On Geomantic Figure Magic Squares

We all know and love magic squares, don’t we?  Those grids of numbers, sometimes called “qamea” (literally just meaning “amulet” or “talisman” generally in Hebrew, קמיע or qamia`), are famous in Western magic for being numerological stand-ins or conceptions of the seven planets, sure, such as the 3×3 square for Saturn, the 5×5 square for Mars, and so forth, but they’re also huge in Arabic magic, too, from which Western magicians almost certainly got the idea.  Sure, magic letter squares are ancient in the West, such as the famous Sator Square from Roman times until today, and have more modern parallels in texts like the Sacred Magic of Abramelin, but magic number squares are fun, because they combine numerical and numerological principles together in an elegant form.

Which is why I was caught off-guard when I saw these two squares online, the first from this French blog post on Arabic geomancy and the other shared in the Geomantic Study-Group on Facebook:

Well…would you take a look at that?  Geomantic magic squares!  It took me a bit to realize what I was seeing, but once it hit me, I was gobsmacked.  It might not be immediately apparent how to make a geomantic magic square, but it becomes straightforward if you consider the figures as numbers of points, such that Laetitia stands in for 7, Puer for 5, Carcer for 6, and so forth.  Sure, it’s not a traditional kind of n × n number square that goes from 1 to n², but there are plenty of other magic squares that don’t do that either in occult practice, so seeing a kind of geomantic figure magic square actually makes a lot of sense when they’re viewed as numbers of points.  In this case, the magic sum of the square—the sum of the columns or rows—is 24.

Consider that first magic square, elegant as it is.  When it’s oriented on a tilt, such that one of its diagonals is vertical, we have the four axial figures (Coniunctio, Carcer, Via, and Populus) right down the middle, and all the other figures are arranged in reverse pairs in their corresponding positions on either side of the square.  For instance, Amissio and Acquisitio are on either side of the central axis “mirroring” each other, as are Tristitia and Laetitia, Fortuna Maior and Fortuna Minor, and so forth.  This is a wonderful geometric arrangement that shows a deep and profound structure that underlies the figures, and which I find particularly beautiful.

Of course, knowing that there are at least two such geomantic figure magic squares, and seeing possibilities for variation (what if you rearranged the figures of that first magic square above such that all the entering figures were on one side and all the exiting figures on the other?), that led me to wonder, how many geomantic magic squares are there?  Are there any structural keys to them that might be useful, or any other numerical properties that could be discovered?  So, late one evening, I decided to start unraveling this little mystery.  I sat down and wrote a quick program that started with the following list of numbers:

[ 4 , 5 , 5 , 5 , 5 , 6 , 6 , 6 , 6 , 6 , 6 , 7 , 7 , 7 , 7 , 8 ]
  • Why this list?  Note that the figure magic squares rely on counting the points of the figures.  From that point of view, Puer (with five points) can be swapped by Puella, Caput Draconis, or Cauda Draconis in any given figure magic square and it would still be another valid magic square that would have the same underlying numerical structure.  There’s only one figure with four points (Via), four figures with five points (Puer, Puella, Caput Draconis, Cauda Draconis), six figures with six points (Carcer, Coniunctio, Fortuna Maior, Fortuna Minor, Amissio, Acquisitio), four figures with seven points (Albus, Rubeus, Laetitia, Tristitia), and only one figure with eight points (Populus).  If we simply focus on the point counts of the figures themselves and not the figures, we can simplify the problem statement significantly and work from there, rather than trying to figure out every possible combination of figures that would yield a magic square from the get-go.
  • How does such a list get interpreted as a 4 × 4 square?  There are 16 positions in the list, so we can consider the first four positions (indices 0 through 3) to be the top row of the square, the second four positions (indices 4 through 7) to be the second row, the third four positions (indices 8 through 11) to be the third row, and the fourth four positions (indices 12 through 15) as the fourth row, all interpreted from left to right.  Thus, the first position is the upper left corner, the second position the uppermost inside-left cell, the third position the uppermost inside-right cell, the fourth position the upper right corner, the fifth position the leftmost inside-upper cell, the sixth position the inside-upper inside-left cell, and so forth.  This kind of representation also makes things a little easier for us instead of having to recursively deal with a list of lists.
  • How do we know whether any permutation of such a list, interpreted as a 4 × 4 square, satisfies our constraints?  We need to add up the values of each row, column, and diagonal and make sure they add up to our target number (in our case, 24).

