This is interesting, but I have to quibble with this:
> If you express this value in any other units, the magic immediately disappears. So, this is no coincidence
Ordinarily, this would be extremely indicative of a coincidence. If you’re looking for a heuristic for non-coincidences, “sticks around when you change units” is the one you want. This is just an unusual case where that heuristic fails.
Not necessarily. One of the things I was taught when studying astronomy is that if you observe periodicity that is similar to a year or a day, that's probably not a coincidence, you probably failed to account for the earth's orbit or rotation.
This is a good example, but actually this is exactly what GP was referring to. It is a coincidence that the thing you're observing is periodic with earth's rotation. Observing a similar thing from a satellite (allegorically the same as "changing bases") would remove the interesting periodicity.
The earths rotation coincides with the phenomenon, so it's likely a coincidence.
Yeah, I read the post. What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general.
For example, the speed of sound is almost exactly 3/4 cubits per millisecond. Why is it such a nice fraction? The magic disappears if you change units… (of course, I just spammed units at wolfram alpha until I found something mildly interesting).
Im wondering is there connection or not? We use distance unit to get to π number, whatever the distance unit is right? We get π from circumference to diameter ratio, so however long the meter is the π in your distance unit is same ratio
I'm surprised at the number of people disagreeing with your quibble. I had the exact same thought as you!
If pi^2 were _exactly_ g, and the "magic" disappeared in different units, THEN we could say "so this is no coincidence" and we could conclude that it has to be related to the units themselves.
But since pi^2 is only roughly equal to g, and the magic disappears in different units, I would likely attribute it to coincidence if I hadn't read the article.
It would be useful if people carried around some card with all the information that they understood on it, since opinions are largely symptoms of this.
In almost all cases any apparent phenomenon specific to one system of measurement is clearly a coincidence, since reality is definable as that which is independent of measurement.
Agreed. Irrespective of how the story is later developed, "So, this is no coincidence", is a baffling thing to put immediately after apparently demonstrating a coincidence!
I agree. But if you remove the "so", there is no contradiction. It is possible the author used "so" not to mean "in other words", but simply as a relatively meaningless discourse marker.
Doesn't the relationship hold if we change units? It seems like it must.
When I worked with electric water pumps I loved that power can be easily calculates from electrical, mechanical, and fluid measurements in the same way if you use the right units. VoltsAmps, torquerad/sec, pressure*flow_rate all give watts.
Nope, it completely vanishes in other units. If you do all your distance measurements in feet, for example, the value of pi is still about 3.14 but the acceleration due to gravity at the earth's surface is about 32 feet s^(-2). If you do your distance measurements in furlongs and your time measurements in hours then the acceleration due to gravity becomes about 630,000 furlongs per hour squared and pi (of course) doesn't change.
This is not quite the same situation, as you are calculating a value having a dimension (that of power, or energy per second) three different ways using a single consistent system of units, and getting a result demonstrating / conforming to the conservation of energy. If you were to perform one of these calculations in British imperial units (such as from pressure in stones per square hand and rate of flow in slugs per fortnight) you would get a different numerical value (I think!) that nonetheless represents the same power expressed in different units. The article, however, is discussing a dimensionless ratio between a dimensionless constant and a physical measurement that is specific to one particular planet.
I think what the author want to convey is that the metric system was designed based on the assumption that pi^2 = g. The assumption pi^2 = g is one of the source of the metric system (at least for the relationship between meter and second). The deviation was due to the size of earth being incorrectly measured by French in the original expedition.)
I seriously doubt you could define any system of units that has zero coincidences, even with significant computational effort. Some things in the real world are just going to happen to line up close to round numbers, or important mathematical constants, or powers or roots of mathematical constants, and then you’ll have some coincidences.
There are just too many physical quantities we find significant, and too many ways to mix numbers together to make expressions that look notable.
It’s really the best and only way to find non-coincidences involving the definition of units, though. All such non-coincidences will have this property
All coincidences involving the definition of units will also have this property. Once you’ve narrowed to that specific domain, invariance to change of units is completely uninformative.
What are you disputing about the explanation given in the post? As far as I can tell, it’s basically accurate (although the pendulum unit was called the toise, and the meter seems to have targeted half a toise). If you accept that account, it’s not a coincidence.
Your quibble seems nitpicky and unwarranted. What the author is saying is that the relationship becomes evident if we consider the units of m/s^2 for gravity. They just didn't quite say it like that.
Obviously it’s nitpicky. That’s what a quibble is. But I don’t think it’s unwarranted. How you reason your way to a conclusion is at least as important a lesson as the conclusion itself. And in this case, the part I quoted is a bad lesson.
> This is just an unusual case where that heuristic fails.
I don't have this heuristic drilled into me, so I saw the point immediately. To be frank, I suspected the general direction of the answer after reading the headline, and this general direction, probably, can be expressed the best by pointing at the sensitivity of the approx. equation from the headline to the choice of units.
So, I think, the reaction to this quote says more about the person reacting, then about this quote. If the person tends to look answers in a physics (a popular approach for techies), then this quote feels wrong. If the person thinks of physics as of an artificial creation filled with conventions and seeking answers in humans who created physics (it is rarer for techies and closer to a perspective of humanities and social sciences), then this quote is the answer, lacking just some details.
No, like objectively, a dimensionless number lining up with a meaningful constant is more likely to be because of some underlying mathematical connection, and a dimensioned number lining up is more likely to be a coincidence. There are only a handful of ways for a unit’s heritage to have a connection to a local physical phenomenon like the post describes, and that’s what it takes to have a unit-dependent non-coincidence. That’s not dependent on your perspective.
The thing that’s interesting in this case is that the meter’s connection to g is obscured by history, whereas most of the time a unit’s heritage is well known. Nobody is going to be surprised by constants coming out of amps, ohms, and volts, for example, because we know that those units are defined to have a clean relationship.
What? The entire point is that it’s no coincidence in this unit set. Saying that changing units indicates a coincidence is like saying that if we see Trump suddenly driving a Tesla after Elon stated throwing money at him, that must be just a coincidence because if we change the car model to a ford then there would be nothing odd about it.
One of my old physics professors said something similar - there are only three numbers in the world - 0, 1, and infinity. No wait, zero is just one divided by infinity, so there are only two numbers, zero and one. So if the answer is not zero, it must be one. (ie, how to justify dimensional analysis and ignore any dimensionaless constant).
Hysterical, especially for the fact that he quotes 'two' and 'three' in the sentence itself.
As a physicist? When we did physics at school, and we were solving problems, the answer was always a number together with its unit. pi² might be 10 because it is a pure number, but g can never be 10, because it is an acceleration, a physical quantity, so it must be 10 of some unit.
Not if you define g as the real number before m/s^2, in the expression '10 m/s^2'.
In middle school physics lessons this makes teachers to hate you (it's their job to ensure that you do not do this), but after that, this has advantages time to time.
.. I remember hearing an anecdote that ancient Greeks did not know that numbers can be dimensionless, and when they tried to solve cubic equations, they always made sure that they add and substract cubic things. E.g. they didn't do x^3 - x, but only things like x^3 - 2*3*x. I don't think this is true (especially since terms can be padded with a bunch of 1s), but maybe it has some truth in it. It is plausible that they thought about numbers different ways than we do now, and they had different soft rules that what they can do with them.
According to the Banach–Tarski paradox, if you accept the Axiom of Choice, you can disassemble a spherical cow and put the parts back together such that you end up with two cows of the original size. How exactly this affects Cow Economics is not well-understood.
I think it was Gauss who proved that any convex cow would work equally well. But we need to assume an infinitesimally thin and infinitely long tail as boundary condition.
1) it isn't circular, although just barely (it's an ellipse)
2) the length of the day is not really related to the length of a year, and the second was defined as 1 / (24 * 60 * 60) = 1 / 86400 of the mean solar day length
So this is really just a coincidence, there is no mathematical or physical reason why this relationship (the year being close to an even power of 10 times pi seconds) would exist.
Another "wonderful coincidence" is that the conversion between miles and kilometers involves this constant of conversion : kilometers = miles * 1.609344. Let's call 1.609344 the "km" constant.
As it happens, km is very close the the Golden Ratio (sqrt(5)+1)/2 = 1.618033989... (call this "gr"). In fact they only differ by about 1/2 of one percent (100 * (gr/km - 1) = 0.54%)! As the author of the original article says, "If you express this value in any other units, the magic immediately disappears. So, this is no coincidence ...". Yeah ... wait, what?
Here's another one. Pi (3.141592654...) is nearly equal to 4 / sqrt(gr) (3.144605511...), call the latter number "ap" for "almost pi". This connects pi to the golden ratio, and they differ by only 0.096% (100 * (pi/ap - 1)). Surely this means something -- doesn't it?
