"History of Ciphers" (part 5) - Mathematical Ciphers

    Dear Reader,

    In the fourth part of the "History of Ciphers" (which was itself divided into 4 parts: 4a, 4b, 4c and 4d) we talked about how the Thelemic ciphers can be used to solve some riddles in the Holy Books of Thelema, particularly Liber AL vel Legis — The Book of the Law.

    In this new fifth part we will talk about a different kind of ciphers. Contrary to all the ciphers we talked about before, these ciphers don't have any specific context to be used on* — instead, they are simply based on mathematical sequences of numbers, and as such I will be calling them the Mathematical ciphers. 

* A word of caution is needed at this point. When I say that these ciphers "don't have a context to be used on", what I mean is that these are simply mathematical ciphers — so technically speaking, they aren't Baconian, Thelemic, or Masonic ciphers, but they do have a "right" context to be used on if you are using them to decode something that was willingly encoded with them. So, in a way, context is still of utmost importance even if you work with these ciphers — as with all ciphers of Gematria.

History of Ciphers

by Luís Gonçalves

Part 5: Mathematical ciphers

    There are many kinds of mathematical ciphers, which in this paper I will divide into three categories:

 

     1. The ones based on lists of specific types of numbers, of which I will highlight the Prime numbers sequence, which is perhaps the most well-known among these. I will also talk about an experimental cipher known as the Fibonacci cipher, which seems to have been devised very recently (i.e. already in the 21ˢᵗ century, possibly during the late 2010's).

    2. The ones based on geometric/polygonal sequences, among which the most well-known are undoubtedly the Trigonal (or Triangular) numbers sequence, and the Tetragonal (or Square) numbers. As far as I can tell, these are probably the oldest of the mathematical ciphers. In my research about this kind of ciphers, I was able to trace them at least to the late 17ᵗʰ century, to Johann Henning's "Cabbalologia" (1683), but it's possible that there are still older sources for these ciphers. Henning's work about ciphers hides another nice surprise, however... and I will also talk about it in this paper.

    3. The ones based on the lists of divisors of certain numbers. These are not very well-known or used in Gematria, even though I think that they can be of great value — besides, there are certain very curious factors about them that are worthy of note, and I will explain all of that to my Readers.

 

    The way how these sequences are turned into Gematria ciphers is pretty straightforward: the first number in the mathematical sequence is assigned to letter "A", the following number to letter "B", and so on and so on, always following the alphabetical order of letters together with the respective numerical sequence.

    With this explanation in mind, let us then explore these different types of ciphers.

Ciphers based on lists of specific types of numbers

    Of these ciphers I will only highlight, in this section, the Primes and Fibonacci ciphers:

 

Cipher table from GEMATRO - Gematria Calculator.

     The logic behind this cipher is pretty sound. As number 2 is by definition the first prime number, it is assigned to letter "A", the first letter of the alphabet; next, the second prime number is 3 which is assigned to letter "B"; and the sequence continues in the same way, so that the 26ᵗʰ letter of the English alphabet is assigned the 26ᵗʰ prime number, 101.

 

    In this cipher, "Prime Number" sums 421, which is a prime number:

 

 

 

     Further facts about this cipher:

 

 

     To be honest with my Readers, at this time of writing I have no actual informations about (1) when was this cipher first used, (2) who devised it, or (3) any historical use of it, besides its presence in some modern calculators (namely the "Gematrinator").

 

     So I'm not really sure of how I could further elaborate on the origins and history of this cipher. As it is based on a natural sequence of numbers (the prime numbers) it is, however, perfectly logical, and I don't see any inconvenience in using it for the purposes of Gematria. Anyway, I always advise my Readers to work with ciphers in a conscious manner, and that not all ciphers are "right" if you want to decode something with them. As always, context is king.

 

 

* * *

 

 

    Next comes the Fibonacci cipher:

 

Cipher table from GEMATRO - Gematria Calculator.

