When I did my recent site redesign and added all those new pages on prayers, rituals, and whatnot, I also consolidated a few pages into ones that fit neatly together, and got rid of a few entirely that didn’t need to be on here anymore.  There weren’t many of those, to be fair, but the main casualties of that effort were my handful of pages on geomancy.  While it may seem odd that I, of all people, would take down pages on the art I love so much, it was partially because I’m continuing to prepare for my book and wanted to rewrite and incorporate the information of those pages in a better way than what was presented there, and partially because the idea for those pages has long since turned stale; I was going to have an entire online “book” of sorts, but I figure that I’ve written enough about geomancy on my blog that it’s probably easier to just browse through the geomancy category and read.  So, if you end up finding a broken link (which I do my utmost to keep from happening), chances are you’re seeing a relic of an earlier age on this blog that connected to those pages.  After all, even though I’d like to keep my blog in perfect running order, I’m also not gonna scroll through 600-odd posts and comb through each and every link.

One of the things that those lost geomancy pages discussed was the mathematical operations of the figures.  I’ve talked about the mathematics behind the Judge and the Shield Chart before, as well as the Parts of Fortune and Spirit, and I’ve discussed a sort of “rotary function” that rotates the elemental rows up and down through the figures before, but there are three big mathematical operations one can do on the figures themselves that reveal certain relationships between them.  I mention them on my De Geomanteia posts of the figures themselves, though now that the original page that describes them is down, I suppose a new post on what they are is in order, if only to keep the information active, especially since every now and then someone will come asking about them.  This is important, after all, because this information is definitely out there, but it’s also largely a result of my own categorization; I haven’t seen anyone in the Western literature, modern or ancient, online or offline, talk about the mathematical relationships or “operations” between the figures in the way I have, nor have I seen anyone talk about one of the operations entirely, so this post is to clear up those terms and what they signify.

First, let me talk about something tangentially related that will help with some of the operation discussion below.  As many students of geomancy are already aware, a common way to understand the figures is in terms of their motion, which is to say, whether a figure is stable or mobile.  Structurally speaking, stable figures are those that have more points in the Fire and Air rows than in the Water and Earth rows (e.g. Albus), and mobile figures are those that have more points in the Water and Earth rows than in the Fire and Air rows (e.g. Puer).  In the cases where the top two rows have the same number of points as the bottom two rows (e.g. Amissio or Populus), the figures are “assigned” a motion based on their general effects.

• Stable figures: Populus, Carcer, Albus, Puella, Fortuna Maior, Acquisitio, Tristitia, Caput Draconis
• Mobile figures: Via, Coniunctio, Rubeus, Puer, Fortuna Minor, Amissio, Laetitia, Cauda Draconis

Stable figures are generally seen as graphically looking like they’re “sitting upright” when viewed from the perspective of the reader, while mobile figures are considered “upside down” or “unbalanced” when read the same way.  In a similar sense, stable figures generally have effects that are slow to arise and long to last, while mobile figures are just the opposite, where they’re quick to happen and quick to dissipate.  Consider mobile Laetitia: a figure of optimism, elevation, hope, and bright-burning joy, but it’s easy to lose and hard to maintain.  This can be contrasted with, for instance, stable Tristitia: a figure of slow-moving depression, getting stuck in a rut, languishing, and losing hope.

The idea of motion, I believe, is a simplification of an older system of directionality, where instead of there being two categories of figures, there are three: entering, exiting, and liminal.  All entering figures are stable, all exiting figures are mobile, and the liminal figures are considered in-between:

• Entering figures: Albus, Puella, Fortuna Maior, Acquisitio, Tristitia, Caput Draconis
• Exiting figures: Rubeus, Puer, Fortuna Minor, Amissio, Laetitia, Cauda Draconis
• Liminal figures: Populus, Via, Carcer, Coniunctio

In this system, entering figures are seen as “bringing things to” the reader or reading, and exiting figures “take things away from” the reader or reading, while liminal figures could go either way or do nothing at all, depending on the situation and context in which they appear.  For instance, consider Acquisitio, the quintessential entering figure, which brings things for the gain of the querent, while exiting Amissio is the opposite figure of loss, taking things away, and all the while liminal Populus is just…there, neither bringing nor taking, gaining nor losing.

