A wonderful coincidence or an expected connection: why π² ≈ g | Hacker News Reader

A wonderful coincidence or an expected connection: why π² ≈ g

roitman.io

509 pointssigna11330 comments

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.

In the example case, the earth's rotation is producing the apparent observation: it's the cause, not a separate phenomenon that happens to coincide, or that might be indicative of a deeper relationship. For something to be a coincidence, it must be otherwise unconnected causally, which is not the case if the reason you found a ~24 hour period is that you forgot to account for the earth's rotation.

I respectfully disagree (without attempting to say you're wrong!) about the definition of coincidence and the requirement of being non causally related. If I'm riding on a bus and the light poles going past line up with my music, that's a coincidence even though they are cause soley by the bus motion BPM matching an essentially random choice of song BPM.

What you're describing is a coincidence.

What I'm describing is an artifact in your data that is caused by the motion of the earth.

To give a more concrete example, suppose you measuring the brightness of a trans-neptunian object, and observe that the brightness changes slightly with a period of about 1 year. You might think it has a non-uniform albedo, and a rotational period of one year, when in reality, it is just brighter when the earth is closer to it.

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).

Alpha brainwaves are almost exactly 10hz, in humans and mice. The typical walking frequency (for humans) is almost exactly 2hz (2 steps per second). And the best selling popular music rhythm is 2hz (120bpm) [1].

Perhaps seconds were originally defined by the duration of a human pace (i.e. 2 steps). These are determined by the oscillations of central pattern generators in the spinal cord. One might suspect that these are further harmonically linked to alpha wave generators. In any case, 120bpm music would resonate and entrain intrinsic walking pattern generators—this resonance appears to make us more likely to move and dance.

Or it’s just a coincidence.

[1] https://www.frontiersin.org/journals/neurorobotics/articles/...

Well, a second is also a pretty good approximate resting heart rate (60 bpm)

> Well, a second is also a pretty good approximate resting heart rate (60 bpm)

I'm sorry to be the kind of person who feels compelled to make this comment, but you mean a Hertz, not a second.

You're right. Thanks for being that guy!

Youre right, you should be sorry. Dont be that guy.

Another bad way to check for non-coincidences is to use a value like g which changes depending on your location.

Pi is the same everywhere in the universe.

g on Earth: 9.8 m/s²

g on Earth's moon: 1.62 m/s²

g on Mars: 3.71 m/s²

g on Jupiter: 24.79 m/s²

g on Pluto: 0.62 m/s²

g on the Sun: 274 m/s²

(Jupiter's estimate for g is at the cloud tops, and the Sun's is for the photosphere, as neither body has a solid surface.)

My physics prof said g is actually a vector field. Because the acceleration has a direction and both magnitude and direction vary from point to point.

Absolutely true on astronomical scales.

An unnecessary complication if you're dropping a brick out of a window.

It's funny how much of physics we do assuming a flat earth.

If you did it "properly" you would calculate the orbit of the brick (assuming earth was a point mass), then find the intersection between that orbit and earth's surface. But for small speeds and distances you can just assume g points down as it would in a flat earth

Your physics Prof is correct of course, and so is GP. "Standard" values for g exist for these bodies, but it also varies everywhere.

This is correct, gravitational constants are a good approximation/simplification since the mass of solar bodies is usually orders of magnitude greater than the other bodies in the problem, and displacement over the course of the problem is usually orders of magnitude smaller than absolute distance between them.

In other words, we assume spherical cows until that approximation no longer works.

I volunteer for the Mars mission as a weight loss tool.

Surviving on Mars will probably involve some mass loss, too.

With the current technology, even getting on a rocket to Mars will involve some weight loss - the mass budget is tight, and each kilogram they can trim off the crew bodies is a kilogram that could be put towards fuel, life support, or scientific equipment.

Fun fact: pi is both the same, and not the same, in all of those places, too.

Because geometry.

If you consider pi to just be a convenient name for a fixed numerical constant based on a particular identity found in Euclidean space, then yes: by definition it's the same everywhere because pi is just an alias for a very specific number.