Starting from this list, I set out to get all the unique permutations.  Originally, I just got all 16! = 20,922,789,888,000 possible permutations, thinking that would be fine, and testing them each for fitting the target number of 24, but after running for twelve hours, and coming up with over 170,000 results with more being produced every few minutes, I realized that I’d probably be waiting for a while.  So, I rewrote the permutation code and decided to get only unique permutations (such that all the 5s in the base list of numbers are interchangeable and therefore equal, rather than treating each 5 as a unique entity).  With that change, the next run of the program took only a short while, and gave me a list of 368 templates.  We’re getting somewhere!

So, for instance, take the last template square that my program gave me, which was the list of numbers [6, 6, 5, 7, 8, 5, 6, 5, 6, 7, 6, 5, 4, 6, 7, 7].  Given that list, we can interpret it as the following template magic square:

6 6 5 7
8 5 6 5
6 7 6 5
4 6 7 7

And we can populate it with any set of figures that match the point counts accordingly, such as the one below:

Fortuna
Minor
Fortuna
Maior
Puer Laetitia
Populus Puella Carcer

Cauda
Draconis

Amissio Albus Acquisitio Caput
Draconis
Via Coniunctio Rubeus Tristitia

Excellent; this is a totally valid geomantic figure magic square, where the point counts of each row, column, and diagonal add to 24.  To further demonstrate the templates, consider the two images of the figure magic squares I shared at the top of the post.  However, although I was able to find the first magic square given at the start of the post (the green-on-sepia one), the second one (blue with text around it) didn’t appear in the list.  After taking a close look at my code to make sure it was operating correctly, I took another look at the square itself.  It turns out that, because although all the rows and columns add to 24, one of the diagonals adds up to 20, which means it’s not a true geomantic figure magic square.  Welp!  At least now we know.

But there’s still more to find out, because we don’t have all the information yet that we set out to get.  We know that there are 368 different template squares, but that number hides an important fact: some template squares are identical in structure but are rotated or flipped around, so it’s the “same square” in a sense, just with a transformation applied.  It’s like taking the usual magic number square of Saturn and flipping it around.  So, let’s define three basic transformations:

  1. Rotating a square clockwise once.
  2. Flipping a square horizontally.
  3. Flipping a square vertically.

We know that we can rotate a square up to three times, which gets us a total of four different squares (unrotated, rotated once, rotated twice, rotated thrice).  We know that we can leave a square unflipped, flipped horizontally, flipped vertically, and flipped both horizontally and vertically.  We know that a square can be rotated but not flipped, flipped but not rotated, or both rotated and flipped.  However, it turns out that trying out all combinations of rotating and flipping gets duplicate results: for instance, flipping vertically without rotating is the same as rotating twice and flipping horizontally.  So, instead of there being 16 total transformations, there are actually only eight other templates that are identical in structure but just transformed somehow, which means that our template count of 368 is eight times too large.  If we divide 368 by 8, we get a manageable number of just 46 root templates, which isn’t bad at all.

What about total possible figure squares?  Given any template, there are four slots for figures with five points, four slots for figures with seven points, and six slots for figures with six points.  The figures of any given point count can appear in any combination amongst the positions with those points.  This means that, for any given template square, there are 4! × 4! × 6! = 414,720 different possible figure squares.  Which means that, since there are 368 templates, there are a total of 152,616,960 figure squares, each a unique 4 × 4 grid of geomantic figures that satisfy the condition that every column, row, and diagonal must have 24 points.  (At least we’ve got options.)

What about if we ignore diagonals?  The blue magic square above is almost a magic square, except that one of its diagonals adds up to 20 and not 24.  If we only focus on the rows and columns adding up to 24 and ignore diagonals, then we get a larger possible set of template squares, root template squares, and figure squares:

  • 5,904 template squares
  • 738 root template squares
  • 2,448,506,880 possible figure squares

So much for less-magic squares.  What about more-magic squares?  What if we take other subgroups of these squares besides the rows, columns, and diagonals—say, the individual quadrants of four figures at each corner of the square as well as the central quadrant, or the just the corner figures themselves, or the bows and hollows?  That’s where we might get even more interesting, more “perfect” geomantic figure magic squares, so let’s start whittling down from least magic to most magic.  Just to make sure we’re all on the same page, here are examples of the different patterns I’m considering (four columns, four rows, two diagonals, five quadrants, four bows, four hollows, one set of corners):

