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.

Solar Grammatomantic Calendar

So, based on that last post where I discussed possibilities of forming a divinatory cycle of days based on the grammatomantic meanings of the Greek letters, I came up with my first draft of a kind of grammatomantic calendar, based on a simple cycle of the letters.  In many ways, this functions much like the tzolk’in calendar of the Maya, but with a little bit of their haab’ thrown in, too.  Essentially, I’ve created a cyclical calendar capable of dating many years into the future or, with some modifications, to the past.  For simplicity, I use the Greek alphabet itself as the core cycle used for this calendar, which is tied to the spring equinox every year.  In effect, I’ve developed a solar grammatomantic calendar, or SGC.  While an interesting little system of noting dates and times in a really obscure fashion, it is at heart a divinatory tool expanding on the methods of grammatomancy applied to a general flow of time, noting how a particular person or event might be affected by the forces at work in the cosmos at that particular time.

So, let’s set some rules and definitions to calculate dates and times in the SGC:

• Letter-day: Duration of time starting at a particular sunrise and the next sunrise. The first value in a cycle of 24 letter-days is Α, then cycles around as expected
• Letter-month: 24 consecutive letter-days. The first value in a cycle of 15 letter-months dependent on letter-year, and cycle around as expected:
• Α if letter-year is Α, Ι, or  Ρ
• Π if letter-year is Β, Κ, or Σ
• Η if letter-year is Γ, Λ, or Τ
• Χ if letter-year is Δ, Μ, or Υ
• Ν if letter-year is Ε, Ν, or Φ
• Δ if letter-year is Ζ, Ξ, or Χ
• Τ if letter-year is Η, Ο, or Ψ
• Κ if letter-year is Θ, Π, or Ω
• Letter-year: 15 consecutive letter-months, or 360 consecutive letter-days plus some number of intercalary days. The first value in a cycle of 24 letter-years is Α, then cycles around as expected. Begins from the first sunrise after or coinciding with the spring equinox
• Intercalary day: Days used to align the cycle of 15 letter-months with the solar year. Not associated with any particular letter, nor are they considered letter-days or belonging to a letter-month. Placed at the end of the letter-year, after the last day of the 15th month of the current year but before the first day of the 1st month of the next year. There are as many intercalary days as needed to fill the gap between the number of letter-days and the number of days in the solar year.
• Letter-age: 24 letter-years, 360 letter-months, or 8640 letter-days plus some number of intercalary days. The first value in a cycle of 24 letter-great-years is Α, then cycles around as expected
• Letter-era: 24 letter-ages, 576 letter-years, 8640 letter-months, or 207360 letter-days plus some number of intercalary days. The first value in a cycle of 24 letter-ages is Α, then cycles around as expected.  At most 13824 years can denoted using only 24 values for the letter-era.

A table for converting one of the larger units of letter-dating into smaller ones shows the relationships between the units.  Note that asterisks in the letter-day column indicate that intercalary days will cause this number to increase as the number of letter-years increases.

 Letter-day Letter-month Letter-year Letter-age Letter-era Letter-day 1 Letter-month 24 1 Letter-year 360* 15 1 Letter-age 8640* 360 24 1 Letter-era 207360* 8640 576 24 1

Of course, even though I’ve listed only five place values for a SGC date, we’d end up with a weird kind of Y2K-esque problem once we finish the ultimate letter-era Ω completely, approximately 13824 years after the first possible date.  Although it’s unlikely to be needed, further spans of time may be indicated by adding larger units, such as a letter-eon which is equivalent to 24 letter-eras; 24 letter-eons would be equivalent to 576 letter-eras, 13824 letter-ages, or 7962624 letter-years.  This easily reaches up into geological or cosmological timeframes, but could be useful for indicating distant, mythological, or astronomical/astrological phenomena.

