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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That extra turn is just what we need for our pair of spheres. The small ball will, as you may have guessed, unwind itself three times onto the large one, while turning four times around itself!

Well okay, we only have a projective sphere, whqich can only accomodate half a revolution, yielding two turns of the little ball. We need two because the little ball is 'spinorial'.

en.wikipedia.org/wiki/Spinor

en.wikipedia.org/wiki/Spin_gro

and

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