Finally, my favorite: 111111111^2 = 12345678987654321. This proves that ... umm ... wait ...
I've got a related one I like. Why are the Avogadro's number and Boltzmann's constant inverses of each other N ~ 1/k? The statement doesn't make sense because the units don't work out, but it is true in mks. It's because they multiply to the gas constant which is ~1. They both are numbers to transfer from the microscopic to human scale units and they cancel for the gas constant, which is about human scale experience of gases.
Funnily Avogadro's constant is actually equal to 1: it's defined as Avogadro's number times mol, but mol is itself a dimensionless quantity equal to the inverse of Avogadro's number.
But it's a coincidence, right? N*k=8.31 is pressure*volume/temperature for a mole of gas.
Temperature has a relatively small range (100-1000) and there's no reason why the range of P*V couldn't be far from that range, for example 0.01–0.1, with a different definition of meter, second or kilogram.
Meter, second, and kilogram were all chosen to be approximately the scale of a human, and the combined multiplicative units like Pascal, m^3, and Kelvin/Celsius are also numerically 1 in these units.
You could not have done a worse job explaining it.
This is written for which audience. For a person who doesn't know physics this is a very long and confusing explanation. Explaining that some units depend on others, and the importance of the ability to reproduce the metric system on your own, is much more important than the whole pre-story of length standards.
There are lots of unanswered questions. What was the second defined as? Don't you measure time using a pendulum? Why was the astronomical definition more reliable?
For a person who does know physics you can write a much shorter and clearer explanation eg.:
"For a universal definition of the meter you need a constant that appears in nature, such as gravity. You could measure the distance an object falls in some amount of time, but it is easier to use a pendulum.
Pendulums swing consistently with a period approximately equal to 2pisqrt(string length/gravity). I you were to use pi^2 for gravity, than after the square root the pis would cancel out, leaving T = 2*sqrt(Length). This is useful because a 1 meter pendulum takes 2 seconds to swing back and forth (1 second per swing.)
Clocks at that time were quite accurate, with the second being reproducible from astronomical measurements. So you could take a pendulum, fiddle with it's length until it does exactly one swing every second, and then use the string or stick to measure whatever you wanted.
That was great so they changed gravitational constant so it would equal pi^2 (9.87 m/s^2). (If you decrease the meter, everything will become longer.)
Then they found out that gravity differs along earth's surface and a perfect mathematical pendulum proved to be difficult to reproduce, so they switched to an astronomy based definition based on the size of earth. That turned out to be broken as well, so they held a physical meter long stick in Paris. A few years ago physicists started using the plank constant which is the smallest possible distance you can measure."
The meter is now (as of 2019) defined as the distance light travels in vacuum during N cycles of an atomic clock[1]. Note that to take into account GR effects you'd need to specify where on Earth you do the measurement, since gravity affects the clock rate. The velocity of light is defined, not measured, now. This is actually quite profound, because our system of units is now based on the validity of special relativity.
Awesome write up and a great surprise in the history of the definition of the meter.
Reading this reminds me a little of mathematicians like Ramanujan who spent a fair amount of time just playing around with random numbers and finding connections, although in this case, I imagine the author knew the history from the beginning.
Anyway, I feel like my math degree sort of killed some of that fun exploration of number relations — but I did like that kind of weird doodling / making connections as a kid. By the time I was done with the degree, I wanted to think about connections between much more abstract primitives I’d learned, but it seems to me there are still a lot of successful mathematicians that work this way — noticing some weird connection and then filling out theory as to why, which occasionally at least turns out to be really interesting.
Relatedly, I recommend reading "The Measure of All Things" by Ken Alder about the origins of the metric system and the first scientific conference ever. It is a surprisingly gripping read.
Totally unrelated to the content, but about the website itself.
The site completely breaks when I visit it. After some investigation, I found out that if I enable Stylus (a CSS injection extension) with any rules (even my global ones), the site becomes unusable. Since it's built using the React framework, it doesn't just glitch; it completely breaks.
After submitting a ticket and getting a quick response from the Stylus dev, it turns out that this website (and any site built with caseme.io) will throw an error and break if it detects any node injected into `<html>`.
I don't currently use Stylus, but it breaks for me too; it looks like CSS isn't applied at all: I get some big logo images, and the text uses standard fonts. Not sure which extension triggers it, probably Dark Reader. I could still read it, so no biggie.
> Sometimes that was even useful. If you needed to buy more cloth, you'd call the tallest person in the village and have them measure the fabric with their cubits.
I highly doubt this bit of strategy would have worked with sellers of said fabric. They may have not had formal measurements but they weren't stupid either.
So if I understand correctly: the meter was defined using gravity and π as inputs (distance a pendulum travels in 1 cycle), so of course g and π would be connected.
On the hand, g is about 32.2 ft/s². So it's suddenly related to π³? I think there's no connection at all, it's just an accident. It would be really weird if some contemporary property of the earth were actually related to a fundamental mathematical constant. It's similar to finding a message among the digits of π that shouldn't be there, statistically speaking.
The bulk of the article is devoted to explaining that g = pi^2 in m/s^2 units (under an old definition of meter) because (that definition of) the meter was not selected arbitrarily, but selected in a way that makes the equation hold on purpose.
This article reasons that it is not a coincidence because of the “seconds pendulum” definition of the meter, which would necessitate the values being equal because of the pendulum time period equation.
That all makes sense to me, and I agree.
But here’s what’s odd to me:
We ended up not choosing the seconds pendulum approach (for reasons mentioned in the article). Instead they chose to use “1 ten-millionth of the Earth’s quadrant”. Now, how is it that that value is so close to the length of the seconds pendulum? Were they intentionally trying to get it close to seconds pendulum length, and it just happened to be a nice round power of ten? Is that a coincidence?
I seriously need an explanation for that too. Never understood how these genius folks just came up with the most outlandish definitions that somehow make perfect sense
I thought they key insight of this article was if he were born on Mars, then the meter would have been defined differently so that gravity would still be 9.8 m/s^2.
I think what you meant to say was that he wouldn't be speaking like this if people were born with 3 fingers.
If the definition of the meter is still wrong disallowing π² = g, how might this affect other calculations like for example thrust and in aerospace engineering?
Another interesting coincidence (or perhaps a decades-long dad-joke troll perpetrated by German-speaking scientists) is that 1 hertz is roughly equal to the frequency at which a human heart (“Herz” in german, with a nearly indistinguishable pronunciation to “Hertz”) beats.
That seems rather low rate. Regular rate in rest is 60 to 100. Which only lower bound is roughly 1Hz, while upper rate is quite far what I would understand German to understand as roughly.
80+ heart rate at rest seems very high to me. Is this based on averages from present-day people? Including lots of sedentary people with hyper tension, overweight and obese people, etc. So "normal" in the statistical sense but maybe not "normal" in the physiological sense?
Athletes have much lower resting rate. So if we take feral humans from 50Ky ago their rest heart rate would have been much closer to the 60 bpm.
The pendulum equation isn't progress at all, it's just another observation of it? Driving it 'backwards' as it were with known values for the parameters it would determine, and seeing that pi squared is roughly g without having to know the actual values of those constants.
And now I've finished the article, nothing more is really offered. Am I missing something? That doesn't explain it/answer the question at all afaict? All we've done is find an equation that uses both pi and g, which shows again the relationship we started from?
Underlying idea of whole metric and SI system is the real start point. You want to define some units that you can easily replicate. Time is reasonable enough one, measure and track length of day and then length of second. Now based on this figure out way to come up with replicable definition for distance, pendulum is good enough gravity is constant enough. Thus linking gravity and meter arises.
From here you can define lot of other units like mass and Volt and Ampere... Everything comes from second which is weird, but does make lot of sense.
If society were tasked one day with reinventing standards and units, it doesn't matter why, what do you think are some things they would change?
For example, I think for human counting, base 12 is about as easy as base 10, but gives good ways to express division by 3, in addition to division by 2 or 4. It also fits better with how we count time, like how there are 60 seconds in a minute, 12 months in a year, etc... but I imagine those might be revised as well.
I think there's an argument for base 16 over 12. It's very slightly worse than base 12 for purely human use, since it's only divisible into halves, quarters and eighths. However, each hex digit maps to an exact number of binary digits, which I think outweighs the benefit of thirds and sixths in a digital world.
Of course, it's all theoretical, because there's no chance this would ever happen short of an apocalypse that takes us back to the stone age, and unless the radiation gives us 12 or 16 fingers, we'd probably just reinvent decimal.
