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How can you play a Rubik's Cube?

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Not play with it, [br]but play it like a piano?

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That question doesn't [br]make a lot of sense at first,

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but an abstract mathematical field[br]called group theory holds the answer,

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if you'll bear with me.

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In math, a group is a particular [br]collection of elements.

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That might be a set of integers,

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the face of a Rubik's Cube,

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or anything,

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so long as they follow [br]four specific rules, or axioms.

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Axiom one:

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all group operations must be closed[br]or restricted to only group elements.

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So in our square, [br]for any operation you do,

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like turn it one way or the other,

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you'll still wind up with [br]an element of the group.

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Axiom two:

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no matter where we put parentheses[br]when we're doing a single group operation,

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we still get the same result.

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In other words, if we turn our square[br]right two times, then right once,

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that's the same as once, then twice,

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or for numbers, one plus two [br]is the same as two plus one.

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Axiom three:

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for every operation, there's an element[br]of our group called the identity.

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When we apply it [br]to any other element in our group,

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we still get that element.

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So for both turning the square[br]and adding integers,

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our identity here is zero,

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not very exciting.

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Axiom four:

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every group element has an element [br]called its inverse also in the group.

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When the two are brought together[br]using the group's addition operation,

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they result in the identity element, zero,

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so they can be thought of [br]as cancelling each other out.

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So that's all well and good,[br]but what's the point of any of it?

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Well, when we get beyond [br]these basic rules,

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some interesting properties emerge.

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For example, let's expand our square[br]back into a full-fledged Rubik's Cube.

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This is still a group [br]that satisfies all of our axioms,

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though now [br]with considerably more elements

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and more operations.

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We can turn each row [br]and column of each face.

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Each position is called a permutation,

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and the more elements a group has,[br]the more possible permutations there are.

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A Rubik's Cube has more [br]than 43 quintillion permutations,

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so trying to solve it randomly [br]isn't going to work so well.

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However, using group theory[br]we can analyze the cube

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and determine a sequence of permutations[br]that will result in a solution.

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And, in fact, that's exactly [br]what most solvers do,

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even using a group theory notation[br]indicating turns.

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And it's not just good for puzzle solving.

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Group theory is deeply embedded [br]in music, as well.

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One way to visualize a chord[br]is to write out all twelve musical notes

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and draw a square within them.

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We can start on any note,[br]but let's use C since it's at the top.

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The resulting chord is called[br]a diminished seventh chord.

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Now this chord is a group[br]whose elements are these four notes.

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The operation we can perform on it[br]is to shift the bottom note to the top.

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In music that's called an inversion,

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and it's the equivalent [br]of addition from earlier.

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Each inversion changes [br]the sound of the chord,

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but it never stops being[br]a C diminished seventh.

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In other words, it satisfies axiom one.

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Composers use inversions to manipulate[br]a sequence of chords

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and avoid a blocky, [br]awkward sounding progression.

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On a musical staff, [br]an inversion looks like this.

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But we can also overlay it onto our square[br]and get this.

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So, if you were to cover your entire[br]Rubik's Cube with notes

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such that every face of the solved cube[br]is a harmonious chord,

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you could express the solution[br]as a chord progression

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that gradually moves [br]from discordance to harmony

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and play the Rubik's Cube,[br]if that's your thing.