Koide’s Formula

Finding a beautiful and simple equation for something in the natural world is fascinating to me – it’s like picking at a corner of loose wallpaper in your room and suddenly seeing the scrolling green text of the Matrix on the wall behind it. Often these relations lead to a deeper understanding, but sometimes an indisputably true and simple relation will remain maddeningly confounding.

In 1981 Yoshio Koide was researching leptons, a family of fundamental particles that includes the familiar electron. There are three leptons which are “charge carriers” (they have mass) – the electron, the muon, and the tauon.

Koide was wondering if there was a way to relate the masses of these three particles with one another. He developed the following equation (related to the eigenvectors of the democratic matrix, here’s a review paper if you want more detail):


Nothing too wild mathematically here. If we assume our three lepton masses are positive (pretty reasonable) then the value of Q can range from 1/3 (all the masses are the same) to 1 (the masses vary wildly from each other). So what is the value of Q? Well, when Koide first proposed this equation, the masses of the leptons were thought to be as follows:

  • Electron: 0.511 MeV/c2
  • Muon: 105.658 MeV/c2
  • Tauon: 1,784.2 MeV/c2

If we plug these values into Koide’s equation, we get a value of 0.667074 – incredibly close to 2/3, which would be precisely halfway between our upper (1) and lower (1/3) bounds we figured out before! This seems like a ridiculous coincidence.

Things like this make you wonder… well, is it exactly 2/3? Or is it just “kind of” close? The mass of the electron and the muon had been measured to a rather high level of accuracy, but the accuracy of the tauon measurements had been lagging behind due to the higher energies required. Perhaps the measurement of the tauon was wrong! It’s a hell of a hunch – but let’s go with it. Assume that the tauon mass has been measured incorrectly, we can set Q = 2/3, input the masses of the electron and muon, and see what the tauon mass “should” be. It turns out that Koide’s equation says the mass of the tauon “should” be 1777 MeV/c2.

Well that’s wonderful, but nature doesn’t seem to care how you think it “should” behave. The only test was to wait for more accurate measurements of the tauon mass and see if this was a neat coincidence based on measurement error or whether there may be something more interesting going on. The mass of the tauon was later revised with better measurements, and… drumroll…

Old Measurement Koide’s Prediction New Measurement
1,784.2 MeV/c2 1,777 MeV/c2 1,776.9 MeV/c2

Whoa. Our simple little equation, using nothing more than grade school arithmetic, has accurately predicted the mass of a fundamental physical particle years in advance of having this measurement confirmed by the best research labs on earth.

And now the question becomes why – why does this work at all? We have three seemingly random lepton masses, measurements of the most complicated physical system we know – our universe. We then input them into a ridiculously simple equation, and the most ridiculously simple answer pops out.

We can gain a tiny bit of insight by figuring out what exactly this equation is telling us.


Basically, we can calculate Q for a given set of three lepton masses. This Q will tell us where a three-dimensional vector specified by the square roots of our three lepton masses will end up.

Q = 1/3 The set of all vectors that form an angle of zero with the unit vector (multiples of the unit vector).
Q = 2/3 The cone seen above which fits perfectly into the “corner” created by our three axes. The set of all vectors that form an angle of pi/4 with the unit vector.
Q = 1 The set of vectors that form an angle of zero with our basis vectors. These vectors lie along one of our three axes.

So it appears that our lepton masses have been chosen in some magical manner as to fall perfectly in the middle of these two extremes. The concept appeals to our perception of the universe as a finely tuned apparatus, but gets us nowhere closer to an interpretation based in physical reality.

It’s a maddening equation. Beautiful. Simple. True. And no one knows what the hell to do with it.

3 thoughts on “Koide’s Formula

  1. Good morning man! I ended up here by Googling how to make your own memory foam bed, and this article caught my eye. I’m no mathematician nor physicist, but just an entrepreneur with a high-school education who craves nerdy stuff like this now that I’m old enough to realize that learning is actually awesome. Should’ve done that in Grade 12 perhaps. šŸ™‚ Anyway…just wanted to share that I have huge respect for those of us who get a kick out of nerdy things like this, especially when nature and math line up, like Fibonacci’s Sequence does with so many organic things in nature. Awesome. Thanks for the rant…it was great.

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