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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Take another copy of SO(3) also looking like the inside of a ball, and glue that onto our original along their spherical boundaries. Leaving a ball turns you over, but reentering turns you upright again!

No matter how you cross Spin(3) you never get back to your starting point turned upside down. The twist is gone! This pair of hemiballs is called 3-sphere or S³. Psst, this object actually lives in ! Now you know a bit more about how that works!

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