As noted above, all the cycles have 24 values, each lettered according to the Greek alphabet starting at Α and ending with Ω, with the exception of the letter-months.  Instead, the cycle of letter-months within a letter-year is dependent on the value of the letter-year itself.  Though this seems arbitrary, this is to preserve the cycle caused by there being 15 letter-months within a letter-year.  For instance, the first letter-month of the overall cycle of letter-months is Α, the first letter in the Greek alphabet; the last letter-month of the same year is Ο, the 15th letter in the Greek alphabet.  The second letter-year continues the pattern of assigning letters to the letter-months: since Ο was the previous letter used, Π is the letter assigned to the first letter-month of the second letter-year.  Continuing this cycle, the first letter-month of the third letter-year is assigned with Η, the first letter-month of the fourth letter-year is assigned with Χ, and so on until the last letter-month of the last letter-year is given to Ω, after which the cycle begins anew with Α.  This produces a cycle of eight letter-years; since there are 24 letter-years in a letter-age, this cycle repeats three times.  By taking the remainder of dividing the letter-year ordinal value by eight (substituting 8 for a result of 0), the table below shows the letters associated with the letter-months for a given letter-year in the cycle.

 Year 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 Α, Ι, Ρ Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο 2 Β, Κ, Σ Π Ρ Σ Τ Υ Φ Χ Ψ Ω Α Β Γ Δ Ε Ζ 3 Γ, Λ, Τ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ 4 Δ, Μ, Υ Χ Ψ Ω Α Β Γ Δ Ε Ζ Η Θ Ι Κ Λ Μ 5 Ε, Ν, Φ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω Α Β Γ 6 Ζ, Ξ, Χ Δ Ε Ζ Η Θ Ι Κ Λ Μ Ν Ξ Ο Π Ρ Σ 7 Η, Ο, Ψ Τ Υ Φ Χ Ψ Ω Α Β Γ Δ Ε Ζ Η Θ Ι 8 Θ, Π, Ω Κ Λ Μ Ν Ξ Ο Π Ρ Σ Τ Υ Φ Χ Ψ Ω

As for the epoch, or the reference date from which the letter-calendar is calculated, I’ve settled on April 3, 1322 BC as the first date in this letter-calender system.  My readers will likely be utterly confused as to why I chose such a distant year and date.  Since I’m a fan of ancient Greek history and civilization, I decided to look back as far as I reliably could, and recalled dimly somewhere in my memory that archaeoastronomers had calculated a date in the Trojan War based on mentions of eclipses in book 17 of the Iliad as well as Hittite and other archaeological records.  As it turns out, such an eclipse happened on November 6, 1312 BC at around 12:35 p.m.  Since the Trojan War took about ten years according to the myths, I wanted to set the epoch date to the day after the spring equinox ten years before the year in which this eclipse occurred.  Looking at an ephemeris for the year 1322 BC, we know that the spring equinox (Sun ingress Aries) occurred sometime on April 2, 1322 BC, making the following dawn of April 3, 1322 BC the start of the first official day of the SGC.  Negative dates, or dates that come before April 3, 1322 BC would not be possible in this system, making the first day “day zero” and anything before prehistory or mythical.  If reverse calculations were desired, the rules to convert dates could be adapted for this, with some kind of inversion applied to the notation (writing it upside down, for instance).

To mark a given date using the SGC, let’s use the notation A.B.C.D.E, where A indicates the letter-era, B indicates the letter-age, C indicates the letter-year, D indicates the letter-month, and E indicates the letter-day.  Each of these could be represented equally well in Greek letters (Α.Ρ.Ψ.Χ.Ε) as they could in Arabic numerals (1.17.23.22.5), so long as one uses the ordinal placement of the letters in the Greek alphabet in mind as well as the funky letter-month 8-year cycle given above.  For intercalary days which don’t belong to any letter-month, a dash, dot, or zero is used for the letter-month position and a Greek letter to indicate the intercalary day.  So, for the fourth intercalary day on the letter-era Α, letter-age Ρ, and letter-year Ψ, we might use the notation Α.Ρ.Ψ.–.Δ with the dash, Α.Ρ.Ψ.•.Δ with the dot, or Α.Ρ.Ψ.0.Δ with the zero.  Arabic numeral representations of the intercalary “month” should use the numeral zero.

