The pointed cubic stone and the Tarot

In an article on the tetraktys and the square of four, René Guénon points out that these two figures together comprise a total of 26 points, which is the value of the Tetragrammaton (yodh he vav he = 10 + 5 + 6 + 5). The 10 point of the tetraktys correspond to the letter yodh, and the 16 points of the square represent the rest of the Name. He illustrates this idea with the following figure.

It occurred to me that the figure looked a bit strange. We would normally think of the nine non-central points of the tetraktys not as being inside a triangle but as forming the edges of one; and the same is true of the 12 exterior points of the square of four. Since in the above diagram, the triangle and the square share an edge, that would be a total of 22 points, not 26, since 4 of the points belong both to the triangle and to the square. That's not the value of the Tetragrammaton, but it is a significant number in its own right, being the number of letters in the Hebrew alphabet and the number of Major Arcana in the Tarot.

At the end of his article, Guénon briefly mentions the three-dimensional analogue of his figure.

Finally, if we consider the solid forms in three dimensional geometry that correspond to the plane figures in question here, the cube corresponds to the square, and the figure that corresponds to the triangle is the quadrangular pyramid with the upper face of the cube as its base. The whole forms what Masonic symbolism designates as the 'pointed cubic stone' and which, in the Hermetic interpretation, is considered to be a figure of the 'philosopher's stone'.

This is the Masonic symbol Guénon is referring to.

The fourth cubic number is 64, and the fourth square-pyramidal number is 30. If we combine these in the same way that Guénon combined the square and the triangle, the total value is 94, a number of no obvious significance. If we combine them my way, though, keeping in mind that the cube and the pyramid share between them a plane of 16 points, the number represented by the pointed ashlar is 94 - 16 = 78.

By my reckoning, then, the plane figure has 22 points, and its solid counterpart has 78. This corresponds perfectly to the Tarot deck, which has 22 Major Arcana and a total of 78 cards in all.

Golden Dawn astrological correspondences for the pip cards

In the Golden Dawn system, 7 of the Major Arcana are associated with astrological planets, and 12 of them with zodiac signs. For the logic behind these correspondences, see my old post "Why Waite switched Justice and Strength"; essentially, it's a slight modification of a tradition based on the Hebrew alphabet and laid out in an appendix to the Jewish mystical text Sepher Yetzirah.

Mapping the 22 Major Arcana to the 22 letters of the Hebrew alphabet and importing traditional Kabbalistic correspondences was a relatively straightforward operation. When it came to the Minor Arcana, though, the Golden Dawn had to create a new system more or less from whole cloth.

The aces and court cards were not given astrological correspondences per se but only elemental ones, as each suit and court rank was associated with one of the four classical elements. Pentacles and Pages mapped to Earth, Swords and Knights to Air, Cups and Queens to Water, and Wands and Kings to Fire. Thus, for example, the Ace of Pentacles was called "The Root of the Powers of Earth" and was mapped to the three Earth signs (Taurus, Virgo, and Capricorn). The Page of Pentacles was "Earth of Earth," the Knight was "Air of Earth," and so on.

There still remained the 36 non-Ace pip cards, 9 in each suit. Each of these was assigned to a particular Planet/Sign combination -- Jupiter in Capricorn, Mars in Taurus, etc. There are 84 such combinations, of which 36 were (seemingly arbitrarily) selected for representation in the Tarot.

This makes for a rather spotty system. For example, if I want to look at the cards that correspond to my birth chart, I find cards for Moon in Libra (Two of Swords), Venus in Aquarius (Five of Swords), and Mars in Pisces (Ten of Cups), but that's all. My other planets (Sun in Pisces, Mercury in Aries, Jupiter in Cancer, and Saturn in Virgo) are not represented at all.

Looking at the table above, it is clear that the correspondences are arbitrary but not random. Certain patterns are immediately obvious. Each suit has already been associated with one of the four elements, and thus with a set of three zodiac signs. Within each suit, the 2, 3, and 4 are mapped to one of those signs; the 5, 6, and 7 to another; and the 8, 9, and 10 to the remaining sign.

