General Knowledge

Group theory 101: How to play a Rubik’s Cube like a piano

How can you play a Rubik’s Cube? Not play with it, but play it like a piano? That question doesn’t make a lot of sense at first, but an abstract mathematical field called group theory holds the answer, if you’ll bear with me. In math, a group is a particular collection of elements. That might be a set of integers, the face of a Rubik’s Cube, or anything, so long as they follow four specific rules, or axioms.

Axiom one: all group operations must be closed or restricted to only group elements. So in our square, for any operation you do, like turn it one way or the other, you’ll still wind up with an element of the group. Axiom two: no matter where we put parentheses when we’re doing a single group operation, we still get the same result. In other words, if we turn our square right two times, then right once, that’s the same as once, then twice, or for numbers, one plus two is the same as two plus one.

Axiom three: for every operation, there’s an element of our group called the identity. When we apply it to any other element in our group, we still get that element. So for both turning the square and adding integers, our identity here is zero, not very exciting. Axiom four: every group element has an element called its inverse also in the group. When the two are brought together using the group’s addition operation, they result in the identity element, zero, so they can be thought of as cancelling each other out.

So that’s all well and good, but what’s the point of any of it? Well, when we get beyond these basic rules, some interesting properties emerge. For example, let’s expand our square back into a full-fledged Rubik’s Cube. This is still a group that satisfies all of our axioms, though now with considerably more elements and more operations. We can turn each row and column of each face. Each position is called a permutation, and the more elements a group has, the more possible permutations there are.

A Rubik’s Cube has more than 43 quintillion permutations, so trying to solve it randomly isn’t going to work so well. However, using group theory we can analyze the cube and determine a sequence of permutations that will result in a solution. And, in fact, that’s exactly what most solvers do, even using a group theory notation indicating turns. And it’s not just good for puzzle solving.

Group theory is deeply embedded in music, as well. One way to visualize a chord is to write out all twelve musical notes and draw a square within them. We can start on any note, but let’s use C since it’s at the top. The resulting chord is called a diminished seventh chord. Now this chord is a group whose elements are these four notes. The operation we can perform on it is to shift the bottom note to the top. In music that’s called an inversion, and it’s the equivalent of addition from earlier.

Each inversion changes the sound of the chord, but it never stops being a C diminished seventh. In other words, it satisfies axiom one. Composers use inversions to manipulate a sequence of chords and avoid a blocky, awkward sounding progression. On a musical staff, an inversion looks like this.

But we can also overlay it onto our square and get this. So, if you were to cover your entire Rubik’s Cube with notes such that every face of the solved cube is a harmonious chord, you could express the solution as a chord progression that gradually moves from discordance to harmony and play the Rubik’s Cube, if that’s your thing.

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