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Physics + MathPhysics & Math

Symmetry: How Beautiful Math Makes Elegant Physics

ByDon LincolnThe Nature of RealityThe Nature of Reality

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What does symmetry mean to physicists?

There is geometrical symmetry. The human body, for instance, has one kind of geometrical symmetry: The left and right side of our bodies are pretty much the same. A typical starfish has a five-fold symmetry, meaning that if you rotate its body through 1/5 of a circle, it looks like it wasn’t rotated at all. A circle has even deeper symmetry: No matter how much you rotate it, the rotated circle looks just like the old one.

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Symmetry: How Beautiful Math Makes Elegant Physics-symmetry_500.jpg

There are many kinds of symmetries, from the left-right symmetry of a face (left), to the five-fold symmetry of a starfish (center), to the complete symmetry of a circle (right). These sorts of symmetries are also seen in physics theories, leading some physicists to describe the equations as beautiful.

Symmetry also has an aesthetic meaning, although this is harder to define; artistic symmetry is beauty found in a pleasing and regular form.

Both of these definitions of symmetry have some place in the meaning used by physicists. Equations are geometrically symmetric if they can be “flip flopped” without changing their meaning. For instance, take the simple sum 3 + 4. If we swap the order, we get 4 + 3. Both of these equations equal 7 and we can thus say that addition is symmetric in this case. Of course, not all equations are symmetric when the order is swapped. For example, in subtraction 4 – 3 isn’t the same as 3 – 4.

These simple symmetries give us an insight into more complex symmetries. These more complex symmetries have a huge impact on theoretical physics. To understand how that is true, we must turn to a physicist who may not be a household name, but should be.

Emmy Noether has been called the most influential woman in mathematics. In an era when women were often expressly forbidden from the academic world, she won the highest respect of leading scientists and mathematicians, including Albert Einstein and David Hilbert.

Before Noether, scientists noticed that certain things, like energy and electrical charge, were “conserved.” That is, the amount of energy in a system is the same before and after an event like a collision. Similarly, electrical charge might move around, but the total charge remains the same. (Note that this only works in “closed” systems, which aren’t gaining or losing energy or charge to external sources.) Exactly why these things were conserved wasn’t understood, but these conservation laws were (and are) taught in all introductory physics classes.

Noether connected these conservation laws with mathematical symmetries that could be expressed in equations. She saw that each symmetry implied a physically conserved quantity. If an equation was unchanged if you swapped it from one point in time to a different point in time, this meant that energy was conserved. If an equation was unchanged if you changed a position with a different position, momentum was conserved.

This observation was a brilliant revelation. Conservation laws weren’t an unexplained phenomenon. They were the measurable manifestation of symmetries in the laws governing the universe. The beauty of the cosmos was the beauty of symmetry.

Noether’s theorem led theoretical physicists to explore the idea of symmetry in natural law more fully, leading to a deeper appreciation of the role of symmetry in the rules that govern the cosmos. Now the symmetry of a particular theory is among the first things physicists consider as they evaluate its merit.

If you talk to a physicist—especially a theoretical physicist—about modern theories and why they are the way they are, the scientist may well wax lyrical about the beauty and simplicity of the equations. Symmetry is the basis for this aesthetic judgment. You need not be a physicist to see the beauty of the stars glittering in a dark midnight sky, the allure of a shimmering rainbow and the delicacy of a snowflake, yet they, too, are inscribed in the symmetry of written formulas, there for all to see—once you know how.

Go Deeper
Author’s picks for further reading

Deep Down Things: The Breathtaking Beauty of Particle Physics
In this book, experimental particle physicist Bruce Schumm connects abstract mathematics with the elegance of the Standard Model of physics.

The Particle at the End of the Universe
Physicist Sean Carroll goes inside the hunt for the Higgs boson.