The liminal figures also serve another purpose: they are also sometimes called “axial” figures, because by taking the upper or lower halves of two axial figures, you can form any other figure.  For instance, the upper half of Populus combined with the lower half of Via gets you Fortuna Maior, the upper half of Coniunctio with the lower half of Carcer gets you Acquisitio, and so forth.  This way of understanding the figures as being composed of half-figures is the fundamental organization of Arabic-style geomantic dice:

Entering figures, like stable figures, look like they’re “coming towards” the reader, while exiting figures look like they’re “going away” from the reader, much like mobile figures.  The reason why the liminal figures (“liminal” meaning “at the threshold”) are considered in-between is that they look the same from either direction, and are either going both ways at once or going in no direction at all.  Populus and Carcer went from liminal to stable due to their long-lasting effects of stagnation or being locked into something, while Via and Coniunctio went from liminal to mobile for their indications of change, movement, and freedom.

Alright!  With the basic structural talk out of the way, let’s talk about operations.  In essence, I claim that there are three primary operations one can do on a figure to obtain another figure, which may or may not be the same as the original figure.  These are:

• Inversion: replace the odd points with even points, and even points with odd points.  For instance, inverting Puer gets you Albus.
• Reversion: flip the figure vertically.  For instance, inverting Puer gets you Puella.
• Conversion: invert then revert the figure, or revert and invert the figure.  For instance, converting Puer gets you Rubeus (Puer →Albus → Rubeus to go the invert-then-revert route, or Puer → Puella → Rubeus to go the revert-then-invert route).

In my De Geomanteia posts, I briefly described what the operations do:

• Inversion: everything a figure is not on an external level
• Reversion: the same qualities of a figure taken to its opposite, internal extreme
• Conversion: the same qualities of a figure expressed in a similar manner

And in this post on a proposed new form of Shield Cart company based on these operations, I described these relationships in a slightly more expanded way:

• Inversion: The two figures fulfill each other’s deficit of power or means, yet mesh together to form one complete and total force that will conquer and achieve everything that alone they could not.
• Reversion: The two figures are approaching the same matter from different directions and have different results in mind, looking for their own ends, but find a common thing to strive for and will each benefit from the whole.
• Conversion: The two figures are similar enough to act along the same lines of power and types of action, but express it in completely different ways from the outside.  Internally, the action and thoughts are the same, but externally, they are distinct.  Think bizarro-world reflections of each other.

These trite descriptions are a little unclear and, now that several years have passed, I realize that they’re probably badly phrased, so it’s worth it to review what these relationships are and how they tie into other conceptions of figure relationships.  After all, inversion and reversion both deal with the notion of something being a figure’s opposite, but we often end up with two separate “opposites”, which can be confusing; and, further, if you take the opposite of an opposite, you get something similar but not quite the same (inversion followed by reversion, or vice versa, gets you conversion).

To my mind, inversion is the most outstanding of the operations, not because it’s any more important than the others, but because it’s so radical and fundamental a change from one figure to the other.  To invert a figure, simply swap the points with their opposites: turn the odd points even and the even points odd.  You could say that you’re turning a figure into its negative, I suppose, like flipping the signs, levels of activity, or polarity of each individual element.  Most notably, the process of inversion is the only one that we can perform through simple geomantic addition of one figure with another; to invert a figure, simply add Via to it, and the result will be that figure’s inversion.  Because inversion is simply “just add Via”, this is probably the easiest to understand: inverting a figure results in a new figure that is everything the original figure isn’t.  We turn active elements passive and passive elements active, male into female and female into male, light into dark and dark into light.  What one has, the other lacks; what one forgets, the other remembers.

So much for inversion.  Reversion is as simple as inversion, but there’s no “just add this figure” to result in it; it’s a strictly structural transformation of one figure based on that figure’s rows.  To be specific and clear about it, to revert a figure, you swap the Fire and Earth lines, as well as the Air and Water lines; in effect, you’re turning the figure upside down, so that e.g. Albus becomes Rubeus or Caput Draconis becomes Cauda Draconis.  Note that unlike inversion where the invert of one figure is always going to be another distinct figure, there are some figures where the reversion is the same as the original figure; this is the case only for the liminal figures (Populus, Via, Carcer, Coniunctio), since rotating them around gets you the same figure.  By swapping the points in the lines of the elements that agree with each other in heat (dry Fire with dry Earth, and moist Air with moist Water), you get another type of opposite, but rather than it playing in terms of a strict swap of polarity like from positive to negative, you literally turn everything on its head.