     The reasoning behind this cipher is this: for the first half of the alphabet we follow the normal Fibonacci sequence, from the 1ˢᵗ Fibonacci number (1) to the 13ᵗʰ (233), while for the second half of the alphabet the order is reversed. The reason for this reversal is that if we followed the Fibonacci sequence until the 26ᵗʰ Fibonacci number we would be left with a table with very large numbers, since the 26ᵗʰ Fibonacci number is 121393:

 

Cipher table from an offline derivation of Gematrinator. Click to enlarge.

     In order to avoid exceedingly high numbers, the choice was made, then, to reverse the Fibonacci sequence for the second half of the alphabet.

 

     Using the Fibonacci cipher, the numerical sum of "Filius Bonacci" (from which came the name "Fibonacci") is 666. This is very curious, to say the least.

 

  

    And even more curious is that the number 666, as written in Revelation 13:18 in the text of the Biblia Sacra Vulgata (the Latin Bible), "Sexcenti Sexaginta Sex", also sums 666:

(click to enlarge)

     Further facts about this cipher:

 

 

     At this time of writing I'm not entirely aware of the circumstances behind the creation of this cipher. I do know, however, that two persons were instrumental in making this cipher come to life:

 

    — The first person is Discord user Barni Yamum, from Germany, whose Fibonacci cipher was added to an offline derivation of the Gematrinator, in which it was called Barni Yamum's Fibonacci Cipher. From some past conversation I've read on Discord, Barni seems to have devised this cipher on his own.

 

Cipher table from an offline derivation of Gematrinator. Click to enlarge.

 

     — The second person is YouTube user Lambda115, who shared the first video about the Fibonacci cipher and was instrumental in making this cipher more widely known. As far as I know, it was because of Lambda115's work that the Fibonacci cipher was added to the Gematrinator.

 

    From what I was able investigate when talking with both Barni Yamum and Lambda115, they both told me that they devised this cipher. While at one side it is true that Lambda115 has explored this cipher a lot on his YouTube videos, and he was technically the first person who ever published a video talking about this specific cipher, it is also true that from some conversations with Barni Yamum on Discord, it became clear to me that Barni also devised this cipher on his own, without knowing about Lambda115's similar idea.

 

    I strongly believe that both individuals may have had the same idea and came up with the same cipher. In fact, that happened to me as well in April 2021, when I devised an experimental system of Gematria inspired on the alphanumeric Base-36 numeration (0-Z), and just some days later I came to find out that the cipher that I thought I had devised, had already been used for many years before — at least since 1999 — under the name of "Alphanumeric Qabbala". So if I say that "I devised" a system of Gematria based on Base-36 numeration, I'm not running away from the truth — indeed I did devise it, since I genuinely didn't know that it existed before — but as later I came to understand, I was simply not the first person who did it. I believe that the same happened in this case, where both individuals came up with the same cipher, without knowing each other's work, and both being honest about having devised it. I don't have any reasons not to trust the words of these two individuals.

    If this assumption of mine is wrong, however, I humbly ask my Readers who might have more to say about this, to please share all the informations you would consider to be relevant in this case. My purpose in this explanation, however, is not to know "who's right" and "who's wrong", but only to share what can be known for sure about the history of this still young cipher.

 

    So, the next ciphers in line are the...

 

 

Ciphers based on Polygonal sequences

 

 

    These are by far, for me, the most interesting of the Mathematical ciphers. I honestly thought until some time ago that, of these ciphers, only the Trigonal and Squares ciphers were used, and they were created very recently (I mean, in the 20ᵗʰ century tops). However, I was wrong in both of these assumptions, since:

 

    — There were other Polygonal ciphers that were used in the past, including (besides the Trigonal and Squares ciphers) the Pentagonal, Hexagonal, Heptagonal, Octagonal, Enneagonal, and Decagonal ciphers (!!!)

