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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Picking a pair of antipodal points alone leads to a projective plane RP(2). As flipping the axis can be compensated by rotating in the opposite direction.

When you fibrate SO(3) with RP² as base space like this, you get full 360° circles as fibers. Think of them as the diameters of the ball.

en.wikipedia.org/wiki/Fibratio

en.wikipedia.org/wiki/Fiber_bu

Now let's double cover that beast:

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