To keep the numbers manageable, let’s focus on root template square counts:

  • Rows and columns only: 738 root templates
  • Rows, columns, and diagonals: 46 root templates
  • Rows, columns, diagonals, and all five quadrants: 18 root templates
  • Rows, columns, diagonals, all five quadrants, bows, and hollows: 2 root templates
  • Rows, columns, diagonals, all five quadrants, bows, hollows, and the four corners: 2 root templates

With each new condition, we whittle down the total number of more-magical root templates from a larger set of less-magical root templates.  I’m sure there are other patterns that can be developed—after all, for some numeric magic squares of rank 4, there are up to 52 different patterns that add up to the magic sum—but these should be enough to prove the point that there are really two root templates that are basically as magical as we’re gonna get.  Those root templates, along with their transformations, are:

  1. [6, 4, 7, 7, 8, 6, 5, 5, 5, 7, 6, 6, 5, 7, 6, 6]
    1. Unflipped, unrotated: [6, 4, 7, 7, 8, 6, 5, 5, 5, 7, 6, 6, 5, 7, 6, 6]
    2. Unflipped, rotated once clockwise: [5, 5, 8, 6, 7, 7, 6, 4, 6, 6, 5, 7, 6, 6, 5, 7]
    3. Unflipped, rotated twice clockwise: [6, 6, 7, 5, 6, 6, 7, 5, 5, 5, 6, 8, 7, 7, 4, 6]
    4. Unflipped, rotated thrice clockwise: [7, 5, 6, 6, 7, 5, 6, 6, 4, 6, 7, 7, 6, 8, 5, 5]
    5. Flipped, unrotated: [7, 7, 4, 6, 5, 5, 6, 8, 6, 6, 7, 5, 6, 6, 7, 5]
    6. Flipped, rotated once clockwise: [6, 8, 5, 5, 4, 6, 7, 7, 7, 5, 6, 6, 7, 5, 6, 6]
    7. Flipped, rotated twice clockwise: [5, 7, 6, 6, 5, 7, 6, 6, 8, 6, 5, 5, 6, 4, 7, 7]
    8. Flipped, rotated thrice clockwise: [6, 6, 5, 7, 6, 6, 5, 7, 7, 7, 6, 4, 5, 5, 8, 6]
  2. [8, 6, 5, 5, 6, 4, 7, 7, 5, 7, 6, 6, 5, 7, 6, 6]
    1. Unflipped, unrotated: [8, 6, 5, 5, 6, 4, 7, 7, 5, 7, 6, 6, 5, 7, 6, 6]
    2. Unflipped, rotated once clockwise: [5, 5, 6, 8, 7, 7, 4, 6, 6, 6, 7, 5, 6, 6, 7, 5]
    3. Unflipped, rotated twice clockwise: [6, 6, 7, 5, 6, 6, 7, 5, 7, 7, 4, 6, 5, 5, 6, 8]
    4. Unflipped, rotated thrice clockwise: [5, 7, 6, 6, 5, 7, 6, 6, 6, 4, 7, 7, 8, 6, 5, 5]
    5. Flipped, unrotated: [5, 5, 6, 8, 7, 7, 4, 6, 6, 6, 7, 5, 6, 6, 7, 5]
    6. Flipped, rotated once clockwise: [8, 6, 5, 5, 6, 4, 7, 7, 5, 7, 6, 6, 5, 7, 6, 6]
    7. Flipped, rotated twice clockwise: [5, 7, 6, 6, 5, 7, 6, 6, 6, 4, 7, 7, 8, 6, 5, 5]
    8. Flipped, rotated thrice clockwise: [6, 6, 7, 5, 6, 6, 7, 5, 7, 7, 4, 6, 5, 5, 6, 8]

That second one, for instance, is the root template of that first figure magic square given above (green-on-sepia), unflipped and rotated clockwise twice.  So, with these, we end up with these two root template squares, from which can be developed eight others for each through rotation and reflection, meaning that there are 16 template squares that are super magical, which means that there are a total of 6,635,520 possible figure squares—414,720 per each template—once you account for all variations and combinations of figures in the slots.