You can also see that, once you know the 2, the 5, and the 8 of each suit, you can derive the remaining correspondences from these. When the planets are put in their Ptolemaic order (as I have done in the chart above), the pattern is obvious. So the question is, How are the 2, the 5,  and the 8 assigned? How is it decided which maps to which of its suit's three signs, and how are the planets chosen?

Well, when you tabulate the correspondences as I have done, with both the planets and the signs in their traditional order, the pattern becomes pretty obvious. Ignore the suit signs for now and just focus on the numbers, starting with 2 at Mars in Aries. Go left, then left again, and then diagonally down-left. Left, left, down-left. Keep doing that, wrapping around to the other side when you reach the left edge of the table, and you've got all your numbers. Then add suit signs based on the elements. Fire signs are Wands, Water signs are Cups, and so on. And that's that; that's the whole system.

The only remaining question is, Why start with Mars in Aries? My guess is that it's simply because Aries is the first sign and is traditionally "ruled" by Mars.

The symmetry of the decad

A key component of the Minor Arcana is the series of integers from 1 to 10 -- the decad. Suppose we divide the decad into two parts (1-5 and 6-10) and think of the second half as the mirror-image of the first. That would mean 10 is the mirror-image of 1, 9 is the mirror-image of 2, and so on. We can depict this graphically thus.

I've marked the numbers for some basic properties. Odd numbers are black, and even numbers are red. Primes are in white boxes, and composite numbers are in yellow boxes. Squares and triangular numbers are marked with the appropriate polygons.

(The number 1, is of course, a special case where any of these properties is concerned. The ancients considered it prime but neither odd nor even; moderns consider it odd but neither prime nor composite. If the figurate numbers are defined by algebraic formulae, 1 is both square and triangular; if they are defined  by the geometrical figures for which they are named, it is neither. Since 1 is somewhat odd and somewhat prime but definitely not even or composite, I have coded it as an odd prime. My decision to mark it as not figurate was an admittedly ad hoc.)

Notice that each number's "mirror-image" is in fact its opposite in terms of all the properties shown on the graphic. Every odd number is paired with an even number; every prime is paired with a composite; every figurate number is paired with a non-figurate one.

I believe the decad is the only series of integers from 1 to n that has this property. I can say this with confidence because 2, as an even prime, must be paired with an odd composite. If it were not paired with 9, 9 would have to be paired with some other even prime -- but there is no other even prime.

Note the way 4 and 7 stand out on the diagram, as the only composite on the left and the only prime on the right. The Pythagoreans connected 4 with 10, and 7 with 1 -- and if we were to draw those connecting lines on our diagram, its symmetry would be preserved. Four was connected with ten because 1 + 2 + 3 + 4 = 10, the basis of the tetractys symbol. Seven was called "hidden unity" because, not counting the trivial sense in which 1 is a factor of everything, 1 and 7 are the only two numbers with neither factors nor multiples within the decad.

Dice and the Minor Arcana: Opsopaus's geometrical approach

In his article "Tarot Divination Without Tarot Cards" (qv), John Opsopaus proposes a system of correspondences between the 56 cards of the Minor Arcana and the 56 possible rolls of three dice. This is analogous to his systems for mapping the 21 trumps to the 21 possible throws of two dice -- but is necessarily more complicated because the Minor Arcana are structured in a way that the trumps are not. While the trumps are numbered in a linear fashion, the Minor Arcana are grouped into four suits, each of which has 10 numbered (or "pip") cards and four face (or "court") cards.

Just as 21 is a triangular number, 56 is a tetrahedral one. Since 10 is also a tetrahedral number, the pips of each suit can be assigned to the smaller tetrahedron at one of the corners of the larger one. Once this is done, there remain 16 points at the center, arranged in the shape of a truncated tetrahedron, and these can be assigned to the courts.

The diagram below (which is my own work but is based closely on Opsopaus) shows how the 6th tetrahedral number can be divided into four smaller tetrahedra (red, yellow, green, blue) and a truncated tetrahedron (purple). Purple points represent court cards, and the other four colors represent the four suits of pips.

