Geomantic Superfigures (Emblems) Revisited

A while back, I mentioned something about geomantic superfigures, which has recently become another focus of mine.  The De Geomanteia posts I’m doing are awesome for getting me to revisit old topics in geomancy, and I feel like the superfigures (I should a devise better name than that, perhaps “geomantic emblems”?) are something to be worked on a little more.  In a nutshell, geomantic superfigures are 19-row figures that, if you take any four consecutive rows, yields one of the 16 geomantic figures, and all 16 selections of four consecutive rows yield all sixteen geomantic figures exactly once.  They’re basically microcosms of the universe represented geomantically.  I suggest reading the old post above on the geomantic superfigures to get an idea of what they are.  There are 256 different geomantic superfigures, which is significant since 256 = 16².

I developed the idea for geomantic superfigures emblems a while ago as an exercise in combining a particular problem from computer science algorithms and DNA sequencing with geomancy, and though it seems useful, I never really developed a use for it, and so the idea and the list of 256 emblems just sat there gathering dust.  Recently on the Geomantic Campus mailing list, the geomancer prunesquallori picked up the idea and did some more analysis on it, and came up with a few awesome observations:

  1. Each geomantic emblem has 19 lines, but the last three lines must always be the same as the first three lines, i.e. line 1 = line 17, line 2 = line 18, line 3 = line 19).
  2. Because of the repetition of lines, we can reduce the size of the geomantic emblem to 16 lines without losing any information.  This is a far more appealing number than 19, geomantically speaking.
  3. One can rotate the 16-line emblems line by line, i.e. old line 1 becomes new line 16, old line 2 becoming new line 1, … old line 16 becoming new line 15.  This, when combined with the above, will yield another valid geomantic emblem.
  4. Each emblem can be rotated a total of 16 times, which produces a cycle.  Because there are 256 emblems, each of which can be rotated into or is rotated from another 16 emblems, we can reduce the 256 emblems to 16 if we ignore what position we begin at.

Consider the geomantic emblem from the last post, which we described as the binary string 0000100110101111000.  This figure contains, taking successive groups of four consecutive bits (0 represents a passive line and 1 an active line), the geomantic figures Populus, Tristitia, Albus, Rubeus, Carcer, Fortuna Maior, Coniunctio, Puer, Amissio, Acquisitio, Puella, Caput Draconis, Via, Cauda Draconis, Fortuna Minor, and Laetitia.  However, notice that the last three binary digits and the first three are the same; we can reduce the figure in size to the emblem 0000100110101111.  We can take for granted that the last three bits are going to be 000 since the first three are 000, so we leave them unwritten.  If we rotate the emblem by two bits to the right, we get 1100001001101011 in 16-bit form, or 1100001001101011110 in 19-bit form, which is another valid geomantic emblem.  This geomantic emblem contains, in order, the geomantic figures Fortuna Minor, Laetitia, Populus, Tristitia, Albus, Rubeus, Carcer, Fortuna Major, Conjunctio, Puer, Amissio, Acquisitio, Puella, Caput Draconis, Via, and Cauda Draconis.

Knowing that we can break down the emblems into 16-bit strings, or 16-line emblems, makes interpreting and using them a good bit easier.  For a simple elemental interpretation of the emblems, consider that a normal 4-row geomantic figure has one row for each of the four elements fire, air, water, and earth from top to bottom.  If we magnify the geomantic emblems into four groups of four rows, the first set of four rows can be assigned to fire as a whole, the second set to air, the third set to water, and the fourth set to earth.  Within these sets, we assign each individual row to an element as we would normally, so the first row of a set is assigned to fire, the second to air, and so forth.  By using this scheme, we can interpret the geomantic emblem as having whole geomantic figures representing how a particular element manifests.