To be honest I'm not a fan, time is cumbersome to do any sort of addition or subtraction to get exact days/hours/minutes (not to mention timezones etc).
Compare to metric units, always base 10 and always easy to convert mm to cm to m and so on.
Now that we live in a digital world - why do we consistently reinvent date/time libraries? To me that's proof enough the concepts are just hard to work with and over a long span of time verify your calculation is correct.
None of those issues with date and time are anything to do with the base, they're to do with date and time as defined by humans being inherently complicated concepts. Specifically, trying to have a single measurement "fit" for a load of different purposes.
If we had based our system around base 12, a base 12 version of the metric system would be just as easy to work with as metric is in decimal, with the added bonus that you can divide powers of the base (10, 100, 1000, etc.) into quarters, thirds and sixths without needing a decimal place, and thirds of 1 would be non-repeating.
I remember in mechanical engineering class we would often use this for exercise sheets. On our calculator we could directly enter π and ², thus it was equally as fast to entering 10.
How do mathematicians handle it when there are tantalizingly close relationships between purely mathematical numbers? I fell down this particular rabbit hole through the musical entrance (just intonation intervals), but I mean look at all this, it's clearly a setup.
This is neat, but still something if a coincidence.
It appears the first definition of a metre is in fact around 1/4e10 the circumference of Earth, and the further coincidence is that a 1m mathematical pendulum has a period of almost exactly 2 seconds.
So there's still a neat relationship between mass/radius of Earth, its diurnal rotation period and the Babylonian division of it into 86,400 seconds.
Our mechanics professor in university told us that Pi squared is ten and 6xPi=20. He said "engineers do not care for the fourth digit after the dot. If you want some number very precisely calculated just hire a mathematician instead because they are cheaper per hour"
Multiple people have brought this up and I just don't get it. When would you ever use Pi squared for anything, and if you'd never use it, who cares what its value is?
The only thing I could come up with is marking out an area by rolling a wheel some number of times to measure each edge.
Long ago, I memorized the square root of pi precisely because it seemed like the least useful number I could think of. (I was frustrated by something. Never mind.) But pi squared seems like it's pretty much the same in terms of usefulness.
If you said pi is the square root of 10, then I could see the value. Maybe that is what is being implied?
There is nothing meaningful about this. I can change the unit system to make g any value I want (this is done all the time in research). I try really hard to ignore all the physics-related articles posted here but this one is too egregious. It's not the usual thing where the author know nothing about the nouns they are using (enter, quantum). In this case the concepts are fairly simple. The fact that units can be freely changed should be taught in middle school.
Somehow I found programmers have a much higher probability of talking about physics than people in other professions. And unfortunately in all cases I've seen, they have no idea what they are talking about. Unlike programming, physics is hard enough that it needs to be studied in classes.
This is addressed in the article, including the fact that you can change the unit system to change the value of g.
The article explains that the coincidence comes from the fact that the meter, as a unit, was defined (by Huygens) based on g and π. It was later redefined several times and the link between the two values became anecdotal. In other words, on another planet the gravitational constant would still have had a value of (approximately) π², and what would have been different is our unit length.
Yeah, I saw the author says it depends on the units. But like, why is this interesting? This is not physics, just some number coincidence in the metric unit system, and I'm sure one can find many more these kind of things by playing around with the constants. The fact the author calls this a "wonderful coincidence" is just... Like, a simple energy conservation or momentum conservation, taught in middle school, is infinitely worth being talked than this.
One philosophy in physics, is that the world and its rules are independent of human. We actively try to eliminate and downplay historical and human factors in the theory, and try to talk about just "the physics", because those factors often obscure the real physics (mechanism) and complicate the calculation. I mean people can find a historical thing interesting, but I guess I just feel disappointed that people find such a trivial thing so interesting, and maybe think that this is what physics is about, while physics is about anything but those pure coincidences.
>“It was contended," says Dr. Peacock, " by Paucton, in his Mẻtrologie, that the side of the Great Pyramid was the exact 1/500th part of a degree of the meridian, and that the founders of that mighty monument designed it as an imperishable standard of measures of length.
>Newton was trying to uncover the unit of measurement used by those constructing the pyramids. He thought it was likely that the ancient Egyptians had been able to measure the Earth and that, by unlocking the cubit of the Great Pyramid, he too would be able to measure the circumference of the Earth.
Having pi^2 = g would annoy me a bit as g is fundamentally a measured value. Depending on the required accuracy of the calculation, you can't even use the idealized value.
g is related to the radius of the earth; the meter is related to the circumference of the earth; and pi is the relationship between the radius and the circumference.
Aside from the fact that the post already explained what the actual historical connection is, your explanation requires some serious hand-waving about the mass of the Earth and the gravitational constant, neither of which were known when the meter was first defined.
Reasonably accurate values for both M_earth and G were known at the time the SI meter was defined.
Also it's not too hard to extend this. M_earth is a function of Earth's radius which goes into the definition of the meter. G is a function of earth's orbital period, which goes into the definition of the second. Further our definition of mass is based on the density of water, which is chosen because it is a stable liquid at this particular orbital distance from a star of our sun's mass.
It must have been brutal to create the first clocks when your smallest external reference is a day long. You first make a huge hour glass or clepsydra and tweak it once per day until it's perfect. Very slow debugging loop.
What an amazing post! Such an interesting investigation. These kinds of write-ups make me realize how truly far we are from AGI. Sure, it can write amazing code, poems, songs, but can it draw interesting conclusions from first principles? I asked both ChatGPT and Claude, the same question, and both failed at pointing out the connection the author states.
This is not to deride feats of AI today, and I am sure it will transform the world. But until it can show signs of human ingenuity in making unexpected and far-off connections like these, I won't be convinced we are nearing AGI.
Does this mean in 400 years it's possible we no longer disagree about how to evaluate things? i.e. we converge on one totalitarian utility function that everyone basically accept answers every possible trolley dilemma?
In 1600, people just took the world as that: measurements are sloppy, and vary culturally and based upon location etc. But we eventually came upon tools and techniques that are broadly accepted as repeatable and standard.
Would this sort of shift be possible? Or desirable?
No, Tau is fundamental. Pi only exists because someone mistakenly thought the formula for circumference involved diameter, when in fact it involves radius. ("Quit factoring a 2 out of Tau!" I tell them.)
Eh, you can find plenty of cases where tau is just as awkward as pi is elsewhere. Right off the bat, the area of a circle becomes more awkward with tau, becoming (tau*r^2)/2, and in general, the volume of an n-ball gains weird powers and roots of two in its denominator as n increases if you switch to tau. In general, I don't think you can really claim either one is "more fundamental". It's just a matter of framing.
Why is this comment section, specifically, such an embarrassing dumpster fire?
It's a serious question; this is the sort of neat derivation that makes for a popular Youtube video, and despite Youtube comments being famously... variable in quality, the comment section on videos about things like this is vastly more literate than the threads here right now.
Is it just a coincidence, the chaotic behavior of uninformed sneer comments (which exist on every post; I've certainly been guilty, to my shame) meaning that some post is going to end up being the one with no other type of comment? Or is there some surprising reason why?
One french royal cubit ≈ one egyptian cubit ≈ about π/6 meters. One royal span ≈ 1/5 meter = 20cm.
I'm wondering whether some of these coincidences could be explained by the anthropic principle, which deals with these quasi-equalities, for instance:
>An excited state of the 12C nucleus exists a little (0.3193 MeV) above the energy level of 8Be + 4He. This is necessary because the ground state of 12C is 7.3367 MeV below the energy of 8Be + 4He; a 8Be nucleus and a 4He nucleus cannot reasonably fuse directly into a ground-state 12C nucleus. However, 8Be and 4He use the kinetic energy of their collision to fuse into the excited 12C (kinetic energy supplies the additional 0.3193 MeV necessary to reach the excited state), which can then transition to its stable ground state. According to one calculation, the energy level of this excited state must be between about 7.3 MeV and 7.9 MeV to produce sufficient carbon for life to exist, and must be further "fine-tuned" to between 7.596 MeV and 7.716 MeV in order to produce the abundant level of 12C observed in nature.
1. A more fundamental aspect under the anthropic principle which underpins the existence of complex life and intelligent observers is the quasi-alignment of values such as the fundamental constants in physics within a short margin.
2. If you consider the universe to be the product of a random sampling process over these constants (either real or virtual, it occurred many times or just once), and given the fact we exist, which implies an abundance of coincidences, the maths seem to tell us that we should expect to observe superfluous coincidences that are non-functional for the appearance of complex life, rather than the strictly minimal set of functional coincidences necessary for its emergence.