Both inversion and reversion get you an “opposite” figure, but there are different axes or scales by which you can measure what an “opposite” is.  As an example, consider Puer.  If you invert Puer, you get Albus; this is an opposite in the sense that the youthful brash boy with all the energy in the world is the “opposite” of the wise old man without energy.  What Puer has (energy), Albus lacks; what Albus has (experience), Puer lacks.  On the other hand, if you revert Puer, you get Puella; this is another kind of opposite in the sense that the masculine is the opposite of the feminine.  What Puer is (masculine, active, emitting), Puella isn’t (feminine, passive, accepting).  This type of analysis, where inversion talks about “has or has not” and reversion talks about “is or is not” is the general rule by which I understand the figures, and holds up decently well for the odd figures.  It’s when you get to the even figures that this type of distinction between the operations by means of their descriptions collapses or falls apart:

• For non-liminal even figures, the inversion of a figure is the same as its reversion.  Thus, “is” is the same thing as “has”.  For instance, Acquisitio is the total opposite of Amissio, since they are both reversions and inversions of each other; gain both is not loss and loss does not have gain.
• For liminal even figures, the reversion of a figure is the same figure as itself.  Thus, “has” makes no sense, because the figure isn’t speaking to anything one “has” or “lacks” to begin with.  For instance, Carcer’s reversion is Carcer; Carcer is imprisonment and obligation, it doesn’t “have” a quality of its own apart from what it already is.  On the other hand, Carcer’s inversion, what Carcer is not, is Coniunctio, which is freedom and self-determination.  Again, Coniunctio describes a state of being rather than any quality one has or lacks.

Between inversion and reversion, we can begin to understand the pattern of how the babalawos of Ifá, the West African development and adaption of geomancy to Yoruba principles and cosmology, organize their sixteen figures, or odu:

Rank	Latin Name	Yoruba Name	Relationship
1	Via	Ogbe	inversion
2	Populus	Oyẹku
3	Coniunctio	Iwori	inversion
4	Carcer	Odi
5	Fortuna Minor	Irosun	inversion- reversion
6	Fortuna Maior	Iwọnrin
7	Laetitia	Ọbara	reversion
8	Tristitia	Ọkanran
9	Cauda Draconis	Ogunda	reversion
10	Caput Draconis	Ọsa
11	Rubeus	Ika	reversion
12	Albus	Oturupọn
13	Puella	Otura	reversion
14	Puer	Irẹtẹ
15	Amissio	Ọsẹ	inversion- reversion
16	Acquisitio	Ofun

With the exception of the even liminal figures, which are grouped in inversion pairs at the beginning of the order, it can be seen that the other figures are arranged in reversion pairs, with the even non-liminal figures grouped in what is technically either inversion or reversion, but which are most likely considered to just be reversions of each other.  Note how the non-liminal even figure pairs are placed in the order: they separate the strict-inversion pairs from the strict-reversion pairs, one at the start of the strict-reversion pairs and one at the end.  While it’s difficult to draw specific conclusions from this alone (the corpus of knowledge of odu is truly vast and huge and requires years, if not decades of study), the placement of the figures in this arrangement cannot be but based on the structure of the figures in their inversion/reversion pairs.

In another system entirely, Stephen Skinner describes some of the relationships of figures in Arabic geomancy in his book “Geomancy in Theory and Practice”, at least as used in some places in northern Africa, where the relationships are described in familial terms and which are all seemingly based on inversion:

• Man and wife
  • Tristitia and Cauda Draconis
  • Laetitia and Caput Draconis
  • Albus and Puer
  • Puella and Rubeus
  • Coniunctio and Carcer
• Brothers
  • Fortuna Minor and Fortuna Maior
  • Acquisitio and Amissio
• No relation
  • Via and Populus