Moreover, in using this scheme, we then can have lines that represent a particular element within a particular element.  The fire row of the fire quartet of lines would be “fire of fire”, or pure fire; the air row of the fire quartet would be “air of fire”, the interactive or mobile force of fire.  The system would continue so there’d be sixteen combinations: fire of fire, air of fire, water of fire, earth of fire, fire of air, air of air, and all the way down to earth of earth at the bottom.  By seeing how the interplay of elements works within the elements themselves, we can get a deeper understanding of the emblem they appear in.  Conversely, if we take the pure elemental lines out of the emblem and combine them, we can get a geomantic figure that can capture the essence of the emblem.  In this manner, there would be 16 emblems per geomantic figure.

As an example, consider the emblem 0000100110101111 from above.  Breaking it down into four groups of four lines, we have Populus (0000), Carcer (1001), Amissio (1010), and Via (1111).  Populus represents the force of fire in the emblem, Carcer the force of air, Amissio the force of water, and Via the force of earth.  If we took the pure elemental lines (fire of fire, line 1; air of air, line 6; water of water, line 11; earth of earth, line 16), we get the figure Tristitia (0001).   If we look at the emblem 1100001001101011, we have Fortuna Minor (1100) for the force of fire, Albus for the force of air (0010), Coniunctio for the force of water (0110), and Puella for the force of earth (1011).  Taking the pure elemental lines, we get the figure Puella (1011).  Fortuna Minor in this emblem would be especially powerful, since it’s a figure ruled by fire appearing as the force of fire in the emblem.

Also, consider that in having a 16-row emblem, we have the same number of rows required to develop a full geomantic chart, which can also help elaborate or expand on the nature of the sequence of figures that combine to form a geomantic emblem.  Given the emblem 1100001001101011, using Fortuna Minor, Albus, Coniunctio, and Puella for the four Mother figures, we find that the Judge is Coniunctio, the Sentence is Amissio, the Via Puncti doesn’t lead anywhere, the sum of the chart is 94, the Part of Fortune is in house 10, and the Part of Spirit is in house 2.  The rest of the chart I leave for the reader to derive, but interpreting this chart could yield even more information on a particular geomantic emblem that would help in unfolding its meaning or core.

Going back a bit, I mentioned above that there are 16 cycles of emblems, where if you rotate a particular emblem 15 times in succession you get another valid emblem.  Repeating this for all 16 emblems yields 256 total emblems.  Starting from an arbitrary point, the 16 16-bit cycles are:

  1. 1000010011010111
  2. 1000010011110101
  3. 1000010100110111
  4. 1000010100111101
  5. 1000010110011110
  6. 1000010110100111
  7. 1000010111100110
  8. 1000010111101001
  9. 1000011001011110
  10. 1000011010010111
  11. 1000011010111100
  12. 1000011011110010
  13. 1000011110010110
  14. 1000011110100101
  15. 1000011110101100
  16. 1000011110110010

Keep in mind that, because emblems can be cycled, you could all start these so that they start with the part of the emblem that goes 1111 and still have valid emblems.  It’s probably better to picture them as rings or bands instead of strings to emphasize their cyclic nature.  They’re presented above so that they start with 10001 for convenience.

Since there are 16 cycles, each of which can produce 16 emblems, I figured that there would be a way to link each cycle to one of the 16 geomantic figures, affording yet another way to classify the emblems but in a helpful non-elemental manner.  This would help in picking out specific emblems from the set of 256, such as by saying “the emblem in the Albus cycle beginning with Puella”.   However, doing this is tricky, since the cycles are independent of starting point, and so using arbitrary lines in the emblem is about as good as labeling two otherwise identical spheres A and B based on where you happened to touch them first.  A lot of the methods I first tried  made use of assigning a “start point” somehow, which defeated the whole purpose.  Other methods I tried allocated multiple emblems to a given figure but skipped over others entirely, which was also unhelpful.

However, in the end, a method of allocating the sixteen emblematic cycles to the sixteen figures of geomancy was found!  How, might you ask?  With a lot of work and structural analysis, that’s for damn sure.  Stay tuned to see what that method is and what the sixteen emblematic cycles are corresponded with.

