So ... how was it???
Quite good but not because it offered any particularly interesting insight into the Rubik's cube or Twisty Puzzles in general.
The lecture was geared towards a very casual audience and was very light on math, theory, and technical terminology.I will try to summarize his lecture based on my crappy memory:Introduction to symmetry:
The lecture was all about "interesting" symmetry. He started out presenting the basic idea of symmetry using the Taj Mahal as an introduction. The building is mirror-symmetrical and has hexagonal tilings and rhombic tilings based on an octahedral symmetry.
Once he'd introduced the concept, his lecture was divided into three examples of symmetry. The Rubik's Cube, MC Escher's work (Prentententoonstelling and Circle limit 1), and then how symmetry on an elliptic curve can be used in cryptography.
He mentioned that operations on a group were associative in that (AB)C is the same thing as A(BC) but that on some things the order matters so AB isn't the same as BA. He didn't mention anything about this being non-commutative.Rubik's Cube:
For the Rubik's Cube, he showed the core of a cube stickered with the mirror of the standard color scheme and a normal color scheme and mentioned offhand that there was no way to go from one to the other. Later in a question from the audience he mentioned that mirror symmetry in N dimensions can be achieved by a rotation in N + 1 dimensions (so you can mirror a Rubik's cube via a rotation in the 4th dimension).
He showed the number of permutations to be (12! * 8! * 2^12 * 3^8) / 12 and said that figuring out that there should be a factor of 12 in the denominator was related to the symmetry of the puzzle and beyond the scope of the lecture. He said that if you used a screwdriver to take apart the puzzle and put it together you'd have a 1/12th chance of it being solvable. (For the record, the factor of 12 is a factor of 2 for the linking of the parity between the corners and edges, a factor of 2 for the twist restriction on the edges, and a factor of 3 for the twist restriction on the corners. He did not explain how the symmetry of the cube and face-turning operations actually enforces these restrictions.)
He also mentioned that the number of positions has nothing to do with the difficulty and used the problem of ordering a random permutation of the alphabet which has 26! positions (many more positions than the Rubik's Cube) as an example.
He then showed a [[1:1],1] commutator: [R, U', R', D', R, U, R', D] (which is [[R:U'],D']).
He mentioned that R U' R' can be called X and D' can be called Y and that because they overlap by a single piece X Y X' Y' would always create a 3-cycle. He did not use the word commutator and did not explain why this construction always creates a 3-cycle. I think the audience had a very hard time following this part of the lecture because his demonstration of the sequence was with static images. An animation would have made it much clearer.
Later an audience member asked if the symmetry discussion for the Rubik's Cube could be extended into higher dimensions. Brian Conrad answered in a roundabout manner but was basically saying "yes". It isn't clear if he was aware of MC4D
or not.MC Escher:
This part of the lecture was really cool but a bit light on the technical ins and outs of symmetry. He showed Circle Limit 1 and discussed the tiling and some aspects of the hyperbolic symmetry:
300px-Hyp-circle-limit-i-tess.png [ 200.02 KiB | Viewed 1984 times ]
He then discussed the strange spiral symmetry of Prentententoonstelling:
escher_prentententoonstelling_orig.jpg [ 147.86 KiB | Viewed 1984 times ]
And then discussed the difficulty of filling in the center and outlined the general idea of how that was done and then showed this:
escher_prentententoonstelling_2.gif [ 7.05 MiB | Viewed 1984 times ]
First he talked about how two people could share a secret in a very simplified Diffie-Hellman way using 2^a = Avalue and 2^b = Bvalue and that 2^ab == 2^ba.
The basic idea is that Alice can publish her 2^a value and call it the "public key", keeping a secret. Bob can do the same with his 2^b value, keeping b secret.
Then they can both compute 2^ab or 2^ba (the same thing) without knowing each other's a and b values because:
(2^a)^b is the same thing as (2^b)^a and they know each other's Avalue / Bvalue without knowing each other's a and b.
He points out that obviously this isn't secure because log2(Avalue) = a and logs are easy. He doesn't mention that in the real world, you do all of this math in a finite field (mod p) and that logarithms in a finite field are "hard".
He then goes on to talk about elliptic curves and how if you pick any two points on an elliptic curve and draw a line through them, the line will cross the curve in a third point. If you mirror this third point across the X (horizontal) axis then you can draw another line, and so on.
He describes this line, mirror, line, mirror... as addition and if you do it N times then its like multiplication by N. He states that because elliptic curves don't have an analogue of division, you can multiply on this curve and inversion via division is hard (because division doesn't exist).
Here is an example:
curveplot.png [ 10.61 KiB | Viewed 1984 times ]
Where the two points, A and B line drawn through and they cross the curve on the upper right point C. By mirroring that point (follow the dotted line) you have a 4th point C'. Then you can draw a line through B and C', get a 5th point, and so on.
Overall the lecture was quite interesting even though it was very light on any technical details of the topics discussed. It would have been great if he'd mentioned Magic Tile
by Roice Nelson with supports many different hyperbolic twisty puzzles: http://www.gravitation3d.com/magictile/