*. while this is true of all magic squares (that they have orientations, and also mirrorimages), this is seldom remarked upon by occultists who identify their selections, since the solution producing the requisite sums suffices to assign the whole. for my purposes, and in order to be thorough, I'll provide what information I am able about traditional cultural usage of these, assigning each to the rectified astronomical standard after the pattern of Agrippa and others in occult traditions, and requesting additions from the interested. my impression is that few take great care in dealing with magic squares as a whole, haphazardly presenting what to them conforms to a logical parameter (true enough, but insufficientlydetailed for scrutinizing occult practice).  (3) PLUTO  traditional: Lo Shu, Saturn thus the Square of Pluto (Lo Shu, formerly Saturn) in its various possible orientations is as follows: (15) (15) (15) (15) (row i) (15) 4 9 2 D I B (row ii) (15) 3 5 7 = C E G (row iii) (15) 8 1 6 = H A F (15) this is the configuration presented as both the Lo Shu (e.g. in Master Huang's book on Numerology) and for the Planet Saturn (in Tyson's Agrippa). I have labelled the rows here (i>iii) which could also be arranged in columns or transposed to iii>i. this latter arrangement (with 8/1/6 at top) is presented in Andrews for the 3x3 square (page 2, figure 1). there are 2 families of 4 orientations, constituting a total of 8 possible arrangements of this 3x3 square if one details its rectilinear positions and mirrorings:  2 7 6 6 7 2 9 5 1 1 5 9 4 3 8 8 3 4  4 3 8 8 3 4 9 5 1 1 5 9 2 7 6 6 7 2  2 9 4 4 9 2 7 5 3 3 5 7 6 1 8 8 1 6  6 1 8 8 1 6 7 5 3 3 5 7 2 9 4 4 9 2  these might be associated with 8 Yijing trigrams for a complex crossreference. how to begin such an assignment will depend on what one takes as a primary sequencing key or mapping translator (e.g. orienting the corners to a 2468 clockwise progression might initiate the sequence, or the 2/7/6 at top might come first as its numbers in the upperleft and topcenter are lowest). that the circling 8 numbers (1>9 excepting 5) are *already* given trigrammatic assignments within the Lo Shu (see Master Huang's Numerology at least for this) may make identifying the particular squares within this scheme easier by keying the numbers to trigrams and then solving for 'net trigram' by configurative quality. substituting numbers allows the following, initially, the following may be used within some formula to conclude as to ordinal value:  2 7 6 6 7 2 __ __ __ __ _____ _____ __ __ __ __ __ __ _____ _____ _____ _____ __ __ __ __ _____ _____ _____ _____ __ __ 9 5 1 1 5 9 _____ __ __ __ __ _____ __ __ . o _____ _____ o . __ __ _____ __ __ __ __ _____ 4 3 8 8 3 4 _____ __ __ _____ _____ __ __ _____ _____ __ __ __ __ __ __ __ __ _____ __ __ _____ __ __ __ __ _____ __ __  4 3 8 8 3 4 _____ __ __ _____ _____ __ __ _____ _____ __ __ __ __ __ __ __ __ _____ __ __ _____ __ __ __ __ _____ __ __ 9 5 1 1 5 9 _____ __ __ __ __ _____ __ __ . o _____ _____ o . __ __ _____ __ __ __ __ _____ 2 7 6 6 7 2 __ __ __ __ _____ _____ __ __ __ __ __ __ _____ _____ _____ _____ __ __ __ __ _____ _____ _____ _____ __ __  2 9 4 4 9 2 __ __ _____ _____ _____ _____ __ __ __ __ __ __ _____ _____ __ __ __ __ __ __ _____ __ __ __ __ _____ __ __ 7 5 3 3 5 7 __ __ __ __ __ __ __ __ _____ . o __ __ __ __ o . _____ _____ _____ _____ _____ 6 1 8 8 1 6 _____ __ __ _____ _____ __ __ _____ __ __ _____ __ __ __ __ _____ __ __ __ __ __ __ __ __ __ __ __ __ __ __  6 1 8 8 1 6 _____ __ __ _____ _____ __ __ _____ __ __ _____ __ __ __ __ _____ __ __ __ __ __ __ __ __ __ __ __ __ __ __ 7 5 3 3 5 7 __ __ __ __ __ __ __ __ _____ . o __ __ __ __ o . _____ _____ _____ _____ _____ 2 9 4 4 9 2 __ __ _____ _____ _____ _____ __ __ __ __ __ __ _____ _____ __ __ __ __ __ __ _____ __ __ __ __ _____ __ __  and once a formula of solution is put into operation, (e.g. root lines value 4, midlines value 2, top 1, leftmost value 4, midtrigram value 2, right 1, solve for each magic square arrangement to 8 values), then these may be arranged and identified accordingly and even used for divinatory and magical enterprises in association with the Planetary influences of Pluto. Andrews makes plain that one might substitute for the numbers in any of these squares a formula whereby the algebra of variables may be used to solve for all the magic square's contents given any initial figure 'X' which is the least number in the square (our solution merely derives the *least* integer, X=1), such that for 'X'=2 (45) (45) (45) (45) (row i) (45) 10 28 07 (row ii) (45) 12 15 18 (row iii) (45) 23 02 20 (45) which will not map to an order A=1...Z=26 system, requiring dualletter combinations such as the most evident Ulian simple combination system: J KH G JB JE JH KC B K through this method one might utilize Plutonian squares with variable least seed numbers to construct any number of alternatives, each having 8 sets of orientations and mirrorings which might be ciphered through an Ulian combination system to string sets. Magic Squares Essay terminates. ============================================================= returning to conversation with Martin, venturing to (4) (notes) # ...most ...fortunate square, ... the square of jupiter # or 4x4.... by traditional Planetary assignment, 4x4 = Jupiter, which is called in astrology 'The Great Beneficent'. for this reason your assertion is valuable within such a context. # 04 14 15 01 # 09 07 06 12 # 05 11 10 08 # 16 02 03 13 = D N O A I G F L E K J H P B C M (A=1... Z=26) through proliferation merely by orientation and mirrorings, we're talking about at LEAST 4 orientations and their 4 mirrors once more, *plus* any variations in composition beyond the obvious row/column reversal (yielding at least 8 possibles) by virtue of formulation of number positions (again where here X=1). why 04 should be placed at top left (rather than 01) is a valuable