Thought Experiments


Symmetry: How Beautiful Math Makes Elegant Physics

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.


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.

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Don Lincoln

    Don Lincoln is a senior experimental particle physicist at Fermi National Accelerator Laboratory and an adjunct professor at the University of Notre Dame. He splits his research time between Fermilab and the CERN laboratory, just outside Geneva, Switzerland. He has coauthored more than 500 scientific papers on subjects from microscopic black holes and extra dimensions to the elusive Higgs boson. When Don isn’t doing physics research, he spends his time sharing the fantastic world of science with anyone who will listen. He has given public lectures on three continents and has authored many magazine articles, YouTube videos and columns in the online periodical Fermilab Today. His book "The Quantum Frontier" tells the tale of the Large Hadron Collider, the world’s preeminent particle accelerator, while his other book "Understanding the Universe" introduces the armchair scientist to particle physics and cosmology and tells how the two fields are intertwined.

    • Jake in Bama

      How do we know that this mathematical symmetry isn’t a blind spot? Isn’t something that biases how we create and interpret experiments? Lead us to only look for symmetric things in data sets?

      Socially at least, beauty and our desire for the peace it gives us often keep us from seeing the ugliness and misery in society.

      • Anonymous

        So, Jake, how are things in Bama? I assume you live there. My recollection of the state is that it is the apex of the ugliness and misery in society.

    • Alex K

      I know this is beside the point, but is there an RSS feed for this blog (and this blog alone)?

    • Rosy Mota

      but neithe beuatiful is in the “symmetry”.the symmetries more pretty
      are linked to asymmetries,that correspond the not equivalence between the left-right systems and others that permit fields with prefered directions.the asymmetry generate the mathematical beautiful,and the hidden reality of the spacetime that is our pure consciouness,or better the 4-dimension ís a symmetry reality given by the time that curve the space generating spacetime continuos that are
      mental operations as the discreteness spectrum of quantum quasntum and gives origin of movement

      • Paul


    • Jason

      Thanks! I appreciate the way you put it for us laymen…I think the universe is nothing but a big mathematical equations, filled with symmetrical values still waiting to be discovered.

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