Abstract Algebra based Tarot Deck

rwcarter

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?
For me that answer could very likely be different from one deck to another because of the art and/or what facet of the "traditional" meaning(s) is/are being highlighted.

Rodney
 

Michael Sternbach

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?

If two people would do this, or you would do it twice, would the results be the same - for every single couple? Hardly.

No, we should be looking for a recognizable pattern that follows a some kind of mathematical logic. And which also makes sense in terms of the meanings of the cards. Something like the Golden Dawn systems connecting the cards to the Tree of Life.
 

Dranorter

If two people would do this, or you would do it twice, would the results be the same - for every single couple? Hardly.

No, we should be looking for a recognizable pattern that follows a some kind of mathematical logic. And which also makes sense in terms of the meanings of the cards. Something like the Golden Dawn systems connecting the cards to the Tree of Life.

Agreed. The structure here is compelling (to me at least); there must be some principled way to use it.

What I want is ideas to sift through. I'm hoping I or someone else can recognize the really general patterns once they show up.

I've happened upon a page that has a lot of really good info on this structure.

http://math.ucr.edu/home/baez/klein.html

I like the quote: "So, we see that far from being arbitrary flukes, the numbers 84 and 168 are burnt deep into the fabric of reality." And that's a mathematician talking! He's not trying to say anything spiritual. (Mathematicians do frequently talk like this, by the way. Notice how the whole reason he presents the proof is that the number 84 doesn't seem immediately elegant. Mathematicians are used to the truth being simple.)

More important though: the page has a MUCH simpler way of explaining the group as a whole. There's a diagram there which is called the Fano Plane: http://math.ucr.edu/home/baez/mathematical/fano_simple.gif The Fano Plane is the simplest possible projective plane. You may remember from geometry that any two points on a plane share a line, and any two lines cross at at most one point. The Fano Plane is, in a way, the smallest structure with those properties. So it's interesting in itself that the smallest possible plane has seven points and seven lines.

The group L2(7) can be described as the set of symmetries of the Fano Plane.

Imagine we take the Fano Plane and put a number on each dot. Symmetries of the Fano Plane are ways of rearranging the numbers such that if three numbers started out on the same line, they end up on the same line. (Of course, they'll switch which line - and keep in mind the circle counts as a line.)

In this view, the numbered major arcana are ways of "flipping" the Fano Plane, in the sense that if you flip again, you end up where you started. Every major arcana affects four points on the Fano Plane, except the Fool, which represents a chosen "original" numbering.

So I think I could use the Fano Plane image to write a much more understandable introduction to exactly what I'm talking about here. But for the time being I'm out of time.
 

witchofglass

[eyes glazing over] math deck math deck math deck

Yeah, this sounds really cool even if the fancy deck math isn't something I use and a bunch of it is over my head.
 

Dranorter

[eyes glazing over] math deck math deck math deck

Yeah, this sounds really cool even if the fancy deck math isn't something I use and a bunch of it is over my head.

At this point I wonder whether I'm making a deck, or just a book about Tarot. (Because... clearly my explanations are becoming book-length.) But it's probably still a deck, since it emphasizes different meanings than other systems.
 

Grizabella

You might like the Tarot of Ceremonial Magic by Duquette. It's full of all kinds of stuff that I haven't studied yet. Mathematics, including algebra, are probably among the intriguing aspects of the deck, although I wouldn't guarantee it.

Algebra and I had a lover's quarrel many, many moons ago when mostly what I learned in Algebra class was that if we could get the teacher (who was also a football coach) to talking about football, we didn't have to bother with the rest of the gibberish of algebraic equations. I could figure out the answers but he insisted we had to do equations to show how we got to the answer and that's where I flopped on my face.
 

Dranorter

I've made some good explanation diagrams and animations, and I've labelled that map I linked to earlier according to what card type goes where. I've been trying to put together something of a careful explanation of the math, but I keep getting distracted just trying to understand it better myself. :) So I'll post more in a bit.

I wanted to make a note of a certain realization, though, for anyone who might be following along. I tried putting the cards in sets of three just according to simple shared ideas, just to see what it would look like. But it didn't add up - my choices were contradictory! It turns out there's more structure than I thought, and now I understand the difference between the "A sets" and the "B sets" that I was talking about in my first post.

Firstly: The A sets are the points of a Fano Plane. The B sets are the lines. (Or, the other way around - it makes no difference which is which.) What this means is, if you pile the cards into seven stacks, one containing each A set, then you can arrange the stacks like the triangular Fano plane diagram, and if three A-sets are on the same line, then there's a B-set which contains one card from each pile on the line.

Once I've chosen the A-sets, I can lay them out in the diagram and decide on three B-sets that I like; after that, the math gives me everything else. I think regardless of whether they're A-sets or B-sets, 10 decisions are all that's needed.

Secondly: The Fano Plane formed of the major arcana sorts the cards naturally into 3 sets overall. (The math is a bit harder to explain here, or, I haven't found the simplest explanation yet.) Basically each stack of cards in the arrangement will have a top card, a bottom card, and the card in the middle. Each A-set obviously contains one top, one middle, and one bottom card. Well it turns out, each B-set *also* contains one top, one bottom, and one middle card.

Much more interestingly, all the "top" cards form a circle; their order in the circle is determined by our choice of sets! And the same is true of the "middle" cards and the "bottom" cards! So by choosing sets of 3 related cards we're actually choosing an ordering. Well, not quite; three circular orderings, each of 7 cards. And the three circles can be lined up concentrically so that the "A" sets or "points" are all aligned, or so that the "B" sets or "lines" are all aligned.

I *think* that it's natural to call the top/middle/bottom cards "beginning", "middle" and "end" cards, and to walk each "line" set from "beginning" to "end" and then find out which "point" you're in and switch back to "beginning" and keep going. This gives a natural ordering to trumps.

There's still a lot of wiggle room to choose a correspondence between points and lines, but I'm much closer to having a single correspondence.

You might like the Tarot of Ceremonial Magic by Duquette. It's full of all kinds of stuff that I haven't studied yet. Mathematics, including algebra, are probably among the intriguing aspects of the deck, although I wouldn't guarantee it.

Thanks for the recommendation!

Algebra and I had a lover's quarrel many, many moons ago when mostly what I learned in Algebra class was that if we could get the teacher (who was also a football coach) to talking about football, we didn't have to bother with the rest of the gibberish of algebraic equations. I could figure out the answers but he insisted we had to do equations to show how we got to the answer and that's where I flopped on my face.

Sounds familiar. I ended up having a couple pretty good teachers but I've heard stories of exactly this before; being a coach and a math teacher seems to be a bad combination.

I do want to mention, algebra and abstract algebra are pretty different! In abstract algebra, the equations are similar but the letters stand for something very different from numbers.
 

rwcarter

When I said earlier that I like math, I meant it. I got up to Differential Calculus in High School. But so far this idea is flying "Superman far" above my head.... I'm still checking in though to see if something will click for me.

Rodney
 

gregory

When I said earlier that I like math, I meant it. I got up to Differential Calculus in High School. But so far this idea is flying "Superman far" above my head.... I'm still checking in though to see if something will click for me.

Rodney
I'm with Rodney. I never made calculus, but I loved math as such...
 

Dranorter

When I said earlier that I like math, I meant it. I got up to Differential Calculus in High School. But so far this idea is flying "Superman far" above my head.... I'm still checking in though to see if something will click for me.

Rodney

Yeah... More explanations will come, I just have to wait for my own understanding to settle a bit.