query. we have been presented the 'Jupiter Square' as selected by Agrippa. in fact, Andrews derives 6 laws (that I am not reproducing here for the sake of simplicity, but a thorough investigation of the 4x4 ought include it) governing all 4x4 magic squares, and produces the 4x4 square below, indicating that squares of *higher* order are too complex to reduce to laws. 01 08 12 13 14 11 07 02 15 10 06 03 04 05 09 16 as we can see just by these 2 examples, there are a large number of possible 4x4 arrangements, before even considering minor variation like orientation or mirroring, which may, as part of their composition, have the 'associated' quality of adding to a specific sum in symmetrical positions (e.g. here the sums equate to 17 on positions across a diagonal). I'm not aware of many cultural designations of the 4x4 magic square comparable to "Lo Shu". Andrews does mention what he calls a "Jaina Square" of 4x4, which is: JAINA SQUARE  07 12 01 14 02 13 08 11 16 03 10 05 09 06 15 04 from an inscription of the 12th or 13th centuries by Jains mentioned by a Professor Smith and passed on by Andrews (pp. 124128) and apparently of a different quality than many of the other possibilities (to be explored later). the qualities of 4x4 squares is only compounded in depth as one progresses to greater orders, plus it may be integrated into larger squares as central portions! # According to the philosophers (agrippa,solomon,pythagoras etc) there's a wealth beyond this, apparently, in the world of philosophy and magic. :> I am only barely touching on this myself by elaborating on your wonderful post. # there # is an intelligence or higher spirit if you like that determine the # outcome of certain events. There is a spirit to what is good and a # spirit to that which is bad. So I gather that the future prediction # of number is dependant on which spirit has the most influence at the # time. However as Agrippa tells us the number of these intelligences # its not neccessary to know who rules, just that there are 2 possible # future outcomes when predicting a number between 116. The # intelligences are both allocated 136 (the total of the square) and # are arrived at by means of the seals and characters of jupiter. my impression (a guess) is that some kind of name was discernable *within* the square being considered as it was substituted with letters and that these were ascribed *after the fact*, then drawn out as sigils accordingly. I see no reason why they would be necessary in any sense, but am open to hearing arguments for this. of course the way Agrippa has it these arrive in Hebrew, but one could construct this within English or other languages. that is certainly my intent, which is in part why I'm providing the Ulian and AlphaZed equivalences in this post, for future reference and construction by the student (myself included!). # the problem has been that no one knows how these seals where arrived # at and therefore can not be assigned to the squares directly. This # seems to be the first road block in it's translation. Is anyone here # familiar with the seals and sigils? this constitutes a secondary level of analysis where magic squares are concerned because they enter into linguistic construction that has no mathematical basis and therefore few real delimiters of form. Martin continues with the Jovian construct analysis: # However Agrippa tells us that they are arrived at # in this fashion: this actually constitutes a description of what was constructed, rather than its construction method per se. # Intelligence to what is good: # Jophiel = IHPHIAL # I=10 + H=5 + PH=80 + I=10 + A=1 + L=30 TOTAL=136 # # A spirit to what is bad: # Hismael = HSMAL # H=5 + S=60 + M=40 + A=1 + L=30 TOTAL=136 one may therefore analyze the *method of determination* to a degree and construct the rules: a) the total of the names of the Intelligences and Spirits must equate to the total of the numbers in the square; b) the letters for these numbers are connected, creating the Seals for magical operations. # ... # # The first step in this unsolvable mess was # the disproval of the 4 elements, our scientits discovered more and # more, and the basic concepts of the elements and their ratios was # lost. the refinement of 'elements' made this possible. modern scientists are merely reductionists, focussing on subatomic elements, whereas previous perspectives of elements weren't reductionist at all but material in a compository manner (examine the early Greeks and we can see that they were struggling to come up with a 'universal element' in the same way that today there is a move toward what has been called 'Unified Field Theory'. whether these early Greeks were actually atomists, precursors to today's materialists is something of a controversy as I understand it (something which I think Heidegger argued against, if I recall correctly). # ...if the # elements where ratios of numbers, like... 1 is to 2 as 2 is to 4 each # one being it's double, then why not the seven planets also? why could # the seven planets also not be ratios of 1 being related to one as the # other?, I mean is it a coincidence that an atom has seven shells, or # the the sacred tetractys (the foundation of all magic and all form) # also has seven points in this fashion? # # THE SACRED TETRACTYS OF PYTHAGORAS: # # POINT 1 UNITY # LINE 2 3 PRIME # SURFACE 4 9 SQUARE (2X2) + (3X3) # SOLID 8 27 CUBE (2X2X2) + (3X3X3) very lovely. the tetraktys is usually: 1 2 3 4 5 6 7 8 9 10 and so while I can see the resemblance, I'm not sure I understand your point here yet. Seyfert1 luckymojo.com@nagasiva =========================================================== refs Agrippa, H.C.: Three Books of Occult Philosophy, Llewellyn Publications, tr. Freake, ed. Tyson, 1997; ISBN 0875428320 Andrews, W.S.: Magic Squares and Cubes, Dover Publications, 1960; ISBN 0486206580 END
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