Stephen Skinner doesn’t elaborate on what “man and wife” or “brothers” means for interpreting the figures, but if I were to guess and extrapolate on that small bit of information alone (which shouldn’t be trusted, especially if someone else knowledgeable in these forms of geomancy can correct me or offer better insight):

• For figures in “man and wife” pairings, the first figure is the “husband” and the second figure is the “wife”.  Though I personally dislike such an arrangement, it could be said that the husband figure of the pair dominates the wife figure, and though they may share certain similarities that allow for them to be married in a more-or-less natural arrangement, the husband figure is more powerful, domineering, overcoming, or conquering than the wife figure.  The central idea here is that of domination and submission under a common theme.
• For figures in “brothers” pairings, the figures are of equal power to each other, but are more opposed to each other than in harmony with each other, though they form a different kind of complete whole.  Thus, they’re like two brothers that fight with each other (in the sense of one brother against the other) as well as with each other (in the sense of both brothers fighting against a third enemy).  The central idea here is that of oppositions and polarity that form a complete whole.
• For the two figures that have no relation to each other, Via and Populus, this could be said that they are so completely different that they operate in truly different worlds; they’re not just diametrically opposed to each other to form a whole, nor is one more dominant over or submissive to the other in the same theme, but they’re just so totally and completely different that there is no comparison and, thus, no relationship.

Of course, all that is strictly hypothetical; I have nothing else to go on besides these guesses, and as such, I don’t use these familial relationships in my own understanding of the figures.  However, these are all indicative ways of how to view “opposites”, and is enlightening on its own.  However, note the specific figures in each set of relationships.  With the exception of Coniunctio and Carcer, all the husband-wife pairs are odd figures, so the only possible relationship each figure could have in their pair is inversion.  For the brother pairs, however, these are the even non-liminal figures, where the figures could be seen as either inversions or reversions of each other.  This could well be a hint at a difference between the meanings of inversion and reversion in an African or Arabic system of understanding the figures.

Alright, so that all deals with inversion and reversion, which leaves us with one final operation.  Conversion, as you might have gathered by now, is just the act of performing inversion and reversion on a figure at the same time: you both swap the parity of each row, and rotate the order of the row upside down (or vice versa, it’s the same thing and doesn’t matter).  In a sense, you’re basically taking the opposite of an opposite, but you’re not necessarily going from point A to point B back to point A; that’d just be inverting an inversion or reverting a reversion.  Rather, by applying both operations, you end up in a totally new state that is at once familiar while still being different.  For instance, consider Puella.  Puella’s conversion is Albus, and at first blush, it doesn’t seem like there’s much in similarity between these two figures except, perhaps, their ruling element (Water, in this case).  But bear in mind that both Puella and Albus don’t like to act, emit, or disturb things; Puella is the kind, welcoming hostess who accepts and nurtures, while Albus is the kind, wizened old man who accepts and guides.  Neither of them are chaotic, violent, energetic, or brash like Puer or Rubeus, and while they don’t do things for the same reason or in the same way, they end up doing things that are highly similar, like the same leitmotif played in a different key.

However, this is a little weird for the liminal figures, because a liminal figure’s reversion is the same as itself; this means that a liminal figure’s conversion is the same as its inversion (because the reversion “cancels out”).  Thus, converting Populus gets you Via, and converting Carcer gets you Coniunctio.  While these are clearly opposites of each other, it speaks to the idea that there’s a sort of “yin in the yang, yang in the yin” quality to these figure pairs.  This is best shown by Populus, which is pure potential with all activity latent and waiting to be sprung, and Via, which is pure activity but taken as a whole which doesn’t, on the whole, change.  Likewise, you can consider Carcer to be restriction of boundaries, but freedom to act within those set parameters, and Coniunctio, which is freedom of choice, but being constrained by the choices you make and the paths you take.

It’s also a little weird for the non-liminal even figures, because the reversion of these figures is the same as its inversion, which means that the conversion of an non-liminal even figure gets you that same figure itself.  While the “opposite of an opposite” of odd figures takes you from point A to B to C to D, the nature of the non-liminal even figures takes you from point A to B right back to A.  This reflects the truly is-or-is-not nature of these figures where there’s only so many ways you can view or enact the energies of what they represent: either you win or you lose, either you gain or you lose.  You might not win using the same strategy as you expected to use, but winning is winning; you may not get exactly what you thought you were after, but you’re still getting something you needed.

