Life In 19x19

From set theory, my favourite result is probably the Banach-Tarski paradox.

Why?

It gives rise to this joke:
q: What's an anagram on the the Banach-Tarski paradox?
a: Click to show

Did someone watch University Challenge last night?

They asked about the Banach Tarski paradox on the show, or just told the joke? (I didn't watch it.)

Banach Tarski was the answer to a question about doubling the sphere, with the clue that it was named after two people; the team of classicists guessed "Smith Wilson".

My favorite theorem ... math, though nestled wholly within physics ... is Noether's theorem. Informally stated, it's that for any symmetry of a physical system, there is a corresponding conservation law. Translational invariance means conservation of momentum, rotational means conservation of angular momentum, invariance in translation through time means conservation of energy, which is a bit mind-blowing. It can be extended to QFT in some way my lack of understanding of which is even beyond me, but which gives a series of "guage symmetries" implying conservation of electric charge and other such quantities.

When I properly read about it, I became very angry at my old college tutors for not doing mechanics with generalized coordinates and the principle of least action in year 1 and QFT in year 3. It's like they told me I'd learned mountain climbing, then drove me up Pike's Peak, without suggesting I give Everest a try. I mean, I could have pursued it all myself, but I was a callow and foolish young man then.

I also have a little affection for the ABC conjecture, but only because Shinichi Mochizuki has apparently proved it by the method of reductio ad nauseam.

Uberdude wrote:Banach Tarski was the answer to a question about doubling the sphere, with the clue that it was named after two people; the team of classicists guessed "Smith Wilson".

I hope Paxman said "ball" rather than "sphere". It's untrue for a sphere - you need to include (some of) the interior.

Smith-Wilson is a technique for extrapolating yield curves.

drmwc wrote:I hope Paxman said "ball" rather than "sphere". It's untrue for a sphere - you need to include (some of) the interior.

Yes, my mistake, 19:30 in http://www.bbc.co.uk/iplayer/episode/b0 ... pisode_34/

I didn't think that was true? The proof of Banach-Tarski that I know does the decomposition for the spherical shell, then extends it to the ball by projecting the decomposition inwards.

I like the teorem that a continuous real function that assumes at least two values also assumes all values in between.
I like the teorem that every continuous bijection from a disk onto itself at least maps one point onto itself.
I like the rule that in projective 2D geometry every teorem remains true if the word "point" is replaced by "line" and vice versa.

e^{i * pi} + 1 = 0

Shaddy wrote:I didn't think that was true? The proof of Banach-Tarski that I know does the decomposition for the spherical shell, then extends it to the ball by projecting the decomposition inwards.

You're right - I was typing rubbish.