 

    — I was able to find a source for all these ciphers that came from the late 17ᵗʰ century: more precisely from 1683, the year when Johann Henning's "Cabbalologia" was published (link to online text here).

 

     All these "new" Polygonal ciphers can be found in Henning's "Cabbalologia", even though in this case the ciphers are adapted to the Elizabethan English alphabet with 24 letters, in which I=J and U=V. As these ciphers are historically relevant, and not just "modern inventions" as I had previously thought, I will present the first two (Trigonal & Squares) using the following model:

    — A link to the website of the OEIS (On-Line Encyclopedia of Integer Sequences) for the corresponding Polygonal sequence;

   

     — The original cipher from "Cabbalologia" for the Elizabethan English alphabet;

     

    — The corresponding cipher for the Modern English alphabet;

    — Some correspondences for these ciphers (Trigonal and Tetragonal). Please be aware that I never duly explored these ciphers except occasionally, so I'll do my best to give some "food for thought" for these ciphers.

 

     For the remaining ciphers, and in order to avoid making this a boresome presentation, I will simply present the polygonal ciphers that are adapted to the modern English alphabet. Their Elizabethan counterpart can be simply reproduced from these tables, while always remembering that in the Elizabethan English alphabet the letters "I" and "J" share the same value (since they were considered to be different shapes of the same letter), the same happening with "U" and "V".

 

    So, in order, we'll begin with the Trigonal or Triangular cipher.

 

OEIS: https://oeis.org/A000217 (Triangular numbers)

 

Trigonal Cabbala for the Elizabethan English alphabet.

 

Trigonal Gematria for the Modern English alphabet.

    This Trigonal or Triangular sequence is also very curious. The way it works may not be very clear at first sight, but it is actually quite logical and sound. In order to find the Nᵗʰ trigonal number, you just add all the natural numbers from 1 to N. For example, the first letter "A" is assigned the first trigonal number, which is 1; next, the letter "B" is assigned the second trigonal number, which is 1+2 = 3; then, "C" is assigned the third trigonal number, which is 1+2+3 = 6, etc, and the sequence continues until the 26ᵗʰ letter "Z" which is assigned the 26ᵗʰ trigonal number, or 1+2+3+...+25+26 = 351.

    In the modern Trigonal cipher, "Maths" sums 528, which is a triangular number itself:

    And in the Trigonal cipher for the Elizabethan English alphabet, "Trigonal Cabbala" (as it is named in "Cabbalologia") shares the same value as "Triangles":

* * *

 

     Next comes the Squares (or Tetragonal) cipher:

   

OEIS: https://oeis.org/A000290 (The squares)

Tetragonal Cabbala for the Elizabethan English alphabet.

Tetragonal Gematria for the Modern English alphabet.

 

    The logic behind this cipher is also easy to grasp. Each Nᵗʰ Square number is equal to N×N. Thus, the first letter "A" is assigned the value 1×1 = 1, the second letter "B" is assigned the value 2×2 = 4, etc, until the 26ᵗʰ letter "Z" which is assigned the value 26×26 = 676.

    In the modern Tetragonal cipher, "Coding" sums 576, which is itself a square number:

    And in the modern Tetragonal cipher, "A Square" sums 1296, which is the square of 36:

    Following these two ciphers, which are by far the most well-known, I'll simply list the remaining polygonal ciphers as listed in Johann Henning's "Cabbalologia". Be aware however, dear Reader, that these are the ciphers for the modern English alphabet and they won't match with the ciphers as listed in "Cabbalologia", since in this book the ciphers were adapted to the Elizabethan English alphabet. In order to use them with that alphabet (which can depend on which historical context you're working on), you just have to remember that in that alphabet, the letters "I" and "J" were considered to be different shapes of the same letter, similarly to what happened with "U" and "V". So for the Elizabethan English alphabet, the numerical correspondences for the letters "I" and "J" will be the same, and likewise with "U" and "V".