With these three operations said, I suppose it’s appropriate to have a table illustrating the three results of these operations for each of the sixteen figures:

Figure	Inversion	Reversion	Conversion
Populus	Via	Populus	Via
Via	Populus	Via	Populus
Albus	Puer	Rubeus	Puella
Coniunctio	Carcer	Coniunctio	Carcer
Puella	Rubeus	Puer	Albus
Amissio	Acquisitio	Acquisitio	Amissio
Fortuna Maior	Fortuna Minor	Fortuna Minor	Fortuna Maior
Fortuna Minor	Fortuna Maior	Fortuna Maior	Fortuna Minor
Puer	Albus	Puella	Rubeus
Rubeus	Puella	Albus	Puer
Acquisitio	Amissio	Amissio	Acquisitio
Laetitia	Caput Draconis	Tristitia	Cauda Draconis
Tristitia	Cauda Draconis	Laetitia	Caput Draconis
Carcer	Coniunctio	Carcer	Coniunctio
Caput Draconis	Laetitia	Cauda Draconis	Tristitia
Cauda Draconis	Tristitia	Caput Draconis	Laetitia

Looking at the table above, we can start to pick out certain patterns and “cycles” of operations that group certain figures together:

• A figure maintains its parity no matter the operation applied to it.  Thus, an odd figure will always result in another odd figure through any of the operations, and an even figure will always yield another even figure.
• A figure added to its inverse will always yield Via.
• A figure added to its reverse will always yield one of the liminal figures.
• A figure added to its converse will always yield another of the liminal figures, which will be the inverse of the sum of the original figure and its reverse.
• If the figure is odd, then its inversion, reversion, and conversion will all be unique figures, but each figure can become any of the others within a group of four odd figures through another operation.
• If the figure is even and liminal, then its reversion will be the same as the original figure, while its inversion and conversion will be the same figure and distinct from the original.
• If the figure is even and not liminal, then its inversion and reversion will be the same figure and distinct from the original, while its conversion will be the same as the original figure.

The odd figures are perhaps most interesting to analyze in their operation groups.  Note that the four figures that result from the operations of a single odd figure (identity, inversion, reversion, and conversion) all, at some point, transform into each other in a neverending cycle, and never transform in any way into an odd figure of the other cycle.  More than that, we can break down the eight odd figures into two groups which have these operational cycles, or “squadrons”, one consisting of Puer-Albus-Puella-Rubeus and the other of Laetitia-Caput Draconis-Cauda Draconis-Tristitia:

Note that the Puer squadron has only figures of Air (Puer and Rubeus) and Water (Puella and Albus), while the Laetitia squadron has only Fire (Laetitia and Cauda Draconis) and Earth (Tristitia and Caput Draconis), and that the converse of one odd figure yields another odd figure of the same element.  Coincidentally, it was this element-preserving property of conversion that led me to the Laetitia-Fire/Rubeus-Air correspondence, matching with the elemental system of JMG and breaking with older literature in these two figures.  More numerologically, also note how each squadron has two figures with seven points and two figures with five points; this was marked as somewhat important in how I allotted the figures to planetary arrangements before, but it could also be viewed under an elemental light here, too.  If each squadron has two figures of the pure elements (Albus and Rubeus in the Puer squadron, Laetitia and Tristitia in the Laetitia squadron), then the converse of each would be the harmonic opposite of the pure element according to their subelemental ruler::

• Laetitia (pure Fire) converts to/harmonizes with Cauda Draconis (primarily Fire, secondarily Earth)
• Rubeus (pure Air) converts to/harmonizes with Puer (primarily Air, secondarily Fire)
• Albus (pure Water) converts to/harmonizes with Puella (primarily Water, secondarily Fire)
• Tristitia (pure Earth) converts to/harmonizes with Caput Draconis (primarily Earth, secondarily Air)

On the other hand, now consider the even figures.  Unlike the odd figures, where the same “squadron scheme” applies for two groups, there are actually two such schemes for even figures, each scheme having one pair of figures.  For the liminal even figures, a figure’s inverse is the same as its converse, and its reverse is the original figure.  On the other hand, for the even entering/exiting even figures, a figure’s inverse is the same as it’s reverse, and its converse is the original figure:

