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Professor of Quantum Mechanical Engineering, MIT; Author, Programming the Universe

The True Rotational Symmetry of Space

The following deep, elegant, and beautiful explanation of the true rotational symmetry of space comes from the late Sidney Coleman, as presented to his graduate physics class at Harvard. This explanation takes the form of a physical act that you will perform yourself. Although elegant, this explanation is verbally awkward to explain, and physically awkward to perform. It may need to be practised a few times. So limber up and get ready: you are about to experience in a deep and personal way the true rotational symmetry of space!

At bottom, the laws of physics are based on symmetries, and the rotational symmetry of space is one of the most profound of these symmetries. The most rotationally symmetric object is a sphere. So take a sphere such as a soccer—or basket-ball that has a mark, logo, or unique lettering at some spot on the sphere. Rotate the sphere about any axis: the rotational symmetry of space implies that the shape of the sphere is invariant under rotation. In addition, if there is a mark on the sphere, then when you rotate the sphere by three hundred and sixty degrees, the mark returns to its initial position. Go ahead. Try it. Hold the ball in both hands and rotate it by three hundred and sixty degrees until the mark returns.

That's not so awkward, you may say. But that's because you have not yet demonstrated the true rotational symmetry of space. To demonstrate this symmetry requires fancier moves. Now hold the ball cupped in one hand, palm facing up. Your goal is to rotate the sphere while always keeping your palm facing up. This is trickier, but if Michael Jordan can do it, so can you.

The steps are as follows:

Keeping your palm facing up, rotate the ball inward towards your body. At ninety degrees—one quarter of a full rotation—the ball is comfortably tucked under your arm.

Keep on rotating in the same direction, palm facing up. At one hundred and eighty degrees—half a rotation—your arm sticks out in back of your body to keep the ball cupped in your palm.

As you keep rotating to two hundred and seventy degrees—three quarters of a rotation—in order to maintain your palm facing up, your arm sticks awkwardly out to the side, ball precariously perched on top.

At this point, you may feel that it is impossible to rotate the last ninety degrees to complete one full rotation. If you try, however, you will find that you can continue rotating the ball keeping your palm up by raising your upper arm and bending your elbow so that your forearm sticks straight forward. The ball has now rotated by three hundred and sixty degrees—one full rotation. If you've done everything right, however, your arm should be crooked in a maximally painful and awkward position.

To relieve the pain, continue rotating by an additional ninety degrees to one and a quarter turns, palm up all the time. The ball should now be hovering over your head, and the painful tension in your shoulder should be somewhat lessened.

Finally, like a waiter presenting a tray containing the pi'ece de resistance, continue the motion for the final three quarters of a turn, ending with the ball and your arm—what a relief—back in its original position.

If you have managed to perform these steps correctly and without personal damage, you will find that the trajectory of the ball has traced out a kind of twisty figure eight or infinity sign in space, and has rotated around not once but twice. The true symmetry of space is not rotation by three hundred and sixty degrees, but by seven hundred and twenty degrees!

Although this excercise might seem no more than some fancy and painful basketball move, the fact that the true symmetry of space is rotation not once but twice has profound consequences for the nature of the physical world at its most microscopic level. It implies that 'balls' such as electrons, attached to a distant point by a flexible and deformable 'strings,' such as magnetic field lines, must be rotated around twice to return to their original configuration. Digging deeper, the two-fold rotational nature of spherical symmetry implies that two electrons, both spinning in the same direction, cannot be placed in the same place at the same time. This exclusion principle in turn underlies the stability of matter. If the true symmetry of space were rotating around only once, then all the atoms of your body would collapse into nothingness in a tiny fraction of a second. Fortunately, however, the true symmetry of space consists of rotating around twice, and your atoms are stable, a fact that should console you as you ice your shoulder.