3. This implies that coincidences and pattern seeking are not just features (or bugs) of our complex minds but are present in the universe latently since it is not just fine-tuned for the emergence of complex life but for the presence of coincidences such as these https://medium.com/@sahil50/a-large-numbers-coincidence-299c....
4. It may be even testable by running computer experiments relying on genetic programming/symbolic regression to see whether there is something special about the value of physical constants in our universe when compared to the value they would have in other universes. I think such experiments should factor the fact that not all equations with the same mathematical complexity (number of operands and operators) have the same cognitive complexity. Indeed, if you look at the big equation in the link above, you'll remark that it can be further compressed into a/b = c/d (where a is the photon redshift radius for instance). So I guess you'd also have to throw into the mix Kolmogorov algorithmic complexity to assess this aspect (which is in fact used in some cognitive theories of relevance to tackle this kind of stuff to the tune of "simpler to describe than to generate")
I disagree… the article talks about defining the meter using the pendulum and the second. Other planets would have their own definition of second, but not their own definition of pendulum. Since one meter is prescribed though a pendulum because of the oscillation formula and dependent on the frequency alone (in this case, 2 seconds), no matter what planet you are on, how strong gravity, is or how long a second is, the pendulum describes a relationship between seconds and meters such that if using this method a planet’s scientists would always define their units such that acceleration of gravity was eerily equal to pi^2.
Makes me think of possible lunar scientists unwittingly making their meter 5/6th shorter(edit: english is hard) and then marveling at the same coincidence…
Your comment is much more rage-bait than the article.
Universal isn't a way we describe numbers. You meant to say dimensionless. Pi is dimensionless constant because it describes a relationship between two measurements of a dimensionless unit circle.
Pi is expressed as a pure ratio between two other dependent numbers. Dimensionless values are special because they don't rely on any particular measurement in any particular location, lending to your misconception of "universal" constant.
This article explains how a particular dimensionful constant (g, the strength of gravity on earth's surface) is related to pi.
They are related because the dimensions in question are both derived from dependent properties of our planet. These dependent properties will be found on any other sphere floating in space if they are derived in the same fashion.
It's good to thoroughly or even marginally understand a topic before adopting a dismissive and authoritative argument against it.
> Universal isn't a way we describe numbers. You meant to say dimensionless.
They probably really meant to say “universal”, since dimensionless values are a less interesting category that includes… well, every number. Pi shows up in math without having to parameterize anything, making it universal in a way that even physical constants of our universe aren’t.
On any planet where you want to define a system of units, you can start by defining a fixed time period (maybe use a fraction of the planetary rotation cycle or something), then make a pendulum that swings with that frequency, and derive a unit of distance from its length.
The local value of g will be roughly pi squared pendulum lengths per tick squared, in that system of measurement.
Nope, it's not a coincidence - it's an interesting exploration of the history of the definition of a metre. Read the article.
As it says, at some point there was an attempt to standardise the length of a metre in terms of a pendulum's length; which related it directly to g through Pi.
Sorry to ruin the party, but g is a quite random number, on other planets the corresponding acceleration is different. So π^2~g is a pure coincidence and not relevant. The Newtonian gravitational constant G is a real constant btw.
Have you read the article? The point is that the definition of the metre, which is used in g, originates from the length of a pendulum that swings once per second in the gravity field around Paris. So it is a matter of definitions, and the length of the metre originates from the duration of the second and the Earth's gravity field. The definitions of 1/40.000 of the Earth's circumference or ~1/300.000.000 of a light second came later.
My intuitive assumption, then, is that on Mars they would have come up with a different meter such that π² ≈ 10 "mars meters" / s².
Or alternatively stated, that the Mars meter would be much shorter than Earth's meter if they used the same approach to defining it (pendulums and seconds).
I have to admit I only read half of the article. Even if there is some historical fact there (but it was not mentioned at the beginning of the article), from a physical standpoint this comparison is already dimensionally wrong and also coincidentally only correct if you choose appropriate units. That was the point I was trying to make. There is not anything "deep" here.
It's not about the values, but the units of measurement. g is in units of meter/second^2. The article discusses the dependency of the meter's original definition on the value of pi.
This is interesting, but I have to quibble with this:
> If you express this value in any other units, the magic immediately disappears. So, this is no coincidence
Ordinarily, this would be extremely indicative of a coincidence. If you’re looking for a heuristic for non-coincidences, “sticks around when you change units” is the one you want. This is just an unusual case where that heuristic fails.
Not necessarily. One of the things I was taught when studying astronomy is that if you observe periodicity that is similar to a year or a day, that's probably not a coincidence, you probably failed to account for the earth's orbit or rotation.
This is a good example, but actually this is exactly what GP was referring to. It is a coincidence that the thing you're observing is periodic with earth's rotation. Observing a similar thing from a satellite (allegorically the same as "changing bases") would remove the interesting periodicity.
The earths rotation coincides with the phenomenon, so it's likely a coincidence.
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Actually no, the whole equation boils down to the definition of meter. Or rather, one of the earlier definitions.
Yeah, I read the post. What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general.
For example, the speed of sound is almost exactly 3/4 cubits per millisecond. Why is it such a nice fraction? The magic disappears if you change units… (of course, I just spammed units at wolfram alpha until I found something mildly interesting).
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Im wondering is there connection or not? We use distance unit to get to π number, whatever the distance unit is right? We get π from circumference to diameter ratio, so however long the meter is the π in your distance unit is same ratio
It does not? Pi has nothing to do with our arbitrary unit system.
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I'm surprised at the number of people disagreeing with your quibble. I had the exact same thought as you!
If pi^2 were _exactly_ g, and the "magic" disappeared in different units, THEN we could say "so this is no coincidence" and we could conclude that it has to be related to the units themselves.
But since pi^2 is only roughly equal to g, and the magic disappears in different units, I would likely attribute it to coincidence if I hadn't read the article.
It would be useful if people carried around some card with all the information that they understood on it, since opinions are largely symptoms of this.
In almost all cases any apparent phenomenon specific to one system of measurement is clearly a coincidence, since reality is definable as that which is independent of measurement.
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Agreed. Irrespective of how the story is later developed, "So, this is no coincidence", is a baffling thing to put immediately after apparently demonstrating a coincidence!
I agree. But if you remove the "so", there is no contradiction. It is possible the author used "so" not to mean "in other words", but simply as a relatively meaningless discourse marker.
Huh, interesting point. Writing unambiguously is ridiculously hard.
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Doesn't the relationship hold if we change units? It seems like it must.
When I worked with electric water pumps I loved that power can be easily calculates from electrical, mechanical, and fluid measurements in the same way if you use the right units. VoltsAmps, torquerad/sec, pressure*flow_rate all give watts.
Nope, it completely vanishes in other units. If you do all your distance measurements in feet, for example, the value of pi is still about 3.14 but the acceleration due to gravity at the earth's surface is about 32 feet s^(-2). If you do your distance measurements in furlongs and your time measurements in hours then the acceleration due to gravity becomes about 630,000 furlongs per hour squared and pi (of course) doesn't change.
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This is not quite the same situation, as you are calculating a value having a dimension (that of power, or energy per second) three different ways using a single consistent system of units, and getting a result demonstrating / conforming to the conservation of energy. If you were to perform one of these calculations in British imperial units (such as from pressure in stones per square hand and rate of flow in slugs per fortnight) you would get a different numerical value (I think!) that nonetheless represents the same power expressed in different units. The article, however, is discussing a dimensionless ratio between a dimensionless constant and a physical measurement that is specific to one particular planet.
No, the equality requires the length of a 2 second period pendulum be g / pi^2. Change your definition of length - that no longer holds true.
g in imperial units is 32 after all. g has units; pi does not
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The "magic" doesn't disappear in "any" other units.
Period = 2π√(length/g)
So the "magic" holds in any units where the unit of time is the period of a pendulum with unit length.
I think what the author want to convey is that the metric system was designed based on the assumption that pi^2 = g. The assumption pi^2 = g is one of the source of the metric system (at least for the relationship between meter and second). The deviation was due to the size of earth being incorrectly measured by French in the original expedition.)
I don’t agree with this. You could literally redefine any unit (as we have done so multiple times in the past) and end up with zero coincidences.
All measurement metrics are “fake” - nothing is truly universal, and can easily be correlated with another human made measure eg Pi.
I seriously doubt you could define any system of units that has zero coincidences, even with significant computational effort. Some things in the real world are just going to happen to line up close to round numbers, or important mathematical constants, or powers or roots of mathematical constants, and then you’ll have some coincidences.
There are just too many physical quantities we find significant, and too many ways to mix numbers together to make expressions that look notable.
