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Abstract Algebra based Tarot Deck


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/P...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!
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hm, more than math we dread long posts!x.x lolz ,
But somehow I manage courage to read it, and it’s interesting idea?
Imo if you watch closely you can notice some common elements that connect all cards/thread that goes trough whole deck?x,x but not sure what would be central card? 0 fool or world/most probably?XD ) and that progression is interesting, but for the plot sake let’s call it level here rather than order!?x,x and I may jump here but can fool be order 0?x,x your starting point, and magician 1? and other mayors lv1, suits lv2, etc

And idea of inversion is interesting, imo if you are familiar with kabalah tree of life concept it have 11 elements and imo it’s possible to draw a line under last/base sephiroth and get extra 11/its reverse?x,x total is 22! Then parallel can be drawn to get 44!x,x ) so this would point at missing 0 card in pips?)

And it’s already possible that every tarot card have it’s reverse in deck?x,x or that is their meaning when in reverse position?x,x

And that have connection with Hebrew alphabet that is interesting because each letter have its meaning and numerological value?x,x so you may want to check that out?^^ ) as it’s closely connected with tarot,

Googled Atlas of Finite Simple Groups, hm it seems like interesting experiment! But imo it may be more interesting as theoretical experiment? Than real Tarot deck as too much basic knowledge is needed for this deck to understand/work/read with?x,x and most people are not interested in math and algorithms?x,x and rely on inspiration/intuition?x,x and not sure what would be meaning of negation in reading?x,x

p.s surprisingly I was wondering same thing!x,x what cards can combine to give magician?x,x we are talking fusion here?x,x lolz so maybe it would make this topic more popular to say; what cards can be combined to give some other card?x,x not easy question but maybe numerology and logic may give answer?x,x

hm, for magician maybe most obvious would be strength and world?x,x but not sure for other two?x,x imo tarot images we see today may not be same as originally imagined?x,x and there is always possibility some elements are changed for time laps or secret society or any other purpose? So maybe you should go with basic keywords instead of images?x,x

looking forward to hear more about what you’ll discover!^^ lolz
best luck with you project!^^)
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Quote:
Originally Posted by reall View Post
hm, more than math we dread long posts!x.x lolz ,
But somehow I manage courage to read it, and itís interesting idea?
Imo if you watch closely you can notice some common elements that connect all cards/thread that goes trough whole deck?x,x but not sure what would be central card? 0 fool or world/most probably?XD )
Heh yep, after messing with it a bit I almost feel more like "world" would be a better choice, but it's mainly because I connect more meanings with the Fool so I have trouble seeing it as neutral.


Quote:
Originally Posted by reall View Post
and that progression is interesting, but for the plot sake letís call it level here rather than order!?x,x and I may jump here but can fool be order 0?x,x your starting point, and magician 1? and other mayors lv1, suits lv2, etc
Hmm, I can understand that. "Order" is definitely the correct math term, but seems wrong when applied to the cards. Maybe "complexity" would make sense.

Quote:
Originally Posted by reall View Post
And idea of inversion is interesting, imo if you are familiar with kabalah tree of life concept it have 11 elements and imo itís possible to draw a line under last/base sephiroth and get extra 11/its reverse?x,x total is 22! Then parallel can be drawn to get 44!x,x ) so this would point at missing 0 card in pips?)
Ah, I hadn't heard of anything like that! But yeah, within this "L2(7)" system, The Fool is sort of the missing zero card for every suite.

Quote:
Originally Posted by reall View Post
And itís already possible that every tarot card have itís reverse in deck?x,x or that is their meaning when in reverse position?x,x

And that have connection with Hebrew alphabet that is interesting because each letter have its meaning and numerological value?x,x so you may want to check that out?^^ ) as itís closely connected with tarot,
Yeah, I have some familiarity with it but not enough. Maybe the answer is there.

Quote:
Originally Posted by reall View Post
Googled Atlas of Finite Simple Groups, hm it seems like interesting experiment! But imo it may be more interesting as theoretical experiment? Than real Tarot deck as too much basic knowledge is needed for this deck to understand/work/read with?x,x and most people are not interested in math and algorithms?x,x and rely on inspiration/intuition?x,x and not sure what would be meaning of negation in reading?x,x
The way I'm thinking of it is, any systematic/structural approach to Tarot just captures certain aspects and connotations of the cards and leaves out others. A deck based on a certain system will have more to offer about the specific set of ideas which the system deals with. In this system I've happened on, each major will emphasize ideas coming from the two "subgroups" which define it.

It's exactly like an equation in math. 2+3 = 1+4 = 5. Yet having 5 things is different from having 2 things and 3 other things. The relationship "=" is not meant to claim 2+3 is exactly the same as 5. It's meant for a situation where we've decided we care about the similarity between 2+3 and 5.