* * *

http://oeis.org/A000326

 

http://oeis.org/A000384

 

http://oeis.org/A000566

 

http://oeis.org/A000567

 

http://oeis.org/A001106

 

http://oeis.org/A001107

 

* * *

    What I found most interesting when I started exploring these ciphers was not noticing which numerical matches between words they supplied; instead, my first instinct when I found these ciphers was to which numbers made part of these polygonal sequences and their order, and I found some curiosities.

 

    One of these coincidences has to do with the year and date when English Qaballa (the "ALW" Cipher) was discovered. As I explained in my first text about the Thelemic ciphers, EQ was discovered by James Lees on November 26ᵗʰ, 1976. Not only was this date highly symbolic, due to its specific connection to the numbers 11 and 26 (standing for the number of letters of the English alphabet, and the 11-letter cycle associated with the "ALW" cipher), but the year was equally relevant, since 1976 is the 26ᵗʰ octagonal number. And not only this, but 1976 also equals 26x76, again confirming a connection between the English alphabet (with 26 letters), the year ('76), and even pointing to the verse in Liber AL vel Legis which was being decoded when this cipher was discovered (verse II:76 — meaning November '76!).

    Another such example is 666, which in Western (i.e. Christian) culture is known as the "Number of the Beast", based on the biblical text of Revelation 13:18. This number is the 36ᵗʰ trigonal number, while 36 is the 8ᵗʰ trigonal number. So there's a clear polygonal connection between these three numbers: 8, 36 and 666. But there's a manuscript version of the book of Revelation where the mentioned number is not 666 but instead 616. And is this number equally relevant from a polygonal or mathematical perspective? Well, in my opinion it isn't as "interesting" as 666, but nevertheless, 616 is both the 16ᵗʰ heptagonal number, and it is also the 9ᵗʰ 13-gonal number. 

    Some polygonal series also contain some interesting numbers. The number 287, for example, which I explored (though not exhaustively) in my text about the Baconian ciphers and which is deeply connected to some mysteries related to Sir Francis Bacon, William Shakespeare and Rosicrucianism, is the 14ᵗʰ pentagonal number, while 1717 (the year when the Great Lodge of England was founded) is the 34ᵗʰ pentagonal. And of course, 34 equals 17+17.

    I've prepared, for quick reference, a spreadsheet with all polygonal series from the "simple" sequence to the 18-gonal numbers, up to the 52nd number in each series. Technically speaking all of these sequences could be used as ciphers, even though the ones which are more frequently used are, as explained before, the "Trigonal" and "Squares" ciphers.

Polygonal sequences — click to enlarge.

    In the community of practitioners of Gematria who use it to "decode the news", as it were, I've seen some people showing their decodes about a possible UFO-related major event in 2022. As sometimes I also like to play the same 'game', I couldn't end this section if I didn't show my own ultimate 'decode' with the polygonal ciphers:

    So there you have it! Haha 😜

* * *

A previously unknown "Extended" alternative cipher

for the English Alphabet, from "Cabbalologia"

 

 

     Henning's "Cabbalologia" is interesting not only because of the polygonal ciphers that it contains, but also because of an alternative "Extended" cipher that can be used for the English alphabet. As far as I know there aren't many people using this cipher, so I believe that this will be a nice surprise, at least, for some of my Readers.

 

    This is the "Alternative Extended" cipher as it appears in "Cabbalologia", which in this case is adapted to the 24-lettered Elizabethan English alphabet:

 

     Adapting this cipher to the modern English alphabet with 26 letters, we would have:

Custom cipher table made with GEMATRO - Gematria Calculator.