It’s really the best and only way to find non-coincidences involving the definition of units, though. All such non-coincidences will have this property
All coincidences involving the definition of units will also have this property. Once you’ve narrowed to that specific domain, invariance to change of units is completely uninformative.
Reading this gave me a chill. Please take my temperature and compare it to the norm temperature of humanity.
Changing units in Electrodynamics for instance comes with unexpected factors in formulas though, indeed containing π. (CGS <-> SI)
Isn't that just the change between rad/s and Hz?
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It is not unusual case. The heuristic you want is working. It's nothing more than a coincidence.
What are you disputing about the explanation given in the post? As far as I can tell, it’s basically accurate (although the pendulum unit was called the toise, and the meter seems to have targeted half a toise). If you accept that account, it’s not a coincidence.
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I think you are right and i had the exact same thought. I think people are misunderstanding you.
Your quibble seems nitpicky and unwarranted. What the author is saying is that the relationship becomes evident if we consider the units of m/s^2 for gravity. They just didn't quite say it like that.
Obviously it’s nitpicky. That’s what a quibble is. But I don’t think it’s unwarranted. How you reason your way to a conclusion is at least as important a lesson as the conclusion itself. And in this case, the part I quoted is a bad lesson.
> This is just an unusual case where that heuristic fails.
I don't have this heuristic drilled into me, so I saw the point immediately. To be frank, I suspected the general direction of the answer after reading the headline, and this general direction, probably, can be expressed the best by pointing at the sensitivity of the approx. equation from the headline to the choice of units.
So, I think, the reaction to this quote says more about the person reacting, then about this quote. If the person tends to look answers in a physics (a popular approach for techies), then this quote feels wrong. If the person thinks of physics as of an artificial creation filled with conventions and seeking answers in humans who created physics (it is rarer for techies and closer to a perspective of humanities and social sciences), then this quote is the answer, lacking just some details.
No, like objectively, a dimensionless number lining up with a meaningful constant is more likely to be because of some underlying mathematical connection, and a dimensioned number lining up is more likely to be a coincidence. There are only a handful of ways for a unit’s heritage to have a connection to a local physical phenomenon like the post describes, and that’s what it takes to have a unit-dependent non-coincidence. That’s not dependent on your perspective.
The thing that’s interesting in this case is that the meter’s connection to g is obscured by history, whereas most of the time a unit’s heritage is well known. Nobody is going to be surprised by constants coming out of amps, ohms, and volts, for example, because we know that those units are defined to have a clean relationship.
you can rule that heuristic out immediately because pi is unitless, surely?
What? The entire point is that it’s no coincidence in this unit set. Saying that changing units indicates a coincidence is like saying that if we see Trump suddenly driving a Tesla after Elon stated throwing money at him, that must be just a coincidence because if we change the car model to a ford then there would be nothing odd about it.
That analogy is so bizarre that I have no idea how to respond to it.
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As a physicist, this makes sense. Pi = 3, pi^2 = 10, which is g
Not sure why everyone is surprised.
Ah, and a year is pi*10e9 seconds (IIRC)
As a computer scientist this is not surprising either. After all there are only three numbers: 0 1 and n.
One of my old physics professors said something similar - there are only three numbers in the world - 0, 1, and infinity. No wait, zero is just one divided by infinity, so there are only two numbers, zero and one. So if the answer is not zero, it must be one. (ie, how to justify dimensional analysis and ignore any dimensionaless constant).
Hysterical, especially for the fact that he quotes 'two' and 'three' in the sentence itself.
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Remember that `i` is also a number. As in `for i = 0 to n`.
Don’t believe those mathematicians when they tell you that `i` is “imaginary”.
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You mean the legendary “i”. There is no n.
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And -0 and NaN
And int_max
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Yea if you're some classical snob.
- Posted by the quantum gang
As a physicist? When we did physics at school, and we were solving problems, the answer was always a number together with its unit. pi² might be 10 because it is a pure number, but g can never be 10, because it is an acceleration, a physical quantity, so it must be 10 of some unit.
Oh, come on, it is inches / (day * "hold on"). Everyone knows this, this is physics for art majors 101.
In guess it's a good thing I left physics after my PhD.
I mean it is pretty common to set c=1 though.
Not if you define g as the real number before m/s^2, in the expression '10 m/s^2'.
In middle school physics lessons this makes teachers to hate you (it's their job to ensure that you do not do this), but after that, this has advantages time to time.
.. I remember hearing an anecdote that ancient Greeks did not know that numbers can be dimensionless, and when they tried to solve cubic equations, they always made sure that they add and substract cubic things. E.g. they didn't do x^3 - x, but only things like x^3 - 2*3*x. I don't think this is true (especially since terms can be padded with a bunch of 1s), but maybe it has some truth in it. It is plausible that they thought about numbers different ways than we do now, and they had different soft rules that what they can do with them.
I assumed the pi / g connection was because cows that accelerate in a vacuum are spherical.
According to the Banach–Tarski paradox, if you accept the Axiom of Choice, you can disassemble a spherical cow and put the parts back together such that you end up with two cows of the original size. How exactly this affects Cow Economics is not well-understood.
I think it was Gauss who proved that any convex cow would work equally well. But we need to assume an infinitesimally thin and infinitely long tail as boundary condition.
Pi seconds is a nano-century. So 1 year = pi*10^7 seconds
as a mechanical engineer, can confirm. also, e ≈ pi ≈ 3
in fact, e = 2, made rather abundantly clear in "finite difference" calculus, and also that in computer science, the "natural" log base 2.
A year is pi 10^7, or pi 1e7.
On the other hand, 10 e9 = 10 * 10^9.
3600 seconds per hour, times 3*8 is only about 80,000 seconds a day. You can’t get to a billion from there.
The pi in pi*10^9 seconds clearly comes from the fact that Earth’s orbit is circular.
Nore sure if serious or not, but anyway:
1) it isn't circular, although just barely (it's an ellipse) 2) the length of the day is not really related to the length of a year, and the second was defined as 1 / (24 * 60 * 60) = 1 / 86400 of the mean solar day length
So this is really just a coincidence, there is no mathematical or physical reason why this relationship (the year being close to an even power of 10 times pi seconds) would exist.
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pi * 1e7
Another "wonderful coincidence" is that the conversion between miles and kilometers involves this constant of conversion : kilometers = miles * 1.609344. Let's call 1.609344 the "km" constant.
As it happens, km is very close the the Golden Ratio (sqrt(5)+1)/2 = 1.618033989... (call this "gr"). In fact they only differ by about 1/2 of one percent (100 * (gr/km - 1) = 0.54%)! As the author of the original article says, "If you express this value in any other units, the magic immediately disappears. So, this is no coincidence ...". Yeah ... wait, what?
Here's another one. Pi (3.141592654...) is nearly equal to 4 / sqrt(gr) (3.144605511...), call the latter number "ap" for "almost pi". This connects pi to the golden ratio, and they differ by only 0.096% (100 * (pi/ap - 1)). Surely this means something -- doesn't it?
Finally, my favorite: 111111111^2 = 12345678987654321. This proves that ... umm ... wait ...
Do miles connect to feet and feet vs. gravity connect to the Golden ratio? 6 feet is like 2 pi?
Not sure, but I am indeed often connected to the earth via gravity by my feet.
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A year is very close to pi * 1e7 seconds, better than half a percent.
ln(5) ~ 1.6094379 is much closer, differing by about half of a percent of a percent.
The best is sum(1, 36)
If the length of a meter were defined as the length of a seconds pendulum [1], then g would equal exactly π². From the pendulum equation:
`T = 2π√(L/g)`
substitute T = 2 s and L = 1 m:
`2 s = 2π√(1 m / g)`
solve for g:
`g = π² m/s²`.
This holds up in any strength of gravity, but the length of a meter would be different depending on it.
[1]. This actually was proposed by Talleyrand in 1790. Imagine the world if this were true!
The article explains that Huygens proposed this in the 17th C, and gives the same derivation :)
I've got a related one I like. Why are the Avogadro's number and Boltzmann's constant inverses of each other N ~ 1/k? The statement doesn't make sense because the units don't work out, but it is true in mks. It's because they multiply to the gas constant which is ~1. They both are numbers to transfer from the microscopic to human scale units and they cancel for the gas constant, which is about human scale experience of gases.
Funnily Avogadro's constant is actually equal to 1: it's defined as Avogadro's number times mol, but mol is itself a dimensionless quantity equal to the inverse of Avogadro's number.
Multiplying by increasingly complicated expressions equivalent to “1” is what I remember doing for almost every problem in Quantum Mechanics.