That there are 16 templates based on two root templates is suggestive that, maybe, just maybe, there could be a way to assign each template to a geomantic figure.  I mean, I was hoping that there was some way we’d end up with just 16 templates, and though I was ideally hoping for 16 root templates, two root templates is pretty fine, too.  With 16 figures, there are at least two ways we can lump figures together into two groups of eight: the planetary notion of advancing or receding (advancing Populus vs. receding Via for the Moon, advancing Albus vs. receding Coniuncto for Mercury, advancing Fortuna Maior and receding Fortuna Minor for the Sun, etc.), or the notion of entering or exiting figures.  Personally, given the more equal balance of figures and the inherently structural nature of all this, I’m more inclined to give all the entering figures to one root template and all the exiting figures to the other.  As for how we might assign these templates to the figures, or which set of templates get assigned to the entering figures or exiting figures, is not something I’ve got up my sleeve at this moment, but who knows?  Maybe in the future, after doing some sort of structural analysis of the templates, some system might come up for that.

More than that, how could these squares be used?  It’s clear that they’ve got some sort of presence in geomantic magic, but as for specifically what, I’m not sure.  Unlike a geomantic chart, which reveals some process at play in the cosmos, these geomantic squares are more like my geomantic emblems project (and its subsequent posts), in that they seem to tell some sort of cosmic story based on the specific arrangement of figures present within the square or emblem.  However, like those geomantic emblems, this is largely a hammer without a nail, a mathematical and structural curiosity that definitely seems and feels important and useful, just I’m not sure how.  Still, unlike the emblems, figure squares actually have a presence in some traditions of geomancy, so at least there’s more starting off there.  Perhaps with time and more concentrated translation and studying efforts, such purposes and uses of figure squares can come to light, as well as how a potential figure rulership of the sixteen most-magical templates can play with the 414,720 different arrangements of figures on each template and how they feel or work differently, and whether different arrangements do different things.  Heck, there might be a way to assign each of the different combinations of figures on the templates to the figures themselves; after all, 414,720 is divisible by 16, yielding 25,920, which itself is divisible by 16, yielding 1620, so there might be 1620 different figure squares for each of the 256 (16 × 16) combinations of figures.  Daunting, but hey, at least we’d have options.

Also, there’s the weird bit about the target sum of the magic squares being 24.  This is a number that’s not really immediately useful in geomancy—we like to stick to 4 or 16, or some multiple thereof—but 24 is equal to 16 + 8, so I guess there’s something there.  More immediately, though, I’m reminded of the fact that 24 is the number of permutations of vowels in my system of geomantic epodes for most figures.  For instance, by giving the vowel string ΟΙΕΑ (omikron iōta epsilon alpha) to Laetitia, if we were to permute this string of vowels, we’d end up with 24 different such strings, which could be used as a chant specifically for this figure:

ΟΙΕΑ ΟΙΑΕ ΟΕΙΑ ΟΕΑΙ ΟΑΙΕ ΟΑΕΙ
ΙΟΕΑ ΙΟΑΕ ΙΕΟΑ ΙΕΑΟ ΙΑΟΕ ΙΑΕΟ
ΕΟΙΑ ΕΟΑΙ ΕΙΟΑ ΕΙΑΟ ΕΑΟΙ ΕΑΙΟ
ΑΟΙΕ ΑΟΕΙ ΑΙΟΕ ΑΙΕΟ ΑΕΟΙ ΑΕΙΟ

From that post, though, Populus only has a three-vowel string, which can be permuted only six times, but if we repeat that chant four times total, then we’d still end up with 24 strings to chant, so that still works out nicely:

ΙΕΑ ΕΑΙ ΑΙΕ ΕΙΑ ΙΑΕ ΑΕΙ
ΙΕΑ ΕΑΙ ΑΙΕ ΕΙΑ ΙΑΕ ΑΕΙ
ΙΕΑ ΕΑΙ ΑΙΕ ΕΙΑ ΙΑΕ ΑΕΙ
ΙΕΑ ΕΑΙ ΑΙΕ ΕΙΑ ΙΑΕ ΑΕΙ

So maybe 24 is one of those emergent properties of some applications of geomantic magic that could be useful for us.  Perhaps.  It’s worth exploring and experimenting with, I claim.

In the meantime, I’ll work on getting a proper list drawn up of all the templates for the various types of geomantic magic squares—ranging from less magic to more magic—at least just to have for reference for when further studies are or can be done on this.  This is more of a curiosity of mine and not a prioritized topic of research, but at least I know it exists and there’s the potential for further research to be done on it for future times.