Visualizing Automorphisms of S6

Summary

The Weird Symmetry of Shuffling Six Items Visualizing the Exceptional Automorphism of S6 When shuffling around items, there's something strangely unique about shuffling precisely six items. In fact, it's so weird, that mathematicians refer to it as "exceptional" or "exotic". This is an interactive post exploring this exceptional property, and trying to visualize just what the heck is going on. We're going to build up to the following diagram, along with the concepts needed to understand it. While this topic is usually encountered in an advanced undergraduate or graduate math course in abstract algebra, I'm only assuming a moderate familiarity with basic algebra. Permutations First, let's talk about shuffling. In the world of math, we call any rearrangement of items (or shuffling) a permutation. Below is an interactive diagram where you can click on any two numbers to swap them. Note that any possible arrangement can be achieved by a series of swaps (which are formally called "transpositions"). Permutations (click on any two numbers to swap them) In the figure above, we're also using a compact notation called "cycle notation" to describe the permutation. Transpositions are cycles that look like (1 2), which means that 1 and 2 switch places. A cycle like (1 2 3) means that 1 goes to where 2 was, 2 goes to where 3 was, and 3 goes to where 1 was. We can get to the cycle (1 2 3) by performing the swap (1 2) followed by (2 3). A permutation can have disjoint cycles too, like (1 2)(3 4), which means that 1 and 2 swap places and 3 and 4 swap places. Note, while the figure is animating, the cycle notation will take on the longer form like (1 2)(2 3), and then when it finishes, it will simplify to the fully reduced form (1 2 3). Group Theory Permutations behave in a way that mathematicians abstract into a concept called a group. If I ask you a concrete algebra problem like 2(5 + x) = 8 and ask you to solve for x, you're taking advantage of a lot of abstract qualities, like th...