But it's a coincidence, right? N*k=8.31 is pressure*volume/temperature for a mole of gas. Temperature has a relatively small range (100-1000) and there's no reason why the range of P*V couldn't be far from that range, for example 0.01–0.1, with a different definition of meter, second or kilogram.
Meter, second, and kilogram were all chosen to be approximately the scale of a human, and the combined multiplicative units like Pascal, m^3, and Kelvin/Celsius are also numerically 1 in these units.
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You could not have done a worse job explaining it.
This is written for which audience. For a person who doesn't know physics this is a very long and confusing explanation. Explaining that some units depend on others, and the importance of the ability to reproduce the metric system on your own, is much more important than the whole pre-story of length standards.
There are lots of unanswered questions. What was the second defined as? Don't you measure time using a pendulum? Why was the astronomical definition more reliable?
For a person who does know physics you can write a much shorter and clearer explanation eg.:
"For a universal definition of the meter you need a constant that appears in nature, such as gravity. You could measure the distance an object falls in some amount of time, but it is easier to use a pendulum.
Pendulums swing consistently with a period approximately equal to 2pisqrt(string length/gravity). I you were to use pi^2 for gravity, than after the square root the pis would cancel out, leaving T = 2*sqrt(Length). This is useful because a 1 meter pendulum takes 2 seconds to swing back and forth (1 second per swing.)
Clocks at that time were quite accurate, with the second being reproducible from astronomical measurements. So you could take a pendulum, fiddle with it's length until it does exactly one swing every second, and then use the string or stick to measure whatever you wanted.
That was great so they changed gravitational constant so it would equal pi^2 (9.87 m/s^2). (If you decrease the meter, everything will become longer.)
Then they found out that gravity differs along earth's surface and a perfect mathematical pendulum proved to be difficult to reproduce, so they switched to an astronomy based definition based on the size of earth. That turned out to be broken as well, so they held a physical meter long stick in Paris. A few years ago physicists started using the plank constant which is the smallest possible distance you can measure."
The meter is now (as of 2019) defined as the distance light travels in vacuum during N cycles of an atomic clock[1]. Note that to take into account GR effects you'd need to specify where on Earth you do the measurement, since gravity affects the clock rate. The velocity of light is defined, not measured, now. This is actually quite profound, because our system of units is now based on the validity of special relativity.
1 - https://en.wikipedia.org/wiki/Metre
Awesome write up and a great surprise in the history of the definition of the meter.
Reading this reminds me a little of mathematicians like Ramanujan who spent a fair amount of time just playing around with random numbers and finding connections, although in this case, I imagine the author knew the history from the beginning.
Anyway, I feel like my math degree sort of killed some of that fun exploration of number relations — but I did like that kind of weird doodling / making connections as a kid. By the time I was done with the degree, I wanted to think about connections between much more abstract primitives I’d learned, but it seems to me there are still a lot of successful mathematicians that work this way — noticing some weird connection and then filling out theory as to why, which occasionally at least turns out to be really interesting.
Relatedly, I recommend reading "The Measure of All Things" by Ken Alder about the origins of the metric system and the first scientific conference ever. It is a surprisingly gripping read.
https://www.simonandschuster.com/books/The-Measure-of-All-Th...
Totally unrelated to the content, but about the website itself.
The site completely breaks when I visit it. After some investigation, I found out that if I enable Stylus (a CSS injection extension) with any rules (even my global ones), the site becomes unusable. Since it's built using the React framework, it doesn't just glitch; it completely breaks.
After submitting a ticket and getting a quick response from the Stylus dev, it turns out that this website (and any site built with caseme.io) will throw an error and break if it detects any node injected into `<html>`.
[1] https://github.com/openstyles/stylus/issues/1803
I don't currently use Stylus, but it breaks for me too; it looks like CSS isn't applied at all: I get some big logo images, and the text uses standard fonts. Not sure which extension triggers it, probably Dark Reader. I could still read it, so no biggie.
> Sometimes that was even useful. If you needed to buy more cloth, you'd call the tallest person in the village and have them measure the fabric with their cubits.
I highly doubt this bit of strategy would have worked with sellers of said fabric. They may have not had formal measurements but they weren't stupid either.
I find this comment delightfully ironic in a contemporary moment of blatant shrinkflation.
This post feels like one of those cork boards with photos and strings connected between them in some prepper’s basement, but in blog post form.
Very interesting!
So if I understand correctly: the meter was defined using gravity and π as inputs (distance a pendulum travels in 1 cycle), so of course g and π would be connected.
On the hand, g is about 32.2 ft/s². So it's suddenly related to π³? I think there's no connection at all, it's just an accident. It would be really weird if some contemporary property of the earth were actually related to a fundamental mathematical constant. It's similar to finding a message among the digits of π that shouldn't be there, statistically speaking.
The bulk of the article is devoted to explaining that g = pi^2 in m/s^2 units (under an old definition of meter) because (that definition of) the meter was not selected arbitrarily, but selected in a way that makes the equation hold on purpose.
This article reasons that it is not a coincidence because of the “seconds pendulum” definition of the meter, which would necessitate the values being equal because of the pendulum time period equation.
That all makes sense to me, and I agree.
But here’s what’s odd to me:
We ended up not choosing the seconds pendulum approach (for reasons mentioned in the article). Instead they chose to use “1 ten-millionth of the Earth’s quadrant”. Now, how is it that that value is so close to the length of the seconds pendulum? Were they intentionally trying to get it close to seconds pendulum length, and it just happened to be a nice round power of ten? Is that a coincidence?
I seriously need an explanation for that too. Never understood how these genius folks just came up with the most outlandish definitions that somehow make perfect sense
He wouldn't be speaking like this if he was born on Mars.
Sure he would, the meter would just be a different length.
I thought they key insight of this article was if he were born on Mars, then the meter would have been defined differently so that gravity would still be 9.8 m/s^2.
I think what you meant to say was that he wouldn't be speaking like this if people were born with 3 fingers.
The pendulum from the Mars pole to Paris would be long indeed!
Finally an easy way to identify aliens!
How? Because they don't have a Venus?
If the definition of the meter is still wrong disallowing π² = g, how might this affect other calculations like for example thrust and in aerospace engineering?
And what would all other natural constants look like, had the meter kept the value derived from the length of the pendulum?
I laughed 3 times reading this article while pondering the novelty of standardization.
Is standardization the sans-serif of civilization?
Another interesting coincidence (or perhaps a decades-long dad-joke troll perpetrated by German-speaking scientists) is that 1 hertz is roughly equal to the frequency at which a human heart (“Herz” in german, with a nearly indistinguishable pronunciation to “Hertz”) beats.
That seems rather low rate. Regular rate in rest is 60 to 100. Which only lower bound is roughly 1Hz, while upper rate is quite far what I would understand German to understand as roughly.
80+ heart rate at rest seems very high to me. Is this based on averages from present-day people? Including lots of sedentary people with hyper tension, overweight and obese people, etc. So "normal" in the statistical sense but maybe not "normal" in the physiological sense?
Athletes have much lower resting rate. So if we take feral humans from 50Ky ago their rest heart rate would have been much closer to the 60 bpm.
The pendulum equation isn't progress at all, it's just another observation of it? Driving it 'backwards' as it were with known values for the parameters it would determine, and seeing that pi squared is roughly g without having to know the actual values of those constants.
And now I've finished the article, nothing more is really offered. Am I missing something? That doesn't explain it/answer the question at all afaict? All we've done is find an equation that uses both pi and g, which shows again the relationship we started from?
The point is that the original definition of the meter was the length such that, when g is expressed in meters, pi^2 = g
Underlying idea of whole metric and SI system is the real start point. You want to define some units that you can easily replicate. Time is reasonable enough one, measure and track length of day and then length of second. Now based on this figure out way to come up with replicable definition for distance, pendulum is good enough gravity is constant enough. Thus linking gravity and meter arises.
From here you can define lot of other units like mass and Volt and Ampere... Everything comes from second which is weird, but does make lot of sense.
If society were tasked one day with reinventing standards and units, it doesn't matter why, what do you think are some things they would change?
For example, I think for human counting, base 12 is about as easy as base 10, but gives good ways to express division by 3, in addition to division by 2 or 4. It also fits better with how we count time, like how there are 60 seconds in a minute, 12 months in a year, etc... but I imagine those might be revised as well.
Anyways, I'm curious to hear what others think.
I think there's an argument for base 16 over 12. It's very slightly worse than base 12 for purely human use, since it's only divisible into halves, quarters and eighths. However, each hex digit maps to an exact number of binary digits, which I think outweighs the benefit of thirds and sixths in a digital world.
Of course, it's all theoretical, because there's no chance this would ever happen short of an apocalypse that takes us back to the stone age, and unless the radiation gives us 12 or 16 fingers, we'd probably just reinvent decimal.