This is a great way of dealing with the pips, but the problem is that the courts are also divided into four suits, and there seems to be no natural way of quartering our central truncated tetrahedron.

One way of dealing with the courts is to associate each of the court ranks with one of the suits -- which has traditionally been done by way of mapping both the court ranks and the suits to the four classical elements. One popular system is Kings/Clubs/Fire, Queens/Cups/Water, Knights/Swords/Air, Knaves/Coins/Earth.

Notice that our truncated tetrahedron is made up of four hexagons, each of which faces one of the four pip-tetrahedra. For example, in the diagram above, the top surface of the truncated tetrahedron is a hexagon, comprising the rolls {334, 344, 335, 345, 355, 346, 356}, and facing the red tetrahedron. If we assign the red tetrahedron to the suit of Clubs, say, which is associated with the court rank of Kings, then the seven rolls on the red-facing hexagon will correspond to the seven court cards which are Clubs and/or Kings. The roll 345, which is in the center and which thus belongs exclusively to the red-facing hexagon, corresponds to the King of Clubs. Each of the remaining six rolls is shared with one of the other hexagons. For example, the rolls 355 and 356 belong to both the red-facing and the blue-facing hexagons. If we assign blue to Cups, corresponding to Queens, then one of these rolls will be the King of Cups, and the other will be the Queen of Clubs; we can perhaps assign the higher toll, 356, to the former on the grounds that a King should outrank a Queen.

These examples are just examples. I have not yet thought out which tetrahedron should correspond to which suit or any of the other details. Nevertheless, Opsopaus's basic schema seems to have a lot going for it.

Dice and the Minor Arcana: Outlining the challenge

In past posts, I have discussed possible ways of pairing the 21 Tarot trumps with the 21 possible rolls of two dice, concluding in the end that the "Air Hexactys" system was the best. In this system, dice rolls are ranked first by total (e.g., any roll that totals 7 outranks any roll that totals 6); and then, among rolls with the same total, by the higher of the two numbers rolled (e.g., among rolls totaling 7, 1-6 outranks 2-5, which in turn outranks 3-4). The dice rolls, thus ranked from lowest to highest, were then paired with the trumps from 1 to 21. This system seems to "work" and to make symbolic sense, as briefly discussed in my post "The root trumps of the Air Hexactys."

Now, the main reason for assuming a link between dice rolls and Tarot cards is the fact -- unlikely to be a coincidence -- that there are 21 possible rolls of two dice (2d6) and 56 possible rolls of three (3d6), corresponding precisely to the 21 trumps (the practice of numbering the Fool and counting it as a trump is a relatively recent development) and the 56 suit cards (Minor Arcana). The next step, therefore, is to try to find a dice-to-cards mapping that works for the Minor Arcana as well as the Air Hexactys works for the trumps.

Here are some of the challenges facing anyone who would attempt such a mapping:

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Four suits: The trumps have a simple linear structure, being numbered from 1 to 21; all that is required is to establish a linear ranking of the dice rolls, which is not difficult. The Major Arcana, on the other hand, are organized in four suits. There is no obvious way to divide up the 56 possible rolls of 3d6 into four equal categories.

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Rank within each suit: In most traditional games played with Tarot cards, the long/black suits (swords and clubs) are ranked, beginning with the lowest, A 2 3 4 5 6 7 8 9 10 J C Q K -- the same ranking used in most modern card games, except that aces are low. However, the round/red suits (cups and coins) rank the pips (but not the court cards) in reverse order: 10 9 8 7 6 5 4 3 2 A J C Q K. Although there are a few exceptions (most notably the French jeu de tarot), I think we can take this idiosyncratic ranking system as a very old and probably original feature of the Tarot pack.

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Relative rank of suits: Due to the quirk just mentioned, it seems that if we want to rank all 56 Minor Arcana linearly, we have to group them by suit first and then by rank within each suit. It wouldn't make sense, for example, to begin with the four aces, then the four deuces, and so on, because long aces are low but round aces are high. I suppose we could begin with the long aces and round tens, then the long deuces and round nines, etc., but this seems very unnatural.