Now that we have the units defined, the cycles understood, the epoch proclaimed, and the notation set up, it’s time to begin our rules for converting dates from this letter-calendar to Gregorian dates and back.  Let’s use E, A, Y, M, and D to indicate the ordinal values of the letter-era, letter-age, letter-year, letter-month, and letter-day in these conversions; in other words, these variables represent the Arabic numerals associated with the place values, bearing in mind the funky ordinal values associated with the Greek letters for the letter-month.

To convert a Gregorian calendar date to a letter-calendar date:

1. Find the number of years elapsed (J) between the Gregorian calendar year (GY) and the epoch year (EY).  If the Gregorian calendar date falls on or after the first sunrise after or coinciding with the spring equinox in its year, J = GY − EY.  If the Gregorian calendar date falls before the first sunrise after or coinciding with the spring equinox in its year, J = GY − EY − 1.
2. Divide J by 576 and take the whole part to find the number of letter-eras that have passed (JW), and take the fractional part to find how much other time has passed (JF).
3. Calculate the letter-era: E = JW + 1.  E should be a whole number between 1 and 24.  Assign E the Greek letter according to its ordinal value.
4. Multiply JF by 24 and take the whole part to find the number of letter-ages that have passed (AW), and take the fractional part to find how much other time has passed (AF).
5. Calculate the letter-age: A = AW + 1.  A should be a whole number between 1 and 24.  Assign A the Greek letter according to its ordinal value.
6. Multiply AF by 24 and take the whole part to find the number of letter-years that have passed (YW), and take the fractional part to find how much other time has passed (YF).
7. Calculate the letter-year: Y = YW + 1.  Y should be a whole number between 1 and 24.  Assign Y the Greek letter according to its ordinal value.
8. Find the number of days that have elapsed (T) between the Gregorian calendar date (GD) and the most recent spring equinox date (ED).
9. If T is greater than 360, this is an intercalary day.  Set the letter-month M = 0 or missing.  Calculate the intercalary day D = T −  D.
10. Otherwise, if T is less than or equal to 360, this is a letter-day.
1. Divide T by 24 and take the whole part to find the number of letter-months that have elapsed (TM), and the fractional part to find the number of days that have elapsed (TD).
2. Calculate the letter-month:  M = TM.  M should be a whole number between 1 and 15.    Assign M the Greek letter according to its ordinal value according to the eight-year cycle above based on Y.
3. Calculate the letter-day: D = TD  × 24.  D should be a whole number between 1 and 24.    Assign D the Greek letter according to its ordinal value.

To convert a letter-calendar date to a Gregorian calendar date:

1. Sum together the year-based units multiplied by their coefficients to get the number of years elapsed since the epoch: S = (576 × E) + (24 × A) + Y
2. If the date refers to an intercalary period, sum the total number of letter-days plus the intercalary days: Z = 360 + D
3. If the date refers to a non-intercalary period, sum the count of letter-days plus the number of letter-months multiplied by the number of letter-days in each month: Z = D + (24 × M)
4. Add the number of elapsed years S to the epoch year to find the year of the Gregorian calendar date.
5. Add the number of elapsed days Z to the date of the first dawn after or coinciding with the spring equinox of the Gregorian calendar year to find the month and day of the Gregorian calendar date.

Since we’ve already done this much work to clarify letter-days, we can focus our attention on dividing up individual days into smaller units.  I don’t think it’ll be necessary to get into the magnitude (or lack thereof) of seconds, but having letter-hours might not be a bad idea.  Since there 24 letters, we can create 24 letter-hours for each day.  The process for this would be nearly the same as calculating planetary hours.  Let’s define a letter-hour to equal either 1/12 of the time between sunrise and sunset of the current letter-day or 1/12 of the time between sunset of the current letter-day and sunrise of the following letter-day, whichever period the letter-hour is found within.  Each letter-hour is assigned to one of the 24 letters in the Greek alphabet, in the order of the Greek alphabet starting with Α.  We might augment our notation of date to also include time using the notation A.B.C.D.E:F, where F indicates the letter-hour.