Geomantic Superfigures

Back in my college days (all those many years ago, all two of them), I studied computer science, as did many of my friends.  Sometimes we’d solve theoretical or coding problems for fun over dinner in the dining hall, or make references to them in jokes or details about our lives.  It made sense, after all, especially when that’s all you’re studying for weeks on end most of the year.  One time, while walking back from class to our apartments, some of my friends and I were talking about strings.  Not strings as in balls of yarn, but ordered sequences of symbols.  For instance, my name, “polyphanes”, is a string composed of several letters in the Roman alphabet.  Other strings are “euphony”, “polygraph”, “cat”, and “polyandrous”; some could be things like “123456”, where the symbols are Arabic numerals; “10001010011”, binary numerals; or “TCAGAGAGCT”, DNA nucleotides.  I think you get the idea.  Strings can have many symbols (“polyphanes”), one symbol (like “p”), or even no symbols (the so-called empty string, or “”).

Strings are very useful in computer science, and are a very robust tool in the hands of someone who knows how to use them.  For instance, you can take a snippet out of a string to make another string, specifically substring: “p” and “poly” are substrings of “polyphanes”.  They both are strings in and of themselves, but appear within the larger string “polyphanes”.  Substrings are strings that are composed of part or all of another string that is at least as large as the substring.  The empty string, “”, is by definition a substring of any string, including itself.  Now, say you have several strings “polyphanes”, “polygraph”, and “polyandrous”.  Say you want to know whether there exists a substring common to all of them.  From above, we know that the empty string is part of all of them, but that’s trivial to find out and doesn’t help us any.  We know that “p” is part of all of them, as is “a”, but “g” only appears in “polygraph”.  Again, this is mostly easy to figure out, but say you want to know what the longest common substring is between them.  In this example, the longest common substring (LCSubS) is “poly”, with a length of four symbols.  Finding the LCSubS is nontrivial, and can be a pretty complex problem in its own right.  The LCSubS problem comes into heavy employ when dealing with DNA sequencing, when you have several strands of DNA and want to find the longest common sequence of nucleotides that they may have in common.  If there aren’t any, we return the empty string, “”, meaning that there is nothing common at all between them.

Let’s flip the idea on its head, now: say you have several strings “rec”, “cyc”, “l” and “ab”.  Is there a word (a string of Roman letters that makes sense in the English language) that makes use of all of these strings?  Sure: “recyclable”: it can be formed by combining the above strings in a particular way.  Viewed from the point of view of substrings, “recyclable” has “rec”, “cyc”, “l”, and “ab” as substrings.  In this case, “recyclable” is a string and the shorter ones are substrings; alternatively, “rec” and the rest are strings and “recyclable” is a superstring.  A superstring is a string that contains several other strings.  Makes sense, right?  Just as, above, we wanted to find the LCSubS, we can also try to find the shortest common superstring (SCSupS).  “Recyclability” is one possible superstring, but it’s not the shortest.

It was this SCSupS problem that my friends were talking about in particular, and I mostly took a backseat to this conversation (I wasn’t in their class that was discussing this problem).  However, it got me to think about one of my hobbies: geomancy.  In geomancy, the geomantic figures like Puer or Albus can be seen as strings of one or two dots, or as I prefer, binary numerals.  If we let single dots (active elements) be represented by the binary numeral 1 and double dots (passive elements) by the binary numeral 0, and order the symbols from left to right as we would a geomantic figure from top to bottom, we get the following strings to represent the geomantic figures:

Figure Binary Representation
Populus 0000
Via 1111
Puer 1101
Albus 0010
Puella 1011
Rubeus 0100
Laetitia 1000
Caput Draconis 0111
Tristitia 0001
Cauda Draconis 1110
Amissio 1010
Acquisitio 0101
Carcer 1001
Coniunctio 0110
Fortuna Maior 0011
Fortuna Minor 1100

I prefer these binary representations, myself, when I need to illustrate the structure of geomantic figures in a text-only medium.

