Torsion in the Twists (Belt/Plate Trick)

This is a follow-up post to the statement at the end of Everyday I'm $\mathbb{RP}^{n+1}$, that $\pi_1\left(\mathbb{RP}^3\right) \cong \mathbb{Z}_2$

$\mathbb{Z}_2$...  Isn't that the finite simple group of order two?

It was only a matter of time before I found an excuse to post this...

Anyway, have some links:

A quick video of the corresponding plate trick* 

Brief discussions here and here of the belt/plate trick, courtesy Wikipedia.
Some nice demonstrations (and also discussions) of the tricks, courtesy Bob Palais.
A quick video of the much-related quaternion handshake**

*In the plate trick, a path in $SO(3)$ corresponding to a $720^\circ$ rotation is traced out over time by your hand, and the homotopy to a $0^\circ$ rotation is traced out 'spatially' as you move up/down your arm.  These roles are reversed in the belt trick, where the path is traced out spatially and the homotopy occurs over time.

**$\mathbb{RP}^3$ is a quotient of $S^3 \subset \mathbb{R}^4$.  The quaternions are isomorphic to $\mathbb{R}^4$ as an $\mathbb{R}$-vector space.
So it shouldn't be too surprising that $SO(3)$ is a quotient of the unit quaternions.

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.

Pony Break

Mmk time to break up the monotony.  With the Apple family!

Great mixed drink for the winter:  Apple Cider and Cinnamon Schnapps (I tend to use Goldschläger).  Add to taste.

Don't forget the new episode! (taken down)

Applebloom goes on an 8-bit adventure

Stay thirsty, my ponies.