[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:06.96,0:00:09.60,Default,,0000,0000,0000,,How can you play a Rubik's Cube?
Dialogue: 0,0:00:09.60,0:00:13.23,Default,,0000,0000,0000,,Not play with it, \Nbut play it like a piano?
Dialogue: 0,0:00:13.23,0:00:15.91,Default,,0000,0000,0000,,That question doesn't \Nmake a lot of sense at first,
Dialogue: 0,0:00:15.91,0:00:20.64,Default,,0000,0000,0000,,but an abstract mathematical field\Ncalled group theory holds the answer,
Dialogue: 0,0:00:20.64,0:00:22.61,Default,,0000,0000,0000,,if you'll bear with me.
Dialogue: 0,0:00:22.61,0:00:26.72,Default,,0000,0000,0000,,In math, a group is a particular \Ncollection of elements.
Dialogue: 0,0:00:26.72,0:00:28.54,Default,,0000,0000,0000,,That might be a set of integers,
Dialogue: 0,0:00:28.54,0:00:30.47,Default,,0000,0000,0000,,the face of a Rubik's Cube,
Dialogue: 0,0:00:30.47,0:00:32.08,Default,,0000,0000,0000,,or anything,
Dialogue: 0,0:00:32.08,0:00:36.57,Default,,0000,0000,0000,,so long as they follow \Nfour specific rules, or axioms.
Dialogue: 0,0:00:36.57,0:00:38.06,Default,,0000,0000,0000,,Axiom one:
Dialogue: 0,0:00:38.06,0:00:43.68,Default,,0000,0000,0000,,all group operations must be closed\Nor restricted to only group elements.
Dialogue: 0,0:00:43.68,0:00:46.60,Default,,0000,0000,0000,,So in our square, \Nfor any operation you do,
Dialogue: 0,0:00:46.60,0:00:48.75,Default,,0000,0000,0000,,like turn it one way or the other,
Dialogue: 0,0:00:48.75,0:00:52.03,Default,,0000,0000,0000,,you'll still wind up with \Nan element of the group.
Dialogue: 0,0:00:52.03,0:00:53.67,Default,,0000,0000,0000,,Axiom two:
Dialogue: 0,0:00:53.67,0:00:57.100,Default,,0000,0000,0000,,no matter where we put parentheses\Nwhen we're doing a single group operation,
Dialogue: 0,0:00:57.100,0:01:00.60,Default,,0000,0000,0000,,we still get the same result.
Dialogue: 0,0:01:00.60,0:01:05.04,Default,,0000,0000,0000,,In other words, if we turn our square\Nright two times, then right once,
Dialogue: 0,0:01:05.04,0:01:08.06,Default,,0000,0000,0000,,that's the same as once, then twice,
Dialogue: 0,0:01:08.06,0:01:12.59,Default,,0000,0000,0000,,or for numbers, one plus two \Nis the same as two plus one.
Dialogue: 0,0:01:12.59,0:01:14.25,Default,,0000,0000,0000,,Axiom three:
Dialogue: 0,0:01:14.25,0:01:18.86,Default,,0000,0000,0000,,for every operation, there's an element\Nof our group called the identity.
Dialogue: 0,0:01:18.86,0:01:21.29,Default,,0000,0000,0000,,When we apply it \Nto any other element in our group,
Dialogue: 0,0:01:21.29,0:01:23.45,Default,,0000,0000,0000,,we still get that element.
Dialogue: 0,0:01:23.45,0:01:26.86,Default,,0000,0000,0000,,So for both turning the square\Nand adding integers,
Dialogue: 0,0:01:26.86,0:01:29.27,Default,,0000,0000,0000,,our identity here is zero,
Dialogue: 0,0:01:29.27,0:01:31.78,Default,,0000,0000,0000,,not very exciting.
Dialogue: 0,0:01:31.78,0:01:33.22,Default,,0000,0000,0000,,Axiom four:
Dialogue: 0,0:01:33.22,0:01:38.30,Default,,0000,0000,0000,,every group element has an element \Ncalled its inverse also in the group.
Dialogue: 0,0:01:38.30,0:01:42.25,Default,,0000,0000,0000,,When the two are brought together\Nusing the group's addition operation,
Dialogue: 0,0:01:42.25,0:01:45.11,Default,,0000,0000,0000,,they result in the identity element, zero,
Dialogue: 0,0:01:45.11,0:01:48.84,Default,,0000,0000,0000,,so they can be thought of \Nas cancelling each other out.
Dialogue: 0,0:01:48.84,0:01:52.44,Default,,0000,0000,0000,,So that's all well and good,\Nbut what's the point of any of it?
Dialogue: 0,0:01:52.44,0:01:55.30,Default,,0000,0000,0000,,Well, when we get beyond \Nthese basic rules,
Dialogue: 0,0:01:55.30,0:01:57.84,Default,,0000,0000,0000,,some interesting properties emerge.
Dialogue: 0,0:01:57.84,0:02:03.04,Default,,0000,0000,0000,,For example, let's expand our square\Nback into a full-fledged Rubik's Cube.