The problem with ranking by suit first is that Tarot games do not rank the suits. Players must always either follow suit, play a trump, or discard a card. In no case can cards from two different suits be played in the same trick, so the question of whether, say, the King of Swords outranks the King of Clubs never arises.

Modern "esoteric" or divinatory tarot is not a trick-taking game, so "rank" as such is not an issue. Nevertheless, it has become customary to think of coins/pentacles as the "lowest" suit, followed by swords, then cups, and finally clubs/wands as the highest. Waite's chapters on the Minor Arcana in The Pictorial Key to the Tarot, for example, begin with the King of Wands as the highest card and proceed down the ranks to the Ace of Wands; then the Cups, from King to Ace; then the Swords, and finally the Pentacles. (Waite makes tens high and aces low for all suits, contrary to the Continental tradition of which he was perhaps ignorant; Tarot has never been a game in England.)

Historically, the four suits do have a natural rank, since they can be traced back to Chinese suits representing different denominations of money. Coins, which have preserved their character as single coins, are the lowest denomination. Next come Clubs, which were originally strings of 100 or 1000 coins. (Ancient Chinese coins had a hole in the center and were strung together.) Cups derive from the Chinese character for 10,000; and Swords, from the character for ten -- meaning, in context, ten myriads, or 100,000. This ancient ranking has been preserved in many modern Anglo-French-suited games, where Spades (Swords; cf. Italian spade) are the highest-ranking suit, followed by Hearts (Cups), Clubs, and finally Diamonds (Coins). (Bridge, in which Diamonds outrank Clubs, is an exception.)

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Relation to trumps, especially root trumps: Because each of the trumps has been associated with a roll of 2d6, each roll of 3d6 will be associated with between one and three of the trumps. For example, the roll 1-4-6 would be associated with 1-4 (Lover), 1-6 (Hanged Man), and 4-6 (Sun) -- or, alternatively, with the three root trumps associated with 1, 4,  and 6 -- namely, Magician, Death, and World. For rolls where the same number occurs three times, only one of the trumps will be linked; for example, 4-4-4 should be related to Death. (The Four of Swords would seem a natural choice for this roll.)

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Sparse symbolism: Compared to the trumps, the Minor Arcana contain relatively little imagery. Prior to Waite's innovative "scenic pips," a card like the Seven of Swords portrayed nothing more nor less than seven swords (stylized as arcs in the Marseille tradition). Such cards have no immediately obvious meaning, and the traditional meanings ascribed to them vary widely. This makes it hard to judge whether or not a particular dice-card pairing is a "good" one. This can be viewed either as a bug (because it offers little guidance) or as a feature (because it allows the cards to accommodate a wide variety of systems).

Is there any rhyme or reason to the Petit Lenormand?

The Petit Lenormand is a deck of 36 cards used for divination. Each card has an image (much simpler and more straightforward than those of the Tarot) such as a dog, a book, a tree, etc., and each is associated with a number, from 1 to 36, and with a playing card from the 36-card piquet deck. (Modern piquet uses a 32-card deck, but in the 16th century it was played with 36 cards -- standard French-suited playing cards with ranks 2 through 5 removed. The modern piquet deck removes the sixes as well.) Here's an example of what the cards look like.

And here is a table of the Petit Lenormand cards according to rank and suit. The parenthetical signs after some of the card titles indicate which cards are generally considered to be positive or negative; those without signs are neutral.

How on earth was this arrangement arrived at? I've spent some time trying to winkle out hidden patterns, but so far as I have been able to determine, the combinations of number, playing card, and image are truly random. The only pattern evident is that positive and negative cards are not randomly distributed. There are no negative Spades or Hearts and no positive Clubs; Diamonds is the only suit to contain both positive and negative cards.

The pairing of the images with playing cards appears to be unrelated to the traditional meanings of those cards in cartomancy. For example, the Ace of Spades (traditionally "death") is paired with the Woman, while the Coffin is paired with the Nine of Diamonds (traditionally "profit"). The Queen of Spades, universally considered a card of ill omen, is given a positive meaning. Spades in general are considered an unlucky suit, but the Petit Lenormand associates Spades only with positive and neutral images. Black suits are traditionally masculine and red suits feminine, but the Man and Woman cards in the Petit Lenormand reverse that association.

