'Play a Rubik's Cube Like a Piano' With Group Theory & TED Ed from Michael Staff
Playing piano can be difficult. Although, as a pianist, I’d argue that solving the Rubik’s cube is more difficult…way more difficult! It all depends on your perspective of course, but music theorist Michael Staff on TED Ed is showing us how those two perspectives are, in fact, one and the same. In his video, “How to Play a Rubik’s Cube Like a Piano” [below], Michael Staff exhibits how each process (playing the piano and solving the Rubik's cube) relies on the same properties of mathematics, properties that make up a system called group theory. The official TED Ed “Rubik’s cube piano lesson”, written by Michael Staff on TED Ed, explains more thoroughly how the two can aid each other, but the video gives us an overview of some basic principles.
A music theorist and mathematician, Michael Staff describes in the “How to Play a Rubik’s Cube Like a Piano” video that there are four basic rules (or axioms) of group theory that apply to the Rubik’s cube as well as to the piano. “Axiom 1: Closure” dictates that the placement of all elements of a particular group are interchangeable, and that rearranging them will not affect their values. “Axiom 2: Associative” dictates that any action applied to two or more elements applies in any order they appear. “Axiom 3: Identity” dictates that when any element is added to a common element (the “identity”, i.e. “zero”), the element remains the same. And finally, there’s “Axiom 4: Inverse”, which dictates that every group houses two inverse elements, and that bringing the two together will result the identity element, “zero”.
The key analogy put forth by Michael Staff is that, while an “element” can apply to any colored square in the Rubik’s cube, it can also apply to any note on the piano. After all, the notes on a piano (or on any instrument for that matter) represent a closed system of values. Certain rules regarding these values are instilled in us from our earliest music lessons, such as how intervals (the building blocks of chords), have their own inverse, and how chords themselves can be inverted to smooth out transitions. The two projects can be tackled in tandem if you simply assign a number to each note, say, in the twelve-tone tradition of “0-11”. The TED lesson goes on to describe how certain chords equate to certain cube patterns. For a similar exercise in the mathematical relationship between the Rubik’s cube and music, check out Swedish producer and musical inventor Hakan Libdo’s Cube Sequencer.