    My Readers will very easily understand the reasoning behind this cipher. Just like in Hebrew Gematria and Greek Isopsephy the values of letters are grouped in units (1 to 9), tens (10 to 90) and hundreds (100 to 900), so does English Extended follow the same pattern. This Extended Alternative cipher, however, has a 'twist', since after 100 the pattern changes — instead of counting in hundreds, first 100, then 200, etc, we continue to count in tens, going from 100 to 110, then 120, 130, etc. So this is the basics of this Extended Alternative cipher.

    Unfortunately I haven't practiced enough with this cipher, so I'll just show the result that I got, by an absolutely random coincidence. The phrase "Six Six Six" sums 777 in Extended Alternative, and it shares the same value with other phrases:

    Remember, however, that this only has meaning if you give it meaning. Coincidental "matches" like these don't have to necessarily "mean" anything, except when you use Gematria to encode your text, by using some specific phrasing that refer to other things that aren't present in your text. That's, in fact, one of the most arcane and genuine historical uses of Gematria, and one that is generally overlooked in modern times. My most sincere advice to my Readers regarding Gematria is this: don't take Gematria too seriously, or as means to "decode the Universe" — Use it, but don't abuse it.

 

    Next ciphers in line are the...

Ciphers based on the divisors of a number

    The first time I read about this type of ciphers was in a book by Maurice Bouisson that a friend offered me many years ago, called "Magia: Os seus Grandes Ritos e a sua História" (in English it is published as "Magic: Its History and Principal Rites").

 

     In this book, Maurice Bouisson quotes the work of Raymond Abellio, "La Bible: Document Chiffré" ("The Bible: A Ciphered Document"), in which he writes, regarding the Hebrew alphabet:

    "There are 22 letters. There are also exactly 22 regular polygons that can be inscribed on the circle of 360 degrees, and whose angle at the centre is an integer number of degrees. Each letter corresponds to a polygon.

    All numerical science is based on the geometric structure of the circle.

    These polygons go from the equilateral triangle (3 sides, angle at the centre 360/3 = 120 degrees) to the polygon with 360 sides, whose angle at the centre is 1 degree. Putting between parentheses the number of sides, we also have, in order, the equilateral triangle (3), the square (4), the pentagon (5), the regular hexagon (6), the regular octagon (8), and following the regular polygons of 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360 sides.

    To each Hebrew letter must be assigned one of these numbers, in the order from 3 to 360."

    I was absolutely blown away when I first read this. Not only was this an extremely ingenious way of decoding the Hebrew alphabet with the help of Geometry (coincidentally, a possible origin of the word Gematria), it also showed quite an intelligent way of attributing alternative values to the Hebrew letters based on the polygons that can be inscribed on a circle with 360 degrees. Based on this brilliant discovery, Raymond Abellio would further state that there would be two main systems of Gematria for the Hebrew alphabet — one of them being exoteric, that is, not secret, or publicly disclosed:

    ... and the other one being esoteric, i.e. secret, undisclosed to the public:

    An interesting correspondence that I found while searching for R. Abellio's work has to do with the values of two Hebrew expressions: השמים (ha-shamaïm, "The Skies") and הארץ (ha-aretz, "The Earth"):

 

    These two values correspond respectively to 38×7 and to 29×7, and using the same cipher, the value of "Eve" חוה is 29:

    ... and the value of Elohim (used as a title of God, even though it is a plural) is 83, the reversal of 38:

 

    Unfortunately I never duly explored this cipher except on this occasion, so I won't be able to show any more curious discoveries with this system — for now. I would recommend, however, if my Readers are able to do it, to try getting a copy either of Maurice Bouisson's book "Magic: Its History and Principal Rites", or alternatively, which can be more difficult, of Raymond Abellio's "La Bible: Document Chiffré". I'm not sure if there is an English translation of this last work, though.

    Eventually this "Hebrew 360" cipher gave me an idea. Well, the number 360 has exactly 24 divisors, but the numbers 1 and 2 aren't included in the "Hebrew 360" cipher because they don't divide the 360° circle into polygons. However, what would happen if I used the complete series of the 24 divisors of 360 with the Classical Greek alphabet — which happens to have exactly 24 letters?