Nor, if the playing cards are set to one side, does there seem to be any pattern in the numbering of the images. If the cards are listed in order from 1 to 36, no overall pattern is evident. Man and Woman are together, as are Sun and Moon, but that's about it. Traditional meanings of the numbers themselves also seem to have been ignored. For example, even numbers are traditionally considered feminine, and the number 28 particularly so because of its association with the moon and the menstrual cycle -- but 28 is assigned to the Man. The unlucky number 13 is given a positive meaning.

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The Petit Lenormand is inherently less interesting than the Major Aracana of the Tarot because its symbols are so simple. This is true even where there is overlap between the two decks. Where the Moon card of the Tarot is an enigmatic scene featuring towers, dogs, a pool of water, and a crayfish, its Petit Lenormand counterpart is just the moon (and an Eight of Hearts). On the other hand, the Petit Lenormand images are considerably more interesting than the (pre-Waite, non-scenic) Tarot pips -- a rather unpromising symbol set consisting of one sword, two swords, three swords, four swords, five swords, etc.

To me, a big part of the interest of such a system as the Petit Lenormand is that it represents a sort of ontology of life. Implicit in its intended use as a fortune-telling system is that any significant life event or situation can be represented by one of the cards or a combination of them. It's interesting to see what kinds of symbols are selected for such an attempted "alphabet of life." (Another such system would be the Alethiometer symbols of Philip Pullman's novels -- also a set of 36 simple images, with considerable overlap with the Petit Lenormand.)

Suits as elemental qualities rather than elements

 Most interpretations of the suit cards (“Minor Arcana,” if you must) in the tarot deck are based on mapping each suit to one of the four classical elements. Typically Clubs (Wands) represent Fire, Swords are Air, Cups are Water, and Coins (Pentacles) are Earth. Sometimes Clubs and Swords are switched. This is pretty standard, but I’ve always found it symbolically unsatisfying. This post explains an alternative system I developed some years ago.

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In Whitley Strieber’s tarot book The Path, he interprets the suits in a way which bypasses the traditional elemental mappings and instead focuses on the suit symbols themselves, drawing on the Gurdjieffian idea of triads. “In the Tarot, the two great opposing forces are symbolized by Swords and Clubs. . . . Masculine violence penetrates, feminine violence crushes. But when the two are in harmony with each other, there is a vast leap into another energetic level entirely. The suit called Money in old Tarots reveals the nature of this energy.” Money, as the power of “codifying wealth into a system of universally recognized symbols,” is “considered to have magical properties” and “actually represents the mind and the energy of seeing, which will be symbolized by the card of the Sun.” As for the remaining suit, Cups, it “contains the three other suits, which, in balance, make a fourth and much greater whole.”

Strieber makes no reference to the classical elements, though his reference to the Sun would seem to link Coins more to Fire than to the traditional Earth. However, in identifying Swords  and Clubs as the two opposing active forces, “penetrating” and “crushing,” respectively, he reminded me of Aristotle’s analysis of the elements in Book II of On Generation and Corruption. In Aristotle’s theory, underlying the four elements are four even more fundamental qualities: hot, cold, dry, and moist.

Hot and cold, and dry and moist, are terms, of which the first pair implies power to act and the second pair susceptibility. ‘Hot’ is that which ‘associates’ things of the same kind (for ‘dissociating’, which people attribute to Fire as its function, is ‘associating’ things of the same class, since its effect is to eliminate what is foreign), while ‘cold’ is that which brings together, i.e. ‘associates’, homogeneous and heterogeneous things alike. And moist is that which, being readily adaptable in shape, is not determinable by any limit of its own: while ‘dry’ is that which is readily determinable by its own limit, but not readily adaptable in shape.

. . .

