Pi squared is nearly 10 | Hacker News Reader

Pi squared is nearly 10

mihai.page

61 pointsfreediver54 comments

I like the 4-5-6 theorem:

    pi^4 + pi^5 = e^6

Well, to five decimal places, anyway. Some other good ones:

    e^pi - pi = 20

    sqrt(2) ln pi = phi

There are also famous "almost integers" such as this one discovered by Ramanujan:

    e^(pi sqrt(163))

Which is an integer to 12 decimal places.

Edit: I just remembered I have public JupyterLite notebooks for both of these:

https://notebooks.oranlooney.com/lab/index.html?path=fake_ma...

https://notebooks.oranlooney.com/lab/index.html?path=heegner...

> Which is an integer to 12 decimal places

this isn't something I was expecting to read today. I guess this works with weak types? /s

That's why the time is Almost<T> instead of just T.

(e^pi - pi)/pi^4 ~= i^i

the Ramanujan one has some relatively high powered mathematical explanation

https://en.wikipedia.org/wiki/Heegner_number

Wikipedia also notes that “Ramanjuan’s constant” was actually discovered by Charles Hermite in 1859 and it was a 1975 April Fools article in Scientific American that attributed it to Ramanujan.

Stigler's Law of Eponymy strikes again!

https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy

The second fact, pi^2 ~= g, is famous enough that it has a separate section in Wikipedia [1].

[1] https://en.wikipedia.org/wiki/Mathematical_coincidence#Gravi...

And https://xkcd.com/1047/

Also https://xkcd.com/217/

I don’t see pi^2 ~= g in there… did I miss it somewhere?

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.

> Similarly, the ratio between the proton mass and the electron mass is approximately 1836

https://en.wikipedia.org/wiki/Proton-to-electron_mass_ratio says it is

  1836.152673426(32)

So, it differs about 0.153 from its nearest integer. Over 30% of all reals are that close or closer to their nearest integer, so I wouldn’t call that extraordinary at all.

For the fine structure constant, that’s about

  1/137.035999177

Closer, but still not extraordinary. The math is different but I think it’s about a 1:15 chance that a random real will be relatively that close to the reciprocal of an integer.

Ironically, xkcd is quasi-referenced in that article:

As discovered by Randall Munroe, a cubic mile is close to 4/3 π cubic kilometres (within 0.5%). This means that a sphere with radius n kilometres has almost exactly the same volume as a cube with side length n miles.

More advanced slide rules typically have a set of “folded” scales, that can sometimes save a calculation from ending up off scale. In theory, these should be offset by half the scale length, i.e. sqrt(10). However, since the folded scales also offer a convenient way to multiply with the offset factor, most slide rules offset them by π instead, since it’s almost the same as sqrt(10), and multiplication by π is a more useful thing to have around.

My first thought was "well of course it is, since pi is a little larger than 3" but it was cool to see an actual derivation of how much pi squared differs from 10 as a nice, closed form series.

As an ex-physicist, pi^2 is 10. Like g.

I get it that this is a nice calculation with the Zeta function and everything, but 3 and a small something squared will be near 10 so it is 10.

My high school physics teacher made a point of doing math on the size of the moon with pi=3. Close enough.

I remember discovering that pi x 10^7 is very close to the number of seconds in a year while at uni.

One of my tutors was convinced this had to be more than coincidence, but I always figured it was just chance and a nice but sometimes useful shortcut...

You might be able to send someone down an amusing (to observers) rabbit hole of wrongness by telling them it is not exact because Earth’s orbit is not perfectly circular.

You're such an evil person :D

Hah, that would be hilarious

Get enough numbers, accept wide error bars, and some of them are going to overlap.

It cannot be anything but coincidence. While 365.25 days in a year is physics, a day consisting of 86,400 seconds is an entirely arbitrary human construct.

I always liked the fact that 10! (10 factorial) is exactly the number of seconds in six weeks.

  6 weeks * 7 days * 24 hours * 60 minutes * 60 seconds:

    6 * 7 * 24 * 60 * 60
  = 6 * 7 * (3 * 8) * (4 * 5 * 3) * (3 * 2 * 10)
  = 6 * 7 * 3 * 8 * 4 * 5 * (3 * 3) * 2 * 10
  = 6 * 7 * 3 * 8 * 4 * 5 * 9 * 2 * 10
  = 1 * 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9 * 10

and the Earth's orbital diameter is 10^3 light seconds

so the Earth's orbital velocity is 10^-4 c

This first became apparent to me when I got a slide rule. Pi is often marked on the various scales and an x^2 scale is often nearby the x scale.

The author wants tau=2*pi, but in the Greek alphabet, tau has one vertical stroke, and pi has two.

So, visually in Greek, pi=2*tau would seem an improvement.

Oh, well.

Tau is tau over 1, pi is tau over 2. See also https://www.tauday.com/tau-manifesto#sec-conflict_and_resist...

pi's prevalence instead of tau is one of the strongest indicators that we live in a suboptimal timeline.

Then, convert the digits of pi to text to find how to achieve interdimensional travel to reach the optimal timeline.

Millions of years from now, a far off alien race will discover the remnants of Earth, go through our maths knowledge, and they will slap their foreheads because we chose pi rather than tau.

They’ll also wonder why humanity has so many references to a video of a young orange haired guy singing a song.

Only reasonable conclusion is that he is worshiped as a deity …

After all, He is never gonna give up…

> he is worshiped as a deity

That actually made me lol irl!!

> In the US and countries with a similar date format

Humm that's like 2 or 3 countries?

6! is the number of minutes in 12 hours and the number of hours in a 30-day month.

I was a little disappointed that the upper range of gravity on earth only goes to 9.8337. Just a little more and there would have been somewhere on earth that was an exact match.

It would have been the ideal (if chilly) place to start a cult.

The initial definition of the meter would have been such that g=pi^2. It was then adjusted but enough for it to remain an interesting (if not too good) approximation.

Couldn't they have just done the adjustment by piling a lot of dirt in one place.

Quite a lot.

Pi^0 is exactly 1.

You could be on something there.

I also noticed that e^(i*pi) gives an integer exactly.

Yea, Euler’s identity is probably the most elegant relationship between e and pi you’re going to find, maybe the most elegant formula in mathematics!

If you don't unblock scripts from cdn.jsdelivr.net.cdn.cloudflare.net, the math code won't work.

need a countdown for when it gets there

Also number of McDonald's in the world divided by number of McDonald's in US is close to pi. Within 1%.

Source?

BBQ

This does not seem to be true currently. In 2025, there were 45356 McDonald's restaurants worldwide and 13706 in the United States, which is about 3.3092.

https://corporate.mcdonalds.com/content/dam/sites/corp/nfl/p...

However, in 2023, the numbers were 41822 / 13457 = 3.1078, which is (almost) within 1 % difference.

https://corporate.mcdonalds.com/content/dam/sites/corp/nfl/p...

That's because the grow like rivers

pi^2 ~ 10, well known to anyone who used slide rules.

at this rate, pi square is close to 'g'

987654321 / 123456789 = 8 (to the 7th decimal place) is another nice one

It's very close to 123456 * 8 + 6 = 987654 and the rest.

I just wrote an article on that (and the fact that this submission on HN exists, I was not expecting that)

https://mihai.page/ramblings-on-a-math-pattern/

Reply on news.ycombinator.com