Geomantic Mathematics

Generating a complete geomantic chart can be a little daunting for people new to the art of geomancy.  I think it’s simple enough to learn, but there’s a fair bit of calculation involved.  It’s definitely more difficult than Tarot, where you just shuffle some cards and lay them out wherever you damn well please, but not as difficult as doing an astrological chart by hand (but then, who does that anymore?).

Still, there are fewer possible geomantic charts one might get than there are Tarot spreads ((78-10)! or (156-10)!, depending on whether you use reversed cards, and that’s just for the Celtic Cross) or astrological configurations (big big big big number, even if you limit yourself to just the seven traditional planets and whole degrees).  Since the four Mothers essentially define the rest of the chart, and since each Mother can be one of the 16 geomantic figures, there are only 16×16×16×16 = 65536 possible geomantic charts.  Any chart not in this set of charts are invalid and impossible to properly calculate.  How might you determine whether a given geomantic chart is valid?  There are three rules to validate a chart:

The Judge must be an even figure.  It is impossible for a well-formed geomantic chart to have an odd Judge; evenness is often called “impartiality”, and Judges as well as judges must be impartial in deciding a case.  Judge figures must be even due to the formation of the Daughters from the Mothers.  The Daughters make use of the same points from the Mothers, transposed so that they’re arranged in a different direction; thus, the number of points in the Mothers are the same as those in the Daughters.  Any number duplicated yields an even number, and the process of adding figures (or distilling them from the Mothers/Daughters to the Nieces to the Witnesses) preserves this kind of parity.  Thus, the Witnesses must be either both odd or both even, and in either case must add to an even figure.  The Judge is the only figure in the chart where this rule must apply.

At least one figure must be repeated in the chart.  As it turns out, no complete Shield chart with 16 geomantic figures can have all 16 distinct figures; there must be at least one repeated figure in the chart somewhere.  It may be possible that the first 15 figures (Mothers, Daughters, Nieces, Witnesses, and Judge) are distinct, but then the Sentence must of necessity repeat one of the other figures.  Consider that the Judge is formed from the two Witnesses, which themselves are formed from the four Nieces, which are formed from the eight Mothers and Daughters combined.  The Judge has eight separate roots, which may very well be distinct.  However, the Sentence is formed from adding the Judge to the First Mother.  Because the Judge also relies on the First Mother (via the Right Witness and First Niece), you’re essentially adding the First Mother to itself, which yields Populus; Populus, when combined with any other figure, repeats that figure.  Because of this “hidden repetition” in the chart, there’s bound to be at least one figure repeated in the chart somewhere, even if it’s just the Sentence.  That said, there are only 16 charts that have the first 15 figures unique, but that’s a topic for another day.

The inseparable pairs must add to the same figure.  This is an idea picked up from the Madagascan tradition of geomancy of sikidy, and shows the validity of the internal structure of the chart.  The idea here is that certain pairs of figures in the chart must add to the same figure: adding the First Niece to the Judge, the Second Mother to the Sentence, and the Second Niece to the Left Witness all yield the same result.  Similarly, the Left Witness added to the Sentence, the Right Witness to the First Mother, and the Second Niece to the Second Mother also yield the same result.  This is because the “units” that add up to any child figure (First and Second Mothers for the First Niece, or all the Mothers and Daughters for the Judge, or all the Mothers and Daughters for the Sentence with the First Mother duplicated) are the same within these groups of inseperables.  Any set of addition of “units” where two figures are repeated cancel each other out, forming Populus; the remaining figures add up to a particular figure that the other inseperables must also add to.

So, as an example, say that we have the following chart, where we have Via, Acquisitio, Coniunctio, and Laetitia as the Mothers.  Carcer, Cauda Draconis, Amissio, and Fortuna Minor are the Daughters; Amissio, Cauda Draconis, Caput Draconis, and Coniunctio are the Nieces; Rubeus and Tristitia are the Witnesses, Acquisitio is the Judge, and Amissio is the Sentence.

Example Geomantic Tableau

The Judge is Acquisitio, which is an even figure, formed from two odd figures; this is good.  There is multiple repetition in the chart (Acquisitio, Coniunctio, Cauda Draconis, and Amissio are all repeated somewhere in the chart), which is also good.  The two sets of inseparables add up the figures as below:

  1. First Set (sum of Third and Fourth Mothers with all the Daughters)
    1. First Niece + Judge = Amissio + Acqusitio = Via
    2. Second Mother + Sentence = Acquisitio + Amissio = Via
    3. Second Niece + Left Witness = Cauda Draconis + Tristitia = Via
  2. Second Set (sum of the Second, Third, and Fourth Mothers)
    1. Left Witness + Sentence = Tristitia + Amissio = Puella
    2. Right Witness + First Mother = Rubeus + Via = Puella
    3. Second Niece + Second Mother = Cauda Draconis + Acquisitio = Puella

Since the two sets of inseparable pairs add up to the same figures, respectively Via and Puella, this also checks out.  We can now rest assured that our geomantic chart is valid and proper for reading.

Do I do all these checks every time I calculate a geomancy chart?  Lol nope.  When I calculate a geomancy chart by hand (I sometimes use a program I wrote for this to automatically give me all the information I want from a chart), I’ll often just check the parity of the Judge and leave it at that.  Still, learning these rules and how the internal structure of the shield chart works is important to geomancy, since it underlies not only the mechanics of getting the divination system to work but also indicates important spiritual and oracular connections between the otherwise disparate symbols used.