The elementary qualities are four, and any four terms can be combined in six couples. Contraries, however, refuse to be coupled: for it is impossible for the same thing to be hot and cold, or moist and dry. Hence it is evident that the ‘couplings’ of the elementary qualities will be four: hot with dry and moist with hot, and again cold with dry and cold with moist. And these four couples have attached themselves to the apparently ‘simple’ bodies (Fire, Air, Water, and Earth) in a manner consonant with theory. For Fire is hot and dry, whereas Air is hot and moist (Air being a sort of aqueous vapour); and Water is cold and moist, while Earth is cold and dry.

. . .

The ‘simple’ bodies, since they are four, fall into two pairs which belong to the two regions, each to each: for Fire and Air are forms of the body moving towards the ‘limit’, while Earth and Water are forms of the body which moves towards the ‘centre’. Fire and Earth, moreover, are extremes and purest: Water and Air, on the contrary are intermediates and more like blends.

Hot and Cold are the active qualities according to Aristotle, and they map readily to Strieber’s Swords and Clubs, respectively. The cutting blade of the Sword represents the separating, centrifugal power of Heat; and the crushing Club (think of crushing not as breaking into pieces, but as mashing skin, flesh, and bone together into a single mass) corresponds to the centripetal, bringing-together force of Cold.

The mapping of the remaining two qualities and suits is less straightforward. One could argue for mapping Cups to Dry, since a cup contains other things and imposes its shape on them. Coins, especially if interpreted, following Strieber, as abstract “Money” rather than physical coins, could them be considered Moist, since money has no proper form of its own. However, this seems inconsistent with the identification of Money with the fiery Sun, and to use Cups as a symbol of Dry seems so counterintuitive as to be perverse.

The other option, which I prefer, is to see Cups as Moist and Coins as Dry. Here the suit of Cups does not represent the cup itself so much as the contents of the cup (in line with the traditional identification of Cups with Water), and as such stands for fluidity and the Aristotelian quality of Moist. Strieber’s Cup represents the surrounding environment in which the Sword-Club conflict takes place, and is thus readily identifiable with the Air and Water (the two Moist elements) in which living things live and move. Coins, as the universal and unchanging unit of value, can then represent Dry. The Coins in tarot are generally pictured as gold, and gold (like the diamond, the French-suit counterpart to Coins) is famously unchangeable and incorruptible — that is, “Dry” in the Aristotelian sense. The two Dry elements are Earth and Fire, allowing this system to accommodate both the traditional identification of Coins with Earth and Strieber’s link between Money and the Sun.

In fact this system is for the most part compatible with more traditional interpretations of the suits. Cups as Moist retain their association with Water, and Swords as Hot can be identified with either Air or Fire. The only radical departure from tradition is the identification of Clubs (which, like Swords, have traditionally been mapped to one of the Hot elements) with Cold.

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Aristotle’s elements have a natural hierarchy. Earth (purely centripetal) is the lowest (heaviest, closest to the center), then Water (relatively centripetal), Air (relatively centrifugal), and Fire (purely centripetal). The Hot elements are the highest, the Moist are in the middle, and the Cold are the lowest. There is also supposed to be Fire at the center of the Earth, so Dry should probably be considered even lower than Cold.

The suits also have a natural hierarchy, since they most likely derive ultimately from Chinese money-suited cards. Coins (single coins) are the lowest, then Clubs (strings of 100 coins), then Cups (tens of thousands), and finally Swords (hundreds of thousands). This order is still mostly preserved in French-suited cards, where Spades (Swords) are generally considered the highest (as in the expression “in spades”) with Hearts (Cups) in second place. In Bridge, Clubs are lower than Diamonds (Coins), but many other card games preserve the original ranking.

The hierarchy of the elemental qualities matches the original order of the suits. From lowest to highest: Dry/Coins, Cold/Clubs, Moist/Cups, Hot/Swords.

Undeveloped idea: Tarot triangles

The figure below includes 78 triangles: 22 "upside down" and 56 "right side up."

The tarot deck includes 78 cards: 22 trumps and 56 suit cards. There should be some convincingly non-arbitrary way of mapping the one set to the other.

My first idea was to divide the figure into four "suits" thus:

For each of the four colors, there are 13 triangles that are only that color -- frustratingly close to being perfect, since there are 14 cards in each suit.