And that sentence already tells us it's not really a "universal" constant: it's a mathematical constant so it's only constant given some very particular preconditions. In this case, it's only our trusty 3.1415etc given the precondition that we're working in Euclidean space. If someone is doing math based on non-Euclidean spaces they're probably not working with the same pi. In fact, rather than merely being a different value, the pi they're working with might not even be constant, even if in formulae it cancels out as if it were.

As one of those "I got called by the principal because my kid talked back to the teacher, except my kid was right": draw a circle on a sphere. That circle has a curved diameter that is bigger than if you drew it on a flat sheet of paper. The ratio of the circle circumference to its diameter is less than 3.1415etc, so is that a different pi? You bet it is: that's the pi associated with that particular non-Euclidean, closed 2D plane.

So is pi the same everywhere in the universe? Ehhhhhhhh it depends entirely on who's using it =D

In non-euclidean spaces, your definition of pi wouldn't even be a value. It's not well defined because the ratio of circumference to diameter of a circle is dependent on the size of the circle and the curvature inside the circle.

It's probably true that it's only well defined in euclidean space. Your relaxed definition, which I have never seen before, is not very useful.

I don't agree, I thought what he said was very interesting. It never occurred to me that pi might vary, and over a non-flat space I can see what they're saying. I think it's intrinsically interesting simply because it breaks one of my preconceptions, that pi is a constant. Talking about it being 'not very useful' just seems far too casually dismissive.

This whole conversation is painful to read:

1. Your parent was talking about projections from one space to another and getting it confused.

2. Pi is pi and their non-Euclidean pi is still pi (unless you want to argue that a circle drawn on the earth’s surface has a different value of pi).

The problem comes down to projections, then all bets are off.

Just sounds like you’ve confused yourself. It’s like spinning in circles and acting like no one else knows which way is up.

That isn’t a different pi. That’s a different ratio. Your hint is that there are ways to calculate pi besides the ratio of a circle’s circumference to its diameter. This constant folks have named pi shows up in situations besides Euclidean space.

Good job, you completely missed the point where I explain that pi, the constant, is a constant. And that "pi, if considered a ratio" (you know, that thing we did to originally discover pi) is not the same as "pi, the constant".

Language skills matter in Math just as much as they do in regular discourse. Arguably moreso: how you define something determines what you can then do with it, and that applies to everything from whether "parallel lines can cross" (what?) to whether divergent series can be mapped to a single number (what??) to what value the circle circumference ratio is and whether you can call that pi (you can) and whether that makes sense (less so, but still yes in some cases).

Modern mathematics is more likely than not going to define pi as twice the unique zero of cos between 0 and 2, and cos can be defined via its power series or via the exp function (if you use complex numbers). None of this involves geometry whatsoever.

What do you mean? Cos is an inherently geometric function.

Or the speed of light being almost a sweet 300 million m/s.

Or after-atmosphere insolation being somewhat on average 1kw/m2.

I always find insolation and insulation to be such an interesting pair of words

I guess the equivelent of "change the units" is "change the language".

French: insolation et isolation

German: Sonneneinstrahlung / Isolierung

Spanish: insolación / aislamiento

Chinese: 日照 / 绝缘

I guess coincidence

insolation < Latin sol, solis m "sun"

insulation < Latin insula, -ae f "island" (apparently nobody knows where this one comes from)

isolation < French isolation < Italian isolare < isola < Vulgar Latin *isula < Latin insula, -ae f

Spanish aislamiento < aislar < isla < Vulgar Latin *isula < Latin insula, -ae f

Oh and the English island never had an s sound, but is spelled like that because of confusion with isle, which is an unrelated borrowing from Old French (île in modern French, with the diacritic signifying a lost s which was apparently already questionable at the time it was borrowed), ultimately also from Latin insula.

