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Saturday, April 20, 2013

Jeu de Cartes

Mathematicians do not deal in objects, but in relations between objects; thus, they are free to replace some objects by others so long as the relations remain unchanged.
 ––PoincarĂ©






The idea that someone with a bit of natural dexterity and a lot of practice can trace any card's path as it travels through several riffle shuffles of a deck is amazing to most of us (and a good reason we amateurs should stay away from Blackjack tables and Poker games). The above video by Kevin Houston gives an excellent demonstration of how card tracking is possible using a "perfect shuffle" and then goes on to explain the math behind why this particular shuffle works as it does. In passing, musicians will recognize one of the techniques since it's based on modular arithmetic––the same elementary principle as the one at work behind the "octave equivalence" and "pitch class" concepts. But what we're interested in here is still mostly the "geometries of permutations."

Another interesting card shuffle is often associated with the mathematician Gaspard Monge (1746-1818), a colorful character active during and after the French Revolution and usually thought of as the inventor of descriptive geometry. What is often called the "Mongean shuffle" works like this. . . .

We'll select only six cards from the deck to make the principle clear, keeping in mind that, once understood, it can be applied to the entire deck of 52 cards. Let's say we have, arranged from top to bottom, the Ace, 2, 3, 4, 5, and 6 of clubs. Now, ignoring the mechanics of how this shuffle might be performed in practice by a dealer or magician (it's rather clumsy compared to the riffle illustrated in the video), we lay out a single Mongean shuffle of these six cards face up. Begin by laying down the top card, the Ace, then place the next card, a deuce, on the table to the left of the Ace, then the three to the right of the Ace, the four to the left of the deuce, the five to the right, and finally the six to the left. The result is the re-ordering 6, 4, 2, Ace, 3, 5:



Flipping this map diagonally and repeating the procedure, we can follow any card through six shuffles until it returns to its original position.



Why does this map look familiar? (As if you haven't guessed already.) But "looks familiar" isn't enough. Let's try applying a procedure that, once again, will be familiar to most musicians – or at least to those who have played/looked at Bach's Musical Offering.[1]

Here again is the basic Mongean shuffle:


Let's get the cards out of the way to follow the moves more easily. This gives us a sort of placeless map .... (sorry) ...




Start by flipping this map over horizontally, resulting in a retrograde map.




Then flip that result over vertically, resulting in an inversion of the retrograde.




Then lay the cards back on the table using the new map to see what this retrograde-inversion shuffle results in.




The retrograde-inversion of the Mongean shuffle produces precisely the same map we encountered in the sestina.




But we're not finished quite yet. Trying to visualize the Mongean shuffle as a transformed version of a spiral (the way we initially described the sestina) is a bit of a stretch. Still, following a curvy path, it almost makes sense to see how the spiral:





might be seen as a tangled/untangled version of Monge, which itself can be seen to resemble a snake twisting back on itself or the action of a wave rebounding from the side of a pool:




If we want to get away from continuous "curving actions" but still stay with some sort of helpful visualization that gives the same results, we might think of these two procedures as things to do with two three-pronged combs. Making one comb out of the left side and one out of the right, we reverse one of them, stretch them both a bit, and then interleave the result. Then the spiraling looks like this in the corresponding comb version:




Starting back at the consecutive numbers once more, this time dividing the positions into odd and even, we separate them again, turn one of them backward again, and then recombine by concatenation.  So the wave in its corresponding comb version looks like this:





*


Ideomorphically, of course, the two worlds can't be mixed––playing cards are not words and card shuffling is not poetry. And neither, again ideomorphically, is a snake or a spiral a card trick or a poem. And that's just the point, because we are searching for sub-surface isomorphic actions–beyond metaphor and analogy–that connect otherwise unrelatable worlds. In the isomorphic underworld, not only can a hair comb be a metaphorically useful object-symbol within a poem, it can also provide a translucent structural basis for the poem. In fact, the first of the two "comb moves" above is precisely the way the sestina was often described, rather than the spiral description I chose to begin with. The technique in medieval times was known as retrogradatio cruciata–"backward crossing."[2]



Next: Just what is multi-tasking parallel processing, anyway? ––––>




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[1] I mention Bach in this passing reference rather than Schoenberg, whom many would prefer to focus on as the more obvious culprit. I did this to keep my background emphasis that all these tricks are ancient & didn't just spring into being starting in the early 20th century. It's relatively easy to identify the inventor of the steam engine, but not so easy to identify the inventor of the wheel. By the time JS Bach came on the scene, many of these tricks/transformations had already long been in use. For a fun time with the Bach canons in the Musical Offering, there are two interesting web sites – one in a math version and one in a non-math version.
[2] Treated as religious symbolism, this term might also suggest anything from dark word play to heresy to outright satanism. But I have never read any suggestion of this elsewhere, so this must remain my personal fantasy. (Still, an interesting potential plot twist for a novelist or playwright, though?)