Beautiful symmetries

While architects and artists have long known of the beauty of symmetry, as demonstrated in the breathtaking Taj Mahal, it was mathematician Emmy Noether who taught physicists to appreciate the beauty and power of symmetries in our equations.

Symmetry has several ordinary meanings. It is often taken to mean aesthetic beauty in a pleasing and regular form. It also has a technical meaning. This meaning is often geometrical in nature, as in the five-way symmetry of a starfish or how the left and right sides of a person’s face are nearly identical. If you look at both a person’s face and his reflection in a mirror, which swaps left and right, it’s difficult to know which one is the reflection—it looks much like the original.

In contrast, if you flip a person upside down, thereby swapping feet and head, it is easy to tell the flipped version from the original. The essence of geometrical symmetry is that when you make a change, the change isn’t obvious.

Emmy Noether was a brilliant early-20th-century mathematician who made some interesting discoveries about symmetries in equations. To give a simple idea of what that means, consider the sum 1+2, which, of course, equals 3. If we swap the order and write 2+1, the sum is unchanged. This equation is symmetrical under reversal of the order.

Studying far more complex symmetries, Noether was able to link mathematical symmetries with conservation laws. Conservation laws describe things that don’t change. One conservation law you usually learn about in science class is the conservation of mass. If you take three buckets of sand, weigh the sand, and then dump the bucketfuls into a bigger bucket, the weight of the sand in the bigger bucket is just the sum of the weights of the sand in the three smaller buckets. The sand’s mass doesn’t go anywhere. (Einstein showed us that this conservation law isn’t absolute, but it works well in ordinary life.)

Noether showed the following: If, in a physical equation, you replace the term for time with a different term for time—one that is offset from the first by some amount—and the altered equation makes the same prediction as the original, then energy must be conserved. That’s a tricky idea, but you can think about it this way: If I define Monday as the start of the week (my “day 0”) and you define Tuesday as the start of the week (your “day 0”), then we will have a one-day offset between the numbers we assign to a particular day. I’d call Wednesday “day 2” while you’d call it “day 1.” If the equations don’t care about this difference in day numbers and make the same predictions, then energy is conserved.

There are many other similar symmetries in nature: If it doesn’t matter what location you call “position 0,” then momentum is conserved. If it doesn’t matter what direction you call “direction 0,” then angular momentum is conserved. Noether’s theorem says that every symmetry is associated with a conservation law and vice versa. The math leads to observable physical phenomena. With this observation, one of the first things a physicist will study when confronted by a new theory is its symmetries.

Don Lincoln

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