This is a “mathemagical” discussion of the Book of Thoth, which is the collection of cards also commonly called the Tarot of the Egyptians, and how the symbolic order in which Aleister Crowley presented the 78 cards of that deck relates to certain arrangements – corresponding to the generators – of the elements of the finite field, or Galois Field, of order 79.
Important Disclaimer and Relevant Information About Myself: Although I have been an avid reader of the works of Aleister Crowley, I am not a member or initiate in Ordo Templi Orientis, nor an actual practitioner of Thelema, Thelemic principles, or Ceremonial Magic(k). My interest in Magick is mainly one of study: Its principles do from time to time resonate with me and my Zen practice, and there is also a connection with Robert Anton Wilson, the late fiction and non-fiction co-writer of “Illuminatus!” and The Schrodinger’s Cat Trilogy, which I greatly enjoy in the original Bantam paperback editions. I do read Tarot cards, on my own as a meditation and for friends when they want me to do so – for free – and my preferred deck is the Aleister Crowley Thoth Tarot. I see them as a vehicle for generating meaningful coincidences – not fortune telling per se – that resonate with conscious and unconscious elements in both the person reading the cards and whoever else participates in the reading. I am also a mathematician by education, avocation, and my former Federal civilian profession, and have become recently a Retired Member of the American Mathematics Society.
The first introductory note I want to make on the topic of this post is a rather remarkable observation that flavors the essence and importance of what follows: In a table of the symbolic meanings of the prime numbers between 11 and 97 which Aleister Crowley wrote and which now appears in the Samuel Weiser volume called “777 and Other Qabalistic Writings of Aleister Crowley,” he made a rather bewildering omission of the number 79 from that list. This would not be so strange were 79 not one more than the number of cards in a Tarot deck. It is also the order of the multiplicative group of the non-zero elements of GF(79). Did Aleister Crowley do this on purpose? Was it a symbolic omission, with an arcane purpose? Or did he just slip up and miss a prime that did not look prime to him?
Isn’t it interesting? Of course it isn’t! 😀 If you are interested in magick and Thelema, you still might not be interested in Finite Field Theory. And vice versa. If you are not, and I say this with all kindness and seriousness, STOP READING NOW. Do Not Hurt Yourself Shall Be The Whole Of My Law Right At This Moment.
Having said that, Here We Go …
Here is the central thrust of what I am blogging about these mathemagical matters:
In mathematics, the finite field, or Galois field, associated with a prime number p, sometimes given the notation GF(p), is the collection of the numbers 0 through p-1, with division and multiplication defined modulo p: that is, when two of these numbers add or multiply together, the result is the sum or product, respectively, modulo p. That is equivalent to the integer remainder when dividing the sum or product by p. This makes the elements of the Galois field of order p, or GF(p), the numbers 0 through p-1. The entire collection of the non-zero elements of GF(p) forms what mathematicians call a multiplicative group. Addition and multiplication work as they did in the finite field GF(p), but leaving out 0 means that every element has an inverse element: another element such that when they both multiply together, the product modulo p is 1.
(N.B.: There are, just so you know, finite fields whose numbers of elements, or orders, are powers greater than two of prime numbers.)
For the prime number 79, the multiplicative group within GF(79) has 78 elements. This is why to me it is fascinating Crowley left 79 out. The multiplicative group of GF(79) is exactly the number of elements, or cards, in a tarot deck: in particular, in Crowley’s Book of Thoth.
Further, there is an order to the tarot cards, explained by Crowley, putting them in a sequence from the first tarot trump, the Fool, all the way down – corresponding to a progression of more or less decreasing energy or influence, or region of influence on the Kabbalistic Tree of Life – to the 10 of Disks/Pentacles.
What a coincidence! The multiplicative group of nonzero elements of GF(79) also has an order, and so does each element of that group: The order of the group is 78 – there that number is again! – and the order of each element is the smallest power to which it is raised for which that element to that power is congruent to 1. Such a power exists for every element in the multiplicative group.
Some of the elements of the multiplicative group have powers that go through all of the values 1 through 78, in some order. Since they generate the entirety of the elements of the group, they are called generators of the group.
How many generators (also called primitive elements – but not to be confused here with Fire, Water, Air, Earth, and Spirit) are in the multiplicative group of GF(79)? I won’t prove this here, but the number is phi(78), where phi() is the Euler Totient Function. Again, I will not prove or demonstrate how phi() works. I will just give the answer to the question directly. phi(78) = 24. This means that there are 24 different integers, or residue classes modulo 79, whose powers run (rather erratically) through all 78 elements of the multiplicative group.
We can use the sequences of powers of each of these generators, once we know what they are, to produce 24 alternate orderings for the cards of the Book of Thoth. All we have to do is assign numbers 1 to 78 to the cards, then present them – and possibly generate a story to go with the sequence – in the order dictated by the powers of one of the generators, until it has cycled through all of the residue classes and come back to 1. This will take us through all of the cards.
This ordering, like Crowley’s, always starts with Trump 0, or The Fool, as card number 1. (Didn’t he write in his Naples Arrangement that 0=1?) But it proceeds very differently depending on the generator one selects.
Don’t despair! I saved you the trouble of figuring out the generators – primitive elements – of GF(79). They are as below, from the output of a Perl script (tiny) that I wrote to generate them:
3 6 7 28 29 30 34 35 37 39 43 47 48 53 54 59 60 63 66 68 70 74 75 77
Now that we have these generators, we need to look at how they relate to the cards in the Book of Thoth. In order to do this, we need to review Aleister Crowley’s order of presenting those cards in his “The Book of Thoth: A Short Essay on the Tarot of the Egyptians – Being The Equinox, Volume III, Number V, by The Master Therion.” This volume is available from Samuel Weiser, Inc. as “The Book of Thoth,” and the edition I own and have used is the 1996 softcover printing.
I have assigned Crowley’s order therein to the nonzero elements of GF(79) – the elements of its multiplicative group – as follows:
- Elements 1 to 22 are the Trumps, in order, from The Fool to The Universe.
- Elements 23 to 38 are the Court cards, in this order: Knight, Queen, Prince, and Princess of Wands; Knight, Queen, Prince, and Princess of Cups; Knight, Queen, Prince, and Princess of Swords; Knight, Queen, Prince, and Princess of Disks (Pentacles). In other Tarot decks such as Rider-Waite, these become the King, Queen, Knight, and Page, and the Disk suit becomes Pentacles. What can I say? Aleister Crowley was a maverick! 😀
- Elements 39 through 48 are the numbered cards, Ace through Ten, of the suit of Wands.
- Elements 49 through 58 are the numbered cards of the suit of Cups.
- Elements 59 through 68 are the numbered cards of the suit of Swords.
- Elements 69 through 78 are the numbered cards of the suit of Disks (Pentacles).
So, as an example – and remember, there are 24 such examples – I shall describe in my next post a Story – and remember, there are 78 elements to it! – using the Book of Thoth in the order given by the powers of the least generator, 3, which are as below (again, generated by another little Perl script):
1 3 9 27 2 6 18 54 4 12 36 29 8 24 72 58 16 48 65 37 32 17 51 74 64 34 23 69 49 68 46 59 19 57 13 39 38 35 26 78 76 70 52 77 73 61 25 75 67 43 50 71 55 7 21 63 31 14 42 47 62 28 5 15 45 56 10 30 11 33 20 60 22 66 40 41 44 53
Stay tuned – how fun! I get to do a tarot reading for my blog’s readers. I hope you will find it edifying.