To convert a modern time to a letter-hour or vice versa for a given date and location:

1. Find the time of sunrise and sunset for the given date and location, and the time of sunrise for the day following the given date and location.
2. Divide the total length of time between sunrise and sunset by 12 to find the length of the diurnal hour (DH).
3. Establish the divisions of the diurnal hours starting at sunrise according to DH, assigning them the letter values Α through Μ or number values 1 through 12.
4. Establish the divisions of the nocturnal hours starting at sunset according to NH, assigning them the letter values Ν through Ω or number values 13 through 24.
5. Locate the time given among the letter-hours to convert the modern time to a letter hour, or establish the time limits on the given letter-hour to find an approximate modern time.

So, examples!  Let’s take September 1, 2013 at 10:35 a.m. for Washington, DC, USA and convert it into SGC date:time notation.

• Letter-day and letter-month: on this year, the spring equinox occurred on March 20, 2013 after dawn; thus, the first day of this year began on March 21, 2013.  There are 166 days between these two dates.  166 ÷ 24 = 6.91666…, indicating that the letter-month is 6 and the letter-day is 0.91666… × 24 = 22, or Χ.
• Letter-era, letter-age, and letter-year: between 2013 AD and 1322 BC, there are 3334 years.  3334 ÷ 576 = 5.78819444…, indicating that the letter-era is 6 (5 + 1).  0.78819444… × 24 = 18.91666…, indicating that the letter-age is 19 (18 + 1).  0.91666… × 24 = 22, indicating that the letter-year is 23 (22 + 1).
• Letter-hour: on this day, sunrise was at 6:37 a.m. and sunset at 7:38 p.m., with the next sunrise at 6:38 a.m.  The length of a diurnal hour in this day was about 65 minutes long, and a nocturnal hour was about 55 minutes long.  10:35 a.m. falls during unequal hour 4.
• Notation: the full Arabic numeral notation for this date is 6.19.23.6.22:4.  The full Greek letter notation for this date is Ζ.Τ.Ψ.Ω.Χ:Δ.  The letter-month is Ω, not Ζ as might be expected for the ordinal value of 6, due to the letter-year being Ψ (see the chart above).

In the opposite way, let’s convert the SGC date Η.Ρ.Λ.Ο.Υ:Α for Washington, DC, USA to Gregorian notation.

• Conversion to Arabic numerals: The date Η.Ρ.Λ.Ο.Υ:Α resolves to 7.17.11.9.20:1, using the table above to resolve the letter-year.letter-month combination Λ.Ο to 11.9.  Since the letter-month is not blank or missing, this is not an intercalary date.
• Sum the years: There have been (576 × 7) + (24 × 17) + 11 = 4451 years since the epoch date.
• Find the year and spring equinox: 4451 years elapsed from the epoch year 1322 BC refers to the year 3130 AD.  The spring equinox occurred at night after March 20 that year, so the first day of the SGC year would be on March 21.
• Sum the days: There have been (9 × 24) + 20 = 236 days since the year’s first dawn after or coinciding with the spring equinox.
• Find the day: 236 days after March 21, 3130 AD leads to November 12, 3130 AD.
• Find the time: The letter-hour Α indicates the first unequal hour of the day, sometime just after dawn.  Sunrise for this day in Washington, DC, USA occurs at 6:47 a.m., and sunset at 4:56 p.m.; an unequal diurnal hour here would be about 49 minutes long, so the letter-hour Α indicates a time between 6:47 a.m. and 7:38 a.m.

Well, this was all well and good, and despite the complexity only took a day to hash out all the major parts of forming a new calendar system from scratch.  However, while this was a fun exercise in computus of a sort, this doesn’t actually say much about why it was made to begin with: divination using the flow of time itself!  Since I’ve ranted on long enough about the minutiae of date conversions, let’s leave that for next time when we start putting the SGC in practice and making use of its mechanisms for divination, as well as seeing how it lines up with other solar or theophanic phenomena.