Eight Ways to Shuffle Sixty-Four Cards

July 25, 2018

These are visualizations of eight different ways of shuffling sixty-four cards. The horizontal lines of dots represent the particular orders of the cards throughout each shuffle, and the vertical curves represent the paths the cards take from start to finish. The cards are restored to the original order in all the eight cases. From left to right, top to bottom, the shuffles are: (1) Six perfect out-shuffles with two piles. (2) Three perfect in-shuffles with sixteen piles. (3) Four sixteen-card cuts from the top. (4) Seven milk shuffles. (5) Four “count-out-and-transfer” shuffles. (6) Twelve perfect in-shuffles with two piles. (7) Six alternating in- and out-shuffles with sixteen piles. (8) Twelve “deal-one-and-skip-one” shuffles.

The inspiration for these visualizations came from attending a talk by Perci Diaconis last year. I wanted to know if I could better understand the mathematics of card shuffling by aesthetically exploring the various permutations underlying the shuffling methods. My motivation was to make these invisible structures visible and create elegant and interesting art in the process. I find the process of experimenting with mathematical structures through computer code both rewarding and exciting, and I am deeply fascinated with how code can be used to visualize, and make tangible, mathematical concepts, and especially with how complexity can arise from simple assumptions. This art complements my Bridges paper “Card Shuffling Visualizations”.