Everyday I'm $\mathbb{RP}^{n+1}$

When I was just a little boy and my dad was tucking me into bed one night, we (somehow) got to talking about the overall shape of the universe.  It just sort of goes on forever... doesn't it?  According to my old man, there were some people who thought that space didn't exactly go on forever.  "When you travel far enough in that direction," he said, pointing to one corner of the room, "you eventually come back to the place you started, but you'll approach it from there," he said as he pointed to the opposite corner.

At the time,  I had imagined that he was describing something like a 3-torus, i.e. $S^1 \times S^1 \times S^1$.  Nowadays, I'm certain he was talking about $\mathbb{RP}^3$ instead.

Hrm... I should ask him about this stuff the next time I see him...

Pinkie Pie's parents must have taught her how to appreciate abstract notions of space as well.

Today we'll construct $\mathbb{RP}^3$ and see why it's homeomorphic to $SO(3)$, a subset of $O(3)$.

Don't fret if you don't know what it means for two spaces to be 'homeomorphic'.  We're just going to visualize each one and convince ourselves that they're both the same shape.

Smale's Paradox

Just stumbled across a nice demonstration of this eversion on youtube:

Part 1  |  Part 2

One of the members of Smale's group, Bernard Morin, is notable for being an accomplished topologist in spite of being blind since the age of six.  He was also the first person to parametrize Boy's Surface, an immersion of the real projective plane in 3-space.  Here's a model of Boy's surface at a mathematical research institute in Oberwolfach, Germany:

Morin, who is still alive, was shown this model one day.  Being blind, he had to touch the surface to determine what it was.  After a minute or two of feeling around, he shook his head and said, "This is not Boy's Surface.  This is a mirror image of Boy's Surface."  Morin was, apparently, the first person to point this out.

So with that, think of all the new perspectives you might get if you blindfolded yourself for a month. Vision being a crutch for your spatial imagination and all that.

Story source: the guy who taught me Algebraic Topology.

Discord in the Axioms

You've all heard about Gödel's Second Incompleteness theorem, right?  If a theory with a finite list of axioms contains the basis for arithmetic, this theory cannot produce statements about its own consistency without contradicting itself.  So we can't ever prove such a theory's consistency; we can only take solace in the notion that we haven't found any contradictions yet.  Kind of a bummer.

Way back in September, the distinguished Ed Nelson, a math professor at Princeton, claimed to have found an inconsistency in Peano Axions--one of the most widely used formalizations of arithmetic--and distributed a sketch of his proof through a math mailing list.  It's hard to get more basic than this: these are axioms that establish the existence of zero, all the common rules for how the equals sign works, your ability to add one to a number that already exists, and (Nelson thought this last one was the culprit) the validity of proof-by-induction.  There might be some question about whether certain bits of modern set theory are inconsistent, but arithmetic?  Nelson was really going for the balls here.