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.
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.
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, that does not apply to gravity and electromagnetism. They fall off exactly the same. Why would that happen?
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 it's not even on the radar.
Then there is MOND. Whether or not MOND is a correct theory, it works surprisingly well at predicting galaxy rotation curves. 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) ?
Even more suggestive is that this acceleration scale is close to c tim...
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.
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.
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 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.
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.
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.
51 comments
[ 3.4 ms ] story [ 46.4 ms ] threadSo, visually in Greek, pi=2*tau would seem an improvement.
Oh, well.
Only reasonable conclusion is that he is worshiped as a deity …
After all, He is never gonna give up…
[1] https://en.wikipedia.org/wiki/Mathematical_coincidence#Gravi...
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, that does not apply to gravity and electromagnetism. They fall off exactly the same. Why would that happen?
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 it's not even on the radar.
Then there is MOND. Whether or not MOND is a correct theory, it works surprisingly well at predicting galaxy rotation curves. 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) ?
Even more suggestive is that this acceleration scale is close to c tim...
https://en.wikipedia.org/wiki/Proton-to-electron_mass_ratio says it is
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
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.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.
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...
6 weeks * 7 days * 24 hours * 60 minutes * 60 seconds:
so the Earth's orbital velocity is 10^-4 c
It would have been the ideal (if chilly) place to start a cult.
Quite a lot.
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.
this isn't something I was expecting to read today. I guess this works with weak types? /s
https://en.wikipedia.org/wiki/Heegner_number
https://en.wikipedia.org/wiki/Stigler%27s_law_of_eponymy
https://corporate.mcdonalds.com/content/dam/sites/corp/nfl/p...
Humm that's like 2 or 3 countries?