Dialogue: 0,0:02:03.04,0:02:06.64,Default,,0000,0000,0000,,This is still a group \Nthat satisfies all of our axioms,
Dialogue: 0,0:02:06.64,0:02:09.82,Default,,0000,0000,0000,,though now \Nwith considerably more elements
Dialogue: 0,0:02:09.82,0:02:12.07,Default,,0000,0000,0000,,and more operations.
Dialogue: 0,0:02:12.07,0:02:16.66,Default,,0000,0000,0000,,We can turn each row \Nand column of each face.
Dialogue: 0,0:02:16.66,0:02:19.04,Default,,0000,0000,0000,,Each position is called a permutation,
Dialogue: 0,0:02:19.04,0:02:23.60,Default,,0000,0000,0000,,and the more elements a group has,\Nthe more possible permutations there are.
Dialogue: 0,0:02:23.60,0:02:28.22,Default,,0000,0000,0000,,A Rubik's Cube has more \Nthan 43 quintillion permutations,
Dialogue: 0,0:02:28.22,0:02:32.45,Default,,0000,0000,0000,,so trying to solve it randomly \Nisn't going to work so well.
Dialogue: 0,0:02:32.45,0:02:35.86,Default,,0000,0000,0000,,However, using group theory\Nwe can analyze the cube
Dialogue: 0,0:02:35.86,0:02:41.00,Default,,0000,0000,0000,,and determine a sequence of permutations\Nthat will result in a solution.
Dialogue: 0,0:02:41.00,0:02:44.47,Default,,0000,0000,0000,,And, in fact, that's exactly \Nwhat most solvers do,
Dialogue: 0,0:02:44.47,0:02:49.57,Default,,0000,0000,0000,,even using a group theory notation\Nindicating turns.
Dialogue: 0,0:02:49.57,0:02:51.60,Default,,0000,0000,0000,,And it's not just good for puzzle solving.
Dialogue: 0,0:02:51.60,0:02:56.58,Default,,0000,0000,0000,,Group theory is deeply embedded \Nin music, as well.
Dialogue: 0,0:02:56.58,0:03:00.98,Default,,0000,0000,0000,,One way to visualize a chord\Nis to write out all twelve musical notes
Dialogue: 0,0:03:00.98,0:03:03.64,Default,,0000,0000,0000,,and draw a square within them.
Dialogue: 0,0:03:03.64,0:03:08.36,Default,,0000,0000,0000,,We can start on any note,\Nbut let's use C since it's at the top.
Dialogue: 0,0:03:08.36,0:03:12.60,Default,,0000,0000,0000,,The resulting chord is called\Na diminished seventh chord.
Dialogue: 0,0:03:12.60,0:03:17.19,Default,,0000,0000,0000,,Now this chord is a group\Nwhose elements are these four notes.
Dialogue: 0,0:03:17.19,0:03:21.88,Default,,0000,0000,0000,,The operation we can perform on it\Nis to shift the bottom note to the top.
Dialogue: 0,0:03:21.88,0:03:24.36,Default,,0000,0000,0000,,In music that's called an inversion,
Dialogue: 0,0:03:24.36,0:03:27.25,Default,,0000,0000,0000,,and it's the equivalent \Nof addition from earlier.
Dialogue: 0,0:03:27.25,0:03:30.17,Default,,0000,0000,0000,,Each inversion changes \Nthe sound of the chord,
Dialogue: 0,0:03:30.17,0:03:33.90,Default,,0000,0000,0000,,but it never stops being\Na C diminished seventh.
Dialogue: 0,0:03:33.90,0:03:37.66,Default,,0000,0000,0000,,In other words, it satisfies axiom one.
Dialogue: 0,0:03:37.66,0:03:41.58,Default,,0000,0000,0000,,Composers use inversions to manipulate\Na sequence of chords
Dialogue: 0,0:03:41.58,0:03:51.33,Default,,0000,0000,0000,,and avoid a blocky, \Nawkward sounding progression.
Dialogue: 0,0:03:51.33,0:03:54.77,Default,,0000,0000,0000,,On a musical staff, \Nan inversion looks like this.
Dialogue: 0,0:03:54.77,0:03:59.99,Default,,0000,0000,0000,,But we can also overlay it onto our square\Nand get this.
Dialogue: 0,0:03:59.99,0:04:04.48,Default,,0000,0000,0000,,So, if you were to cover your entire\NRubik's Cube with notes
Dialogue: 0,0:04:04.48,0:04:09.54,Default,,0000,0000,0000,,such that every face of the solved cube\Nis a harmonious chord,
Dialogue: 0,0:04:09.54,0:04:13.10,Default,,0000,0000,0000,,you could express the solution\Nas a chord progression
Dialogue: 0,0:04:13.10,0:04:16.95,Default,,0000,0000,0000,,that gradually moves \Nfrom discordance to harmony
Dialogue: 0,0:04:16.95,0:04:20.58,Default,,0000,0000,0000,,and play the Rubik's Cube,\Nif that's your thing.