    And so, the experimental "Greek 360" cipher was born:

 

 

     Remembering that the Elizabethan English alphabet also had 24 letters, the same principle could be applied to it (I'll label this the "Bacon 360" cipher, even though it could more correctly be called "Elizabethan 360"):

Note that I=J and U=V, in the Elizabethan English alphabet.

    In this case, however, we should remember two things: (1) that this is an experimental cipher, which means that I don't have any proofs that this cipher was used before; and (2) that it would only make sense to use this cipher in the right context, which could either be related to the historical period involved in our research, or to the subject that we're studying. It would make sense, for example, to explore this cipher in the context of the Elizabethan Era, or in the contexts of Rosicrucianism or Freemasonry, for example, since the Elizabethan ciphers have been consistently used in these contexts for centuries.

* * *

    My work with this kind of ciphers didn't end up here, though. Regarding the Hebrew and Greek alphabets, I also thought that there could be yet another cipher for the alternative forms of these alphabets: one for the Hebrew alphabet including the 'soffit' (final) forms of the letters Kaph, Mem, Nun, Phe & Tzaddi, and one for the Greek alphabet which includes the obsolete letters Vau/Digamma, Qoppa and Sampi. Notice that these alternative forms of these alphabets contain exactly 27 different values...

    So a question was raised in my mind: what is the smallest number that has exactly 27 divisors, including itself? And the answer to this question is the number 900, which by a coincidence happens to be the maximum value in both of these ciphers. So with this data in our hands, we can safely build two alternative ciphers for the Hebrew and Greek alphabets, where instead of having the units, the tens and then the hundreds, we'll have instead all 27 divisors of 900, from 1 to 900:

    With a bit more imagination (which is already too fertile, I would say 😅) we could also apply this list of the 27 divisors of 900 to Agrippa's Latin cipher, which also uses the numbers from 1 to 900 (even though in the usual groups of units 1-9, tens 10-90, and hundreds 100-900):

 

    In the future I may dedicate some time to finding qabbalistic patterns with these ciphers. For now, as I only devised them but never actually explored their potential, they will have to remain as references for my Readers. However, if anyone among my Readers finds some value in these experimental ciphers and decides to share their work with me... well, I will appreciate very much if you show me some of your own discoveries with them!

    However, a final word is needed (again) about these ciphers based on the divisors of numbers. My Readers should be aware that these are all modern & experimental ciphers, which means that (as far as I know) they don't have any historical origins in the past — at least, excepting the cipher I called "Hebrew 360", which was first mentioned by Raymond Abellio. So in our work with them we should always take this into consideration.

 

    To conclude...

 

 

Some Final Notes

 

 

     I really hope that this presentation of the Mathematical Ciphers will give you, my dear Reader, a more solid knowledge on the origins of some of these ciphers. While some of them can be tracked to at least the 17ᵗʰ century, like the Polygonal ciphers, or the Extended Alternative cipher, others seem to be modern inventions and so should be properly called experimental ciphers. In fact, there are no limits to experimental ciphers, and anyone can create their own experimental cipher, be it Mathematical or not. In fact, I had many ideas that I didn't include in this presentation, or even in some of my previous texts, but I will expose some of them in some future papers.

    For now, I will make a short pause in this series of articles about the History of Ciphers. Instead, I think that the time has come to write shorter, more specific articles — not only about the ciphers themselves, but also about the Practice of Gematria, as well as some examples of discoveries and curiosities that I've found in my work with Gematria and its many different kinds of ciphers.

    If any of my Readers would wish to exchange a word or two with me, I humbly ask you to either use the "Comments" form at the end of this text, or the "Contact" form in the PC version of this blog (right menu).

 

    I bid all my Readers farewell... for now.

    Hopefully the next post will be a pleasant surprise. 😁 

 

 

    Luís Gonçalves