G2 shall be my favorite group! Well, for now. For a while. Or maybe for longer. It's the smallest of the exceptional Lie groups. Found 1893 by Élie Cartan. It was he who suggested to think about it in terms of rolling balls.
en.wikipedia.org/wiki/G2_%28ma

Picture two spheres, one three times larger than the other. Imagine them rolling on another without slipping nor twisting. Rolling surfaces have their own branch of mathematics: contact geometry!

en.wikipedia.org/wiki/Contact_

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Well, not quite: for one, we only want half of the big sphere. If the small one rolls off the rim, it magically reappears on the opposite side. That's a projective plane!

en.wikipedia.org/wiki/Real_pro

en.wikipedia.org/wiki/Projecti

The figure you get when you sew opposite points of the boundary of a hemisphere together is called a cross cap. Before you try that at home: it can't be embedded in 3-space without self-intersection.

en.wikipedia.org/wiki/Cross-ca

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