To be honest I'm not a fan, time is cumbersome to do any sort of addition or subtraction to get exact days/hours/minutes (not to mention timezones etc).
Compare to metric units, always base 10 and always easy to convert mm to cm to m and so on.
Now that we live in a digital world - why do we consistently reinvent date/time libraries? To me that's proof enough the concepts are just hard to work with and over a long span of time verify your calculation is correct.
None of those issues with date and time are anything to do with the base, they're to do with date and time as defined by humans being inherently complicated concepts. Specifically, trying to have a single measurement "fit" for a load of different purposes.
If we had based our system around base 12, a base 12 version of the metric system would be just as easy to work with as metric is in decimal, with the added bonus that you can divide powers of the base (10, 100, 1000, etc.) into quarters, thirds and sixths without needing a decimal place, and thirds of 1 would be non-repeating.
I remember in mechanical engineering class we would often use this for exercise sheets. On our calculator we could directly enter π and ², thus it was equally as fast to entering 10.
That's one way of setting up a standard deviation!
How do mathematicians handle it when there are tantalizingly close relationships between purely mathematical numbers? I fell down this particular rabbit hole through the musical entrance (just intonation intervals), but I mean look at all this, it's clearly a setup.
https://en.wikipedia.org/wiki/Mathematical_coincidence
This is neat, but still something if a coincidence.
It appears the first definition of a metre is in fact around 1/4e10 the circumference of Earth, and the further coincidence is that a 1m mathematical pendulum has a period of almost exactly 2 seconds.
So there's still a neat relationship between mass/radius of Earth, its diurnal rotation period and the Babylonian division of it into 86,400 seconds.
According to the article the 1/4e10 circumference definition came second
Wikipedia says the Earth circumference definition comes from around Copernican times.
Also my reading of TFA is that the pendulum definition was in fact a redefinition that didn't catch on.
I wonder how many numbers they checked until they arrived on this one. As to me it seems picked as something close enough for committee work.
Our mechanics professor in university told us that Pi squared is ten and 6xPi=20. He said "engineers do not care for the fourth digit after the dot. If you want some number very precisely calculated just hire a mathematician instead because they are cheaper per hour"
Multiple people have brought this up and I just don't get it. When would you ever use Pi squared for anything, and if you'd never use it, who cares what its value is?
The only thing I could come up with is marking out an area by rolling a wheel some number of times to measure each edge.
Long ago, I memorized the square root of pi precisely because it seemed like the least useful number I could think of. (I was frustrated by something. Never mind.) But pi squared seems like it's pretty much the same in terms of usefulness.
If you said pi is the square root of 10, then I could see the value. Maybe that is what is being implied?
More fun arithmetic coincidences: https://en.m.wikipedia.org/wiki/Almost_integer
Some of these are test vectors for math systems.
There is nothing meaningful about this. I can change the unit system to make g any value I want (this is done all the time in research). I try really hard to ignore all the physics-related articles posted here but this one is too egregious. It's not the usual thing where the author know nothing about the nouns they are using (enter, quantum). In this case the concepts are fairly simple. The fact that units can be freely changed should be taught in middle school.
Somehow I found programmers have a much higher probability of talking about physics than people in other professions. And unfortunately in all cases I've seen, they have no idea what they are talking about. Unlike programming, physics is hard enough that it needs to be studied in classes.
This is addressed in the article, including the fact that you can change the unit system to change the value of g.
The article explains that the coincidence comes from the fact that the meter, as a unit, was defined (by Huygens) based on g and π. It was later redefined several times and the link between the two values became anecdotal. In other words, on another planet the gravitational constant would still have had a value of (approximately) π², and what would have been different is our unit length.
Yeah, I saw the author says it depends on the units. But like, why is this interesting? This is not physics, just some number coincidence in the metric unit system, and I'm sure one can find many more these kind of things by playing around with the constants. The fact the author calls this a "wonderful coincidence" is just... Like, a simple energy conservation or momentum conservation, taught in middle school, is infinitely worth being talked than this.
One philosophy in physics, is that the world and its rules are independent of human. We actively try to eliminate and downplay historical and human factors in the theory, and try to talk about just "the physics", because those factors often obscure the real physics (mechanism) and complicate the calculation. I mean people can find a historical thing interesting, but I guess I just feel disappointed that people find such a trivial thing so interesting, and maybe think that this is what physics is about, while physics is about anything but those pure coincidences.
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> was actually proposed back in the 17th century
Pretty sure it was done back in Sumer first.
https://www.ukbiblestudents.co.uk/Great%20Pyramid/chapter%20...
>“It was contended," says Dr. Peacock, " by Paucton, in his Mẻtrologie, that the side of the Great Pyramid was the exact 1/500th part of a degree of the meridian, and that the founders of that mighty monument designed it as an imperishable standard of measures of length.
https://www.theguardian.com/science/2020/dec/06/revealed-isa...
>Newton was trying to uncover the unit of measurement used by those constructing the pyramids. He thought it was likely that the ancient Egyptians had been able to measure the Earth and that, by unlocking the cubit of the Great Pyramid, he too would be able to measure the circumference of the Earth.
A standard free measure for distance? Sounds dubious.
No. The other way around. Two seconds is the period of a pendulum with a length of two Sumerian cubits.
(One meter is thus two Sumerian cubits, but that's an artifact due to us still using Sumerian time measurements.)
P.S. I don't know why Sumerians used a factor of two. Americans still divide the day into two 12 hour spans, according to Sumerian fashion.
P.P.S. One second is 1/(2*12*60*60) of a solar day. 12 and 60 were "round numbers" in Sumer; they used sexagesimal counting.
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Pretty sure this is just a coincidence.. I mean pi literally appears everywhere if you look hard enough lol
Having pi^2 = g would annoy me a bit as g is fundamentally a measured value. Depending on the required accuracy of the calculation, you can't even use the idealized value.
We simply wouldn’t need “g” and would say that acceleration due to gravity on earth is sqrt(pi).
g is related to the radius of the earth; the meter is related to the circumference of the earth; and pi is the relationship between the radius and the circumference.
Aside from the fact that the post already explained what the actual historical connection is, your explanation requires some serious hand-waving about the mass of the Earth and the gravitational constant, neither of which were known when the meter was first defined.
Reasonably accurate values for both M_earth and G were known at the time the SI meter was defined.
Also it's not too hard to extend this. M_earth is a function of Earth's radius which goes into the definition of the meter. G is a function of earth's orbital period, which goes into the definition of the second. Further our definition of mass is based on the density of water, which is chosen because it is a stable liquid at this particular orbital distance from a star of our sun's mass.
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Regarding "Catholic meter": its definition depends on time measurement. How did they ensure that "seconds" of different clocks were equal?
The traditional definition of the second before modern timekeeping was 1/86400 of a day. I’m guessing that was precise enough for their purposes.
It must have been brutal to create the first clocks when your smallest external reference is a day long. You first make a huge hour glass or clepsydra and tweak it once per day until it's perfect. Very slow debugging loop.
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What an amazing post! Such an interesting investigation. These kinds of write-ups make me realize how truly far we are from AGI. Sure, it can write amazing code, poems, songs, but can it draw interesting conclusions from first principles? I asked both ChatGPT and Claude, the same question, and both failed at pointing out the connection the author states.
This is not to deride feats of AI today, and I am sure it will transform the world. But until it can show signs of human ingenuity in making unexpected and far-off connections like these, I won't be convinced we are nearing AGI.
> Sure, it can write amazing code, poems, songs, but can it draw interesting conclusions from first principles?
Can you?
https://youtu.be/KfAHbm7G2R0?si=oAPrNGylo7pUcRMZ
Arguably most humans can't do this either.
The only thing with more bollocks in it than this article is your aibro comment.
Can we do the same with 432Hz please
432: yes it's a super fun and interesting number 432 cycles per second: seconds are not in fact special
Would readers well-versed in physics be happier if the equation in the title were `g ≈ π² m/s²` instead of `π² ≈ g`?
I knew that historically meter was related to the size of the earth somehow, but I had never had about the pendulum definition!
Philosophy time:
Does this mean in 400 years it's possible we no longer disagree about how to evaluate things? i.e. we converge on one totalitarian utility function that everyone basically accept answers every possible trolley dilemma?
In 1600, people just took the world as that: measurements are sloppy, and vary culturally and based upon location etc. But we eventually came upon tools and techniques that are broadly accepted as repeatable and standard.
Would this sort of shift be possible? Or desirable?
Personal preferences continue to exist. I don't see that happening.
Might be interesting if were true in Planck units.