Due to how the squadrons “collapse” from groups of four into groups of two for the even figures, the same elemental analysis of harmonization can’t be done for the even figures as we did above for the odd figures.  However, it’s also important to note that each element has four figures assigned to it, two of which are odd (as noted above) and two of which are even:

• Fire: Fortuna Minor (primarily Fire, secondarily Air), Amissio (primarily Fire, secondarily Water)
• Air: Coniunctio (primarily Air, secondarily Water), Acquisitio (primarily Air, secondarily Earth)
• Water: Via (primarily Water, secondarily Air), Populus (primarily Water, secondarily Earth)
• Earth: Carcer (primarily Earth, secondarily Fire), Fortuna Maior (primarily Earth, secondarily Water)

By looking at the inverse relationships of the even figures (which is also converse for liminal figures and reverse for non-liminal figures), we can also inspect their elemental relationships:

• Carcer (primarily Earth, secondarily Fire) inverts to Coniunctio (primarily Air, secondarily Water).  Both the primary and secondary elements of each figure are the opposite of the other, making these two figures a perfect dichotomy in every way.
• Via (primarily Water, secondarily Air) inverts to Populus (primarily Water, secondarily Earth).  Though both these figures share the same primary element, the secondary elements oppose each other.  In a sense, this is a more bland kind of opposition that Carcer and Coniunctio show.
• Acquisitio (primarily Air, secondarily Earth) inverts to Amissio (primarily Fire, secondarily Water).  Unlike Carcer and Coniunctio, and despite that these figures are reversions-inversions of each other, their elemental natures complement each other in both their primary and secondary rulers by heat, as Air and Fire (primary rulers) are both hot elements, and Earth and Water (secondary rulers) are both cold elements.
• Fortuna Maior (primarily Earth, secondarily Water) inverts to Fortuna Minor (primarily Fire, secondarily Air).  Similar to Acquisitio and Amissio, these two figures are reversions-inversions of each other, but their elemental natures complement each other in moisture, as Earth and Fire (primary rulers) are both dry elements, and Water and Air (secondary elements) are both moist elements).

Note that Carcer and Coniunctio along with Via and Populus (the liminal figures) show a more rigid opposition between them based on their inversion pairs than do Acquisitio and Amissio along with Fortuna Maior and Fortuna Minor (the non-liminal even figures).  Liminality, in this case, shows a forceful dichotomy in inversion, while actually possessing motion suggests completion of each other in some small way.  In this post I wrote on how the natures of the elements complement or “agree” each other based on the element of figure and field in the Shield Chart, these could be understood to say something like the following:

• Disagree (Carcer and Coniunctio, Via and Populus): Undoing and harm to the point of weakness and powerlessness, force and constriction from one into the other unwillingly.  This is more pronounced with Carcer and Coniunctio than it is Via and Populus, since Via and Populus still agree in the more important primary element, in which case this is more a complete undoing for strength rather than weakness, an expression of transformation into an unknown opposite rather than a forced march into a known but undesired state.
• Agree in heat (Acquisitio and Amissio): Completion and aid to both, but transformation in the process for complete change in goals and intent.
• Agree in moisture (Fortuna Maior and Fortuna Minor): Balance and stabilization that lead to stagnation and cessation of action, but with potential that must be unlocked or initiated.

Admittedly, this post took a lot longer to write than I anticipated, largely because although the mathematics behind the operations is pretty easy to understand, the actual meaning behind them is harder to nail down, and is largely a result of introspection and reflection on the figures involved in these operations.  For my own part, I don’t claim that my views are the be-all-end-all of these mathematical or structural relationships between the figures, and I would find this a topic positively begging for more research and meditation by the geomantic community as a whole, not just to flesh out more of the meanings and the relationships of the figures themselves, but also how they might be applied in divination as part of divinatory technique rather than just symbolism, like how I suggested using them for a mathematical/structural form of Shield Chart company.

So, what about you?  Do you think anything of these operation-based relationships of the figures?  Are there any insights you’d be willing to share regarding these operations and relationships?  Is there anything you can thread together from the observations I’ve made above that makes things flow better or fit together more nicely?  Feel free to share in the comments!