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_

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

We can make this cross cap go away by picturing two small balls instead, moving in lock step, one always opposite the other. When one rolls 180° from pole to pole, how much has it turned? Measured with respect to the stars, not the big one's center!

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

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

My plan to obtain G2 has secretly and all along been to combine a spinor with a projective plane! The latter is a half-sphere made seamless, and a spinor is the opposite: a double cover of SO(n)!

en.wikipedia.org/wiki/Orthogon

A spinor is like an electron! If you turn an electron by 360° it's orientation somehow becomes negative. Only when you turn it another full round its orientation returns to the starting position.

Attach a ribbon to your electron, and to the ground. As a reminder of the EM field that connects the electron to the universe. Turn it once, and you get a twist. Twice and... well, did you know: you can undo a doubly twisted ribbon by itself!

Dirac's belt trick shows how to undo double twists without moving the electron. Or the ground. It also works with many ribbons at once! And no matter what the rotation axis was! That's a spinor! You need two full turns to get the identity.

en.wikipedia.org/wiki/Plate_tr

Puzzle: Starting at a pole, roll an electron around our 3 times bigger ball. After a quarter turn its orientation became the negative –s of our starting orientation s. Can we get back to s by rolling along the equator?

Spinors are double covers for rotation groups. Double covering sounds like it would become more complicated. But on the contrary, it really makes things simpler: Take SO(3): the space of rotations in 3-space! Pick an axis of a sphere: a pair of antipodal points. We can then rotate around that axis. The result is a closed 3-space.

en.wikipedia.org/wiki/3D_rotat

It looks like a ball where, if you try to leave through the wall, you reappear on the opposite side, turned upside down! Not mirrored, turned.

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:

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!

So, let's solve the puzzle: Start at the center s of our, say, white hemiball. After a quarter turn we get to -s the center of the black hemiball. Now all directions point back at s! Any other quarter turn leads back!

This spinor rolling on a projective plane, where the latter has three times the spinor's radius, models the real valued lie algebra g2. However, this also works for complex values. This is what Cartan originally described.

Fun fact about G2: it's 5-dimensional, but the little ball has only two dimensions of infinitesimal freedom! You could get to any 5d point by navigating as if you were walking on a 2d surface! Orienting a ball by rolling it on the ground is similar...

Also funny: since the big ball isn't really a ball, but a projective plane, that that also played a role in my construction of SO(3) above. No rotation group corresponds to RP² it somehow seems to sit between SO(2) and SO(3)!

I'm indebted to @johncarlosbaez for his talk about the relation of G2 to the octonions, from which I learned all I know about G2. Find out more here:

math.ucr.edu/home/baez/ball/in

Thanks, John!

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