But also 2Pi is fundamental, who defines a ratio of something to 2 of something (radius)?
No, Tau is fundamental. Pi only exists because someone mistakenly thought the formula for circumference involved diameter, when in fact it involves radius. ("Quit factoring a 2 out of Tau!" I tell them.)
Eh, you can find plenty of cases where tau is just as awkward as pi is elsewhere. Right off the bat, the area of a circle becomes more awkward with tau, becoming (tau*r^2)/2, and in general, the volume of an n-ball gains weird powers and roots of two in its denominator as n increases if you switch to tau. In general, I don't think you can really claim either one is "more fundamental". It's just a matter of framing.
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Any examples of textbooks or papers where the author used tau instead of pi (and the topic was not tau or pi)?
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That was (obliviously?) my point.
Another wonderful coincidence:
- Speed of light: 299,792,458 m/s
- Great Pyramid of Giza: 29.9792458°N
Link is broken for me?
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Try refreshing the page
g is the distance an earth skimming satellite falls to earth in one second along the tangent. But I don't know how it is related to pi.
Hence the 2√l formula for the pendulum haha
Why is this comment section, specifically, such an embarrassing dumpster fire?
It's a serious question; this is the sort of neat derivation that makes for a popular Youtube video, and despite Youtube comments being famously... variable in quality, the comment section on videos about things like this is vastly more literate than the threads here right now.
Is it just a coincidence, the chaotic behavior of uninformed sneer comments (which exist on every post; I've certainly been guilty, to my shame) meaning that some post is going to end up being the one with no other type of comment? Or is there some surprising reason why?
I wonder if this is how astrology was born. You can draw arbitrary connections between things if you stare at them long enough.
> And yet, no matter how you look at it, this can’t just be a simple coincidence.
why not?
Okay so this one has an explanation. But what about these ?
https://en.wikipedia.org/wiki/Mathematical_coincidence
See also this blog: https://martouf.ch/tag/coudee-royale-egyptienne/
One french royal cubit ≈ one egyptian cubit ≈ about π/6 meters. One royal span ≈ 1/5 meter = 20cm.
I'm wondering whether some of these coincidences could be explained by the anthropic principle, which deals with these quasi-equalities, for instance:
>An excited state of the 12C nucleus exists a little (0.3193 MeV) above the energy level of 8Be + 4He. This is necessary because the ground state of 12C is 7.3367 MeV below the energy of 8Be + 4He; a 8Be nucleus and a 4He nucleus cannot reasonably fuse directly into a ground-state 12C nucleus. However, 8Be and 4He use the kinetic energy of their collision to fuse into the excited 12C (kinetic energy supplies the additional 0.3193 MeV necessary to reach the excited state), which can then transition to its stable ground state. According to one calculation, the energy level of this excited state must be between about 7.3 MeV and 7.9 MeV to produce sufficient carbon for life to exist, and must be further "fine-tuned" to between 7.596 MeV and 7.716 MeV in order to produce the abundant level of 12C observed in nature.
Source: https://en.wikipedia.org/wiki/Triple-alpha_process#Improbabi...
The idea goes like this:
1. A more fundamental aspect under the anthropic principle which underpins the existence of complex life and intelligent observers is the quasi-alignment of values such as the fundamental constants in physics within a short margin.
2. If you consider the universe to be the product of a random sampling process over these constants (either real or virtual, it occurred many times or just once), and given the fact we exist, which implies an abundance of coincidences, the maths seem to tell us that we should expect to observe superfluous coincidences that are non-functional for the appearance of complex life, rather than the strictly minimal set of functional coincidences necessary for its emergence.
3. This implies that coincidences and pattern seeking are not just features (or bugs) of our complex minds but are present in the universe latently since it is not just fine-tuned for the emergence of complex life but for the presence of coincidences such as these https://medium.com/@sahil50/a-large-numbers-coincidence-299c....
4. It may be even testable by running computer experiments relying on genetic programming/symbolic regression to see whether there is something special about the value of physical constants in our universe when compared to the value they would have in other universes. I think such experiments should factor the fact that not all equations with the same mathematical complexity (number of operands and operators) have the same cognitive complexity. Indeed, if you look at the big equation in the link above, you'll remark that it can be further compressed into a/b = c/d (where a is the photon redshift radius for instance). So I guess you'd also have to throw into the mix Kolmogorov algorithmic complexity to assess this aspect (which is in fact used in some cognitive theories of relevance to tackle this kind of stuff to the tune of "simpler to describe than to generate")
Thoughts ?
Bullshit! One is a number, the other is an acceleration. Who cares that numeric values are similar for a given set of measurement units?
> this can’t just be a simple coincidence.
uhhhhhh yes it can?
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I disagree… the article talks about defining the meter using the pendulum and the second. Other planets would have their own definition of second, but not their own definition of pendulum. Since one meter is prescribed though a pendulum because of the oscillation formula and dependent on the frequency alone (in this case, 2 seconds), no matter what planet you are on, how strong gravity, is or how long a second is, the pendulum describes a relationship between seconds and meters such that if using this method a planet’s scientists would always define their units such that acceleration of gravity was eerily equal to pi^2.
Makes me think of possible lunar scientists unwittingly making their meter 5/6th shorter(edit: english is hard) and then marveling at the same coincidence…
Your comment is much more rage-bait than the article.
Universal isn't a way we describe numbers. You meant to say dimensionless. Pi is dimensionless constant because it describes a relationship between two measurements of a dimensionless unit circle.
Pi is expressed as a pure ratio between two other dependent numbers. Dimensionless values are special because they don't rely on any particular measurement in any particular location, lending to your misconception of "universal" constant.
This article explains how a particular dimensionful constant (g, the strength of gravity on earth's surface) is related to pi.
They are related because the dimensions in question are both derived from dependent properties of our planet. These dependent properties will be found on any other sphere floating in space if they are derived in the same fashion.
It's good to thoroughly or even marginally understand a topic before adopting a dismissive and authoritative argument against it.
> Universal isn't a way we describe numbers. You meant to say dimensionless.
They probably really meant to say “universal”, since dimensionless values are a less interesting category that includes… well, every number. Pi shows up in math without having to parameterize anything, making it universal in a way that even physical constants of our universe aren’t.
On any planet where you want to define a system of units, you can start by defining a fixed time period (maybe use a fraction of the planetary rotation cycle or something), then make a pendulum that swings with that frequency, and derive a unit of distance from its length.
The local value of g will be roughly pi squared pendulum lengths per tick squared, in that system of measurement.
If you read the article you would see that it’s not a coincidence because the meter was defined such that pi^2 = g at the surface of the earth.
Nope, it's not a coincidence - it's an interesting exploration of the history of the definition of a metre. Read the article.
As it says, at some point there was an attempt to standardise the length of a metre in terms of a pendulum's length; which related it directly to g through Pi.
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Numerologists unite!
No mention of ‘meter’ being the unit of measurement, make this like saying 3:14pm is related to pi.
There’s no correlation between a continuous number and a unit of measure. That’s truly apples to oranges comparison.
g can easily be expressed in ‘feet’ as ~32.1 ft/s^2
What do you mean by “no mention”? The entire article is about why this is specifically due to how the meter was first defined.
It isn't equal to g even in SI units except at some very few spots on the surface of the earth.
Change the units to any other system and it's not even roughly true.
Edit: Now that i have read the article i see that it is no coincidence at all that it is close the pi squared. very interesting.
Sorry to ruin the party, but g is a quite random number, on other planets the corresponding acceleration is different. So π^2~g is a pure coincidence and not relevant. The Newtonian gravitational constant G is a real constant btw.
Have you read the article? The point is that the definition of the metre, which is used in g, originates from the length of a pendulum that swings once per second in the gravity field around Paris. So it is a matter of definitions, and the length of the metre originates from the duration of the second and the Earth's gravity field. The definitions of 1/40.000 of the Earth's circumference or ~1/300.000.000 of a light second came later.
My intuitive assumption, then, is that on Mars they would have come up with a different meter such that π² ≈ 10 "mars meters" / s².
Or alternatively stated, that the Mars meter would be much shorter than Earth's meter if they used the same approach to defining it (pendulums and seconds).
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I have to admit I only read half of the article. Even if there is some historical fact there (but it was not mentioned at the beginning of the article), from a physical standpoint this comparison is already dimensionally wrong and also coincidentally only correct if you choose appropriate units. That was the point I was trying to make. There is not anything "deep" here.
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It's not about the values, but the units of measurement. g is in units of meter/second^2. The article discusses the dependency of the meter's original definition on the value of pi.
You are correct but the point is the way the meter is calculated, g in meters per second should come to pi squared.