I dislike most of the usual physical “coincidence” examples, because many of them are really the result of human choices: historical conventions, unit definitions, or the order in which things happened to be discovered. They can be explained, and at most there is some interesting scientific history behind them. They are not deep mysteries.

The more interesting cases are different. They are unit-independent, or they connect parts of physics that seem as though they should have nothing to do with one another. These are the coincidences that feel less like accidents and more like hints that we are missing some deeper structure.

A basic example is the apparent fine-tuning of physical constants. Life as we know it seems to depend on certain dimensionless combinations of constants falling into fairly narrow ranges. The anthropic argument can be used here: perhaps there are many universes with different constants, and only in the rare ones where the constants allow stars, planets, chemistry, and observers would anyone be around to ask the question. But that is still speculative.

Even more interesting are cases where the anthropic argument does not obviously help, or at least does not feel like a complete explanation.

For example, the fine-structure constant is approximately 1/137. It is dimensionless, so this is not an artifact of units. Why does the strength of electromagnetism have that value?

Similarly, the ratio between the proton mass and the electron mass is approximately 1836. Again, this is dimensionless. It is not just a matter of choosing kilograms or electronvolts. The proton mass mostly comes from QCD binding energy, while the electron mass comes from the Higgs mechanism, so the ratio connects very different parts of physics. Why is it that number?

All the fundamental forces of the universe (granted, that's only 4) differ radically in how their strength varies over distance. Except, not gravity and electromagnetism. Their behavior changes in exactly the same way as distance changes, at least when we're talking distances large compared to the sizes of elementary particles (we don't know otherwise, it looks like it is the same over all distance scales but it may not be). Why?

Another way of stating that might be, all forces differ. But both electromagnetism and gravity, if they both have a force carrier, have a massless one. Why would all force carriers differ so much except for those 2? This emphasizes that it's not so much the inverse square law that's surprising but the match between the forces.

Another thing that's special about this match is that while all these coincidences are strange, this one has been measured to entirely absurd precision, and the match is absolutely exact as deep as we have ever measured. The other values are close, but not quite. This one is an absolutely exact match as far as we have ever measured. Both values for the mass of the force carriers are not "both really small but different", but "both exactly zero".

There is also the strange empirical relation among the electron, muon, and tau masses, known as the Koide relation. Roughly speaking, if you combine their masses in a particular way involving their square roots, you get almost exactly 2/3. It is suspiciously accurate and has never been convincingly explained.

Cosmology has similar mysteries. The average density of the universe is very close to the critical density: the value separating a spatially closed universe from a spatially open one. This is the flatness problem. Even more strangely, the matter density and dark-energy density are of the same order of magnitude today, even though they evolve very differently over cosmic time. This is the cosmic coincidence problem. Why should we happen to live at the time when the two are comparable?

Another famous example is the carbon resonance involved in the triple-alpha process. Stars produce carbon through a chain involving helium-4 nuclei, unstable beryllium-8, and an excited state of carbon-12 called the Hoyle state. The energy of this state is just right to greatly enhance carbon production. Without something like this resonance, the universe would have produced far less carbon, or produced it much later, and life as we know it would never have developed in the timeframe it has. Presumably this must follow from the strong nuclear force, but calculating the structure of a carbon-12 nucleus from first principles is so far beyond current capabilities ...

Then there is MOND. Whether or not MOND is a correct theory, it works surprisingly well at predicting galaxy rotation curves, with a particular constant. The striking thing is that the transition seems to occur at a universal acceleration scale of about 1.2 x 10^-10 meters per second squared. Why would there even be a single number at all? Why would that determine the falloff point in every galaxy for the dark matter cloud of that galaxy (the current scientific consensus) ? Why doesn't every galaxy have a different mass distribution? Simulations with our gravity laws allow for quite a big range that we never see anywhere in practice.

Even more suggestive is that this acceleration scale is close to c times H0 divided by 2 pi, where c is the speed of light and H0 is the Hubble constant. Since H0 measures the expansion rate of the universe, cH0 has units of acceleration. Why should the dynamics of stars inside galaxies care about the expansion speed of the universe? Is this just a coincidence now, or will it change as the universe gets older? (because the Hubble constant is predicted to change)

There is a related cosmological coincidence involving the cosmological constant. In natural units, the observed dark-energy density is roughly set by the Planck mass, which comes from gravity, and the Hubble scale, which comes from the size and expansion rate of the universe. Equivalently, the cosmological constant is roughly the inverse square of the Hubble radius. This links the largest observable scale in the universe with the scale set by gravity. Again, it is not obvious why these should be related.

Taken together, these coincidences suggest that our current theories may be missing something. These coincidences are dimensionless patterns, universal scales, and unexpected links between particle physics, nuclear physics, gravity, and cosmology.

In other words: they really evoke a "what are we missing here?" feeling. Because even if only one isn't a coincidence, we're missing quite a bit.

Pi squared is nearly 10