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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If earth had exactly 365 noons a year, how much would it have turned with respect to the stars? Earth's own rotation is in the same direction as it's turning around the sun. Most of the solar system is prograde like that: everything rotates forwards.

In that case, and also in fact, noons add up with the full rotation around the sun. So the 'sidereal year' has one extra rise of your zodiac or your favorite star.

en.wikipedia.org/wiki/Sidereal

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