So during a reading, negation could mean different things, just like "+" can mean different things during arithmetic. "+" can mean you're combining things which come from different sources toward one purpose, or just that you've conceptually divided things and now are putting them back together. Negation could indicate two cards struggling against one another, or just calmly deflecting each other; or it could just indicate a possibility of interaction which may or may not occur.

Quote:
Originally Posted by reall View Post
p.s surprisingly I was wondering same thing!x,x what cards can combine to give magician?x,x we are talking fusion here?x,x lolz so maybe it would make this topic more popular to say; what cards can be combined to give some other card?x,x not easy question but maybe numerology and logic may give answer?x,x
Ha, maybe you're right! I wasn't thinking about what would make a popular topic.

Quote:
Originally Posted by reall View Post
hm, for magician maybe most obvious would be strength and world?x,x but not sure for other two?x,x imo tarot images we see today may not be same as originally imagined?x,x and there is always possibility some elements are changed for time laps or secret society or any other purpose? So maybe you should go with basic keywords instead of images?x,x
Yep! As with any occult/system-based revisionist, part of my idea here is that focusing on the "natural" structures between the ideas would tend to correct any asymmetries caused by changes over time.

I guess, though, that I'm more on the analytical than the intuitive side. I look at the number on the card and treat the image as a memory aid for the number's various meanings.

Quote:
Originally Posted by reall View Post
looking forward to hear more about what youíll discover!^^ lolz
best luck with you project!^^)
Thanks!
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My head hurts. And as a rule all concepts excite me
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I've found a page with a diagram of the group!

http://www.weddslist.com/groups/genu...%282,7%29.html

In reality there are a lot more connections between the cards of course; every card is connected in some way! But that doesn't make for a very good diagram, so this diagram just uses two types of connections, which are all that's needed for everything to get interconnected.

The little black triangles are cycles of three cards which are caused by a chosen minor arcana card. So, just to make up an example, you might start at the Magician and combine it with some ace. This will take you to a Cycle of Arcana, one of the set of 48 cards which I mentioned having order 7. If you combine that Cycle card with the same ace, you get to a different Cycle card. If you add the same ace a third time, you return to the Magician.

For the sake of clarity I'll just say all the black lines in the diagram refer to combining a card with the Ace of Wands. (Different minor arcana cards would work, but I haven't made any assignment so for the most part I can't say which.)

The green lines are between cards which are related by some major arcana card. For the sake of clarity I'll claim all the green lines represent the Magician. (Again, I haven't actually assigned cards to group elements, but this seems pretty safe.)

So I guess you can think of that image as a map of the whole extended Tarot deck in terms of two cards, the Magician and the Ace of Wands. Notice that the Fool isn't specifically marked when the group is drawn this way. In fact, there isn't any one card which appears at the center of the drawing or gets connected with more cards than any other. That has to do with what I was saying about the specific choice of identity element not doing much harm in the long run.

So probably everyone's familiar with the idea of the Tarot as describing the fool's journey. Given the symmetry here, it seems reasonable to claim that any card could be chosen as the identity, and the whole deck could be described as that card's journey. And each card's journey would be different, though I don't know to what extent the system itself (the group structure I mean) provides the details. Describing the Fool's Journey is actually kind of a weakness of this specific structure I've found. These group elements really aren't in a specific order, or else I could just assign Tarot cards to elements based on the ordering.
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I love math. I really do. But you have completely lost me with this one. Hopefully once you show the connections between some of the cards as an example, things will begin to make more sense to me.

Rodney
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WoW! I'm thankfull for diagram!^^ it looks amazing!XD
(and I'm sure your ideas would be much more appreciated in some more inteligent society?x,x where algebra is common knowlage?x,x like India and Old China?x,x lolz )

@rwcarter I think we may only help with this if we focus on simple task;
find cards that make other cards? like
VI lovers may be made from XV and XIV Temperance,
and fool may be paired with chariot and hermit? as they are all on journey?
Magician may be Strength +World?
Sun can be tower/lightning and star?
HP can be Moon and? etc

and I noticed there may be system here?
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Quote:
Originally Posted by rwcarter View Post
I love math. I really do. But you have completely lost me with this one. Hopefully once you show the connections between some of the cards as an example, things will begin to make more sense to me.

Rodney
Heh, that seems to be the consensus I suppose. Maybe I should have posted the question without the math. But a connection appearing between Tarot and somewhat advanced mathematics... That's just too cool not to share, you know?

Once I line up the cards to the mathematical structure even a little bit, it will get a lot easier to talk about; I'll just be asking questions like "Is there a minor arcana card whose relation to the Chariot in the same as the Hermit's relation to the Priest?"

EDIT: Thanks for the ideas, Reall!
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np!^^ that's all I have! x.x
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I've had a thought. Another way of thinking about this idea is to draw two cards and ask, what card would lead from the first to the second? What card would represent the changes which occur?
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