this was proved in 1992 by persi diaconis and david bayer. diaconis and bayer identified a key signature of a shuffled deck of cards: the number of rising sequences. let's look at an example. i start here with an ordered deck of 10 cards, riffle shuffle them once, and take a look. see how it's now composed of two rising sequences, two sub-sequences of increasing cards. and as i continue to shuffle, the order dissolves. after i do another shuffle, i expect four rising sequences, and then after that eight, and so on and so forth. the way in which permutations of a given number of rising sequences compose their algebra provides the key to understanding how the mathematical model of riffle shuffling works. this is just one example of what is in fact a very intriguing intersection of the world of cards, and even card tricks, with mathematics. but here's another trick. did you know that we can use symmetry to uncover the virtually invisible worlds of the atoms in a crystal? well, we can. let's take a look. >> so crystallography is the study of how