#### Dranorter

*Scary math warning:*I've been told that math, along with big walls of text, can scare people off. So as a quick summary: I'm asking for advice on making sets of 3 Trump cards which 'support' one another in the sense that any two generate the meaning of the third, intuitively speaking.

So, something I've been thinking about for a while is finding a way to express or explore the structure of the deck using group theory. Cards combine to create meanings, and group theory studies structures where elements are combined to make other elements. Not that group theory is necessarily the clearly correct way of studying Tarot structure - for example in a group, there is always an "identity element" which, combined with the other elements, does nothing to them - like adding zero or multiplying by one. Taking that seriously, we'd end up having one Tarot card which is central to the structure of the deck, which every other card leads back to (you'll see what I mean shortly), but which doesn't add meaning on its own - combined with any other card, it disappears.

I'm tentatively choosing The Fool as the identity element. If I were only looking at the major arcana, this would be the clear choice; the Fool is kinda ambiguously either 0 or 22, and when you look at the cyclic group with 22 elements, the identity element can be called either 0 or 22.

But as much interesting structure as that group might provide, I decided to take a different approach, looking for an elegant structure which could encompass the whole deck.

At first I was looking for groups with 78 elements, but didn't find anything that looked promising, so bearing in mind that some esoteric decks add cards, I decided to look higher.

I ended up looking at "simple groups", not for any reason except that they are more easily available online. (Search for Atlas of Finite Simple Groups.) Simple groups are the building blocks of everything else though, so this makes sense.

So a few days ago I settled on a group with 168 elements called L2(7).

(You can take a look at it here:

http://brauer.maths.qmul.ac.uk/Atlas/v3/lin/L27/ or here: http://groupprops.subwiki.org/wiki/Projective_special_linear_group:PSL(3,2) ) I wasn't going on much when I chose it; glancing through pages on it, I liked that it involved the numbers 7 and 14, and the fact that it has so many names (L3(2) etc.) seemed to indicate it can be arrived at many ways so is an elegant object. So I was kinda planning on assigning meanings to each of the 168 elements, just vaguely in line with the progressions that interconnect Tarot carts, and see where that got me.

However, taking a second look through the group properties, I noticed that the elements can be classified by their "order": 1 of order 1, 21 of order 2, 56 of order 3, 42 of order 4, and 48 of order 7. Pretty cool!

What this means is I can take the elements of order 1 and order 2, and call them the 22 major arcana; and the elements of order 3, and call them the minor arcana! Which is perfect, because the element of order 1 is the identity - which I had already decided was The Fool, a major arcana.

What is this "order", you might be wondering? The order is how quickly the element leads back to the identity. Normally we only ever combine a Tarot card with a *different* Tarot card, but in a group structure, elements can be combined with themselves.

The Fool being order 1 means that 1 copy of the Fool leads back to the Fool... pretty simple. But the other trumps being order 2 means that each of them is "self-inverse", and if we somehow drew, for example, Hierophant twice in a row, they should cancel each other out, leading back to the Fool.

This makes sense to me. Every card will have its opposite somewhere in the deck, but Trumps are special in that they are their own opposite.

(The Fool is no exception - it's its own opposite even before it gets

started!)

The 56 minor arcana are order 3, meaning for example if you get the 5 of cups twice, then they would combine to represent another minor arcana card, namely whichever card negates the 5 of cups. But if you drew it 3 times in a row, it would return to The Fool - the first two would cancel the third.

So that's how the order works. In a way it can't do too much harm to choose one card, ie The Fool, as the identity - we can transform the group to an "identityless" structure called a Heap, and should get the same Heap even if we chose the wrong Identity, but I'll have to leave that exercise for after I figure out all the meanings here!

As for the 42 elements of order 4, these are the "Roots of the Majors"

or Radical Arcana (Radix Arcana? Radica Arcanarum?) if you will.

They're literally the square roots of the 21 non-identity arcana. And, it's always cool to see the number 42 show up!

The 48 elements of order 7 I call the Cycles of Arcana. (For now at least. I haven't looked closely at their structure.) Individually, they don't connect back to the other types of cards, except the Fool of course. But combine them with each other and you can get any of the other majors or minors.

In retrospect, it makes a lot of sense to have a fairly large group, of which the Tarot proper are only a subset. After all, combining two Tarot cards doesn't always get you a Tarot card! The sets of 42 and 48 are the extra meanings which you get by combining several of the 78 Tarot cards.

Of course it's still possible to get meanings which fall outside of the 168. The way I look at it is this: The Universe is infinite, but we sort of wrap infinity around itself to get a set of cards we can deal with. Normally people talk about this in a linear way, wrapping the number line around into the numbers on the cards. A group structure is just a different way of having things "wrap around"; whenever several cards "cancel" to make the Fool, that's a point where we're simplifying to keep things within our reach.

So, potentially *any* group structure could be used to do this, but I happened on a surprisingly Tarot-like one without even trying to get literal Tarot structures.

What remains to be done is assign specific cards to specific elements! That's where I'd like to ask for some help. Within this group structure L2(7), each trump interacts with two "subgroups" of four cards. For lack of a better term, I'll call these A and B. (I'm sure they have some deeper meaning from which I can get a name, but I don't want to shade their meaning prematurely by calling them "Left" and "Right" or anything.) I'll just talk about the Magician as an example. The Fool is in every subgroup; so for the Magician, I need to choose the two trump cards which are in its set A, and the two which are in its set B.

How these "subgroups" work is, the 3 cards other than the Fool kind of support each other. Any two combine to make the third. So what I'm really asking for is two cards which combine to form the Magician, and then two other cards which also combine to form the Magician. Or of course suggestions for combining other cards would be more than welcome.

Within this system (if the system works out), since the 22 trumps are all "self-inverse", common elements cancel each other; so two cards should "combine" by losing whatever is in common, leaving only what is different. (But this is only true of the trumps! In fact it's only true of trumps which combine to make other trumps. Most random pairs of trumps will produce minor arcana or a non-card meaning.)

The distinction between the "A" sets and the "B" sets points out a larger structure: the 21 numbered trumps end up sorted into seven "A" sets which don't overlap and seven "B" sets which don't overlap. These are like two separate classification systems, giving sets which interact two different ways.

Once I've got about seven or eight of these "subgroups" identified, the meanings for the whole deck should fall into place just by using the group rules. I've studied the Tarot somewhat but I'd really appreciate any advice. Of course, the meanings emphasized by this system will be very different from the usual ones, but I'd really like to see how it all adds up!