Anyway, you have these four-bit strings of binary numbers that correspond to the sixteen geomantic figures.  They’re each strings.  Now, normally in geomancy we’re concerned with adding figures together (such that Puer and Albus make Via, or 1101 + 0010 = 1111).  But in the context of strings, why not try applying this idea of SCSupS to the geomantic figures?  For instance, there are works of geomancy that use five-lined figures, or even six-lined ones.  Heck, the I Ching, a kind of Chinese geomancy, with its 64 hexagrams can be considered in a way similar to this.  Let’s take Hexagram 22, 賁 (Adorning), which looks like the following:

We can consider it to have the binary structure of 100101 (using a 1 to represent a solid Yang line and a 0 to represent a broken Yin line).  If we want to take a substring of length 4 from this string, we have four choices: 1001 (characters 1 through 4), 0010 (characters 2 through 5), or 0101 (characters 3 through 6).  In this case, we could say that the strings of length 4 are substrings of the string of length 6, or we could say that the strings of length 4 have the string of length 6 as their superstring.  The overlap provided by the figures helps 100101 to be the SCSupS for these strings; we could put them end to end, like 100100100101 (read: 1001-0010-0101), but that’s really long with a length of 12; we could overlap a few of them to make 00101001 (read: 0010-0101-1001, with overlap between the 010 of the first two figures and the 1 of the last two figures), but that has length 8.  100101 is the shortest common superstring with a length of 4, because the overlap between the substrings is at their max (three symbols between the first two and the last two figures).  It’s kinda nifty.

So, if you can overlap some of the figures to form a superstring, why not try to make a superstring that contains all 16 geomantic figures to make a geomantic superfigure?  This took some doing, and I eventually wrote a program to cycle through and figure out the minimum length of the superstrings that do this, but it turns out that there are 256 possible geomantic superfigures of length 19: a “geomantic figure” of a sort that has 19 rows of dots, with each substring of 4 rows being one of the sixteen geomantic figures.  Why is 19 the minimum length for a geomantic superfigure?  The math is easy: from the example above using Hexagram 22 which has a length of 6 and has three substrings of length 4, where each figure overlaps with the next using three of its rows, we can say that “for a geomantic superfigure, the shortest possible length is the number of figures contained in it n + 3″.  Since there are 16 figures, a geomantic superfigure must be that plus three, or 19.

Consider an example:

0000 0001 0010 0100 1001 0011 0110 1101 1010 0101 1011 0111 1111 1110 1100 1000
Populus, Tristitia, Albus, Rubeus, Carcer, Fortuna Maior, Coniunctio, Puer, Amissio, Acquisitio, Puella, Caput Draconis, Via, Cauda Draconis, Fortuna Minor, Laetitia

In this case, we overlap Populus with Tristitia with Populus’ lower three lines and Tristitia’s upper three lines (0000 : 0001 = 00001), then we repeat the process with the next figure (00001 : 0010 = 000010), and so on.  We end up with the string 0000100110101111000 of length 19 which, for every possible substring of length 4, contains all 16 geomantic figures.  In other forms of representation, such as dot or line notation, we can get the following visual forms:


The third style, the intersecting lines, is interesting, and brings to mind bindrunes (special “master” runes that are composed of other runes) or supersigils that combine other sigils.  The central idea is the same: combine several shapes, strings, or figures to form a more cohesive larger one.  Here, with this geomantic superfigure, it composes the essence of all the geomantic figures in a particular order (Populus to Tristitia to Albus to Rubeus…).  That order of the figures, how different states flow from one to another, is encapsulated in a single glyph.  Further, if you connect the ends of the Laetitia on the bottom to the top of the Populus, you have a neverending loop, a cycle of manifestation or transformation in the figures.

As I mentioned before, there are 256 of these geomantic superfigures, or 16² superfigures.  Some can be looped back onto itself, and some can’t, but each has the same 16 figures as its components in different orders.  Each order and superfigure has meaning, like a kind of geomantic universe of its own, and can be employed in some kinds of geomantic sigil magic.  This isn’t something I’ve actively used or dealt with in my own work yet, but it’s interesting to note, and if nothing else, is an interesting academic and mystical exercise.  However, I feel like this is too interesting to not put up for aspiring geomantically-minded mages to never have heard about.  If you want the complete list of 256 geomantic superfigures, let me know and I’ll send you a file listing them.