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Subject: Magic Square Solutions?
From: Seyfert-1
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50031029 viii om
first installment, 0, 1, 2x2, 3x3 (+ 4x4 notes)
# ...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 well-worn constructions of the past, even
though their knowledge-base is superceded by plain contradiction,
is to set into Golden Age remembrance what should be oriented to
the present and personal as a mapping one-to-one 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 Seyfert-1 (nagasiva)
------------------------------------------------------------
Proto-Squares
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 pre-Ketherian 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
mirror-images), 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
insufficiently-detailed 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 cross-reference. 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 2-4-6-8 clockwise
progression might initiate the sequence, or the
2/7/6 at top might come first as its numbers in
the upper-left and top-center 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, mid-lines value 2, top 1,
left-most value 4, mid-trigram 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 dual-letter 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 in Sacredlandscapelist, 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. 124-128) 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 1-16. 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.
a 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.
Seyfert-1
luckymojo.com@nagasiva
===========================================================
refs
Agrippa, H.C.: Three Books of Occult Philosophy,
Llewellyn Publications, tr. Freake,
ed. Tyson, 1997; ISBN 0-87542-832-0
Andrews, W.S.: Magic Squares and Cubes,
Dover Publications, 1960; ISBN 0-486-20658-0
===========================================================
END -- apologies for any duplications.