> Oh and the English island never had an s sound, but is spelled like that because of confusion with isle

So the German cognate (I assume) Eiland probably hasn't had one either. Makes sense, since the Nordic Eya / Øy / Ö never felt like they should, and they must be just northern variants that lack the -land (literally the same in English) suffix.

Usefully, the speed of light is extremely close to one foot per nanosecond. This makes reasoning about things like light propagation delays in circuits much easier.

I really wish we had known this back before it was way too late to seriously change our units around. It would mean that our SI length units wouldn't have to have some absolutely ridiculous denominator to derive them from physical constants, and also the term "metric foot" is pretty fun.

I use the term "natural foot." It's very useful in simulations.

Fun, and poetic too

See, the issue with "foot" is that different people use different body parts to measure length. Germany used the "Elle", which is the distance between wrist and elbow, or roughly one foot. Other regions used the foot or the cubit instead.

The primary advantage of the SI system is that it has only ONE length unit that you add prefixes to.

> Or after-atmosphere insolation being somewhat on average 1kw/m2.

I’m kind of inclined to say that this one isn’t so much of a coincidence as it is another implicit “unit” in the form of a rule of thumb. Peak insolation is so variable that giving a precise value isn’t really useful; you’re going to be using that in rough calculations anyway, so we might as well have a “unit” which cancels nicely. The only thing that’s missing is a catchy name for the derived unit. I propose “solatrons”.

units(1) calls it `solarconstant` or `solarirradiance` but that's the quantity above the atmosphere. the same term is sometimes used for the quantity below the atmosphere: https://en.wikipedia.org/wiki/Solar_constant and of course that depends on exactly how much atmosphere you're below

in that sense, oddly enough, the solar constant is not very constant at all

There is relationship between the metric system and the French royal system. The units used in this system have a fibonacci-like relationship where unit n = unit n-1 + unit n-2.

    palm : 7,64 cm
    span : 12,36 cm
    handspan : 20 cm
    foot : 32,36 cm
    cubit: 52,36 cm

cubit/foot =~ 1,618 =~ phi, el famoso Golden ratio. foot/handspan =~ phi too. And so on.

From this it turns out that 1 meter = 1/5 of one handspan = 1/5 x cubit/phi^2

Another way to get at it is to define the cubit as π/6 meters (= 0.52359877559). From this we can tell that

1cbt = π/6m

π meters = 6 cubits

Source: https://martouf.ch/crac/index.php?title=Quine_des_b%C3%A2tis...

32 meters is 35 yards, to within about an eighth of an inch. How's that grab you ?

I wonder if this is related, but imperial measurements with a 5 in the numerator (and a power of two in the denominator) are generally just under a power of two number of millimeters.

The reason is fun, and as far as I know, historically unintentional. To convert from 5/(2^n) inches to mm, we multiply by 25.4 mm/in. So we get 5*25.4/(2^n) mm, or 127/(2^n) mm. This is just under (2^7)/(2^n) mm, which simplifies to 2^(7 - n) mm.

This is actually super handy if you're a maker in North America, and you want to use metric in CAD, but source local hardware. Stock up on 5/16" and 5/8" bolts, and just slap 8 mm and 16 mm holes in your designs, and your bolts will fit with just a little bit of slop.

So the error is 1.6%. Acceptable for everyday hardware I guess.

My favorite is 1 mile = phi kilometers with <1% error

That one’s useful too. If you know a few Fibonacci numbers you can convert miles to kilometres and vice versa with ease.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55 …

21 km is ~13 miles, 13 km is ~8 miles, etc.

A 26 mile marathon? Must be ~42km.

Same for speed limits too; 34 mph is ~55 kmh

I use that approximation, via the Fibonacci sequence, to translate between miles and km. 13 miles ~ 21 km (actually 20.921470).

My favorite approximation is π·E7 = 31415926.5... , which is a <1% error from the number of seconds in a year.

... you all realize that phi is barely a better approximation than 8/5, right? 1.6 vs 1.609 (km in a mile) vs 1.618?

(8/5)/(1 mile/1 km) = 0.9942; (1 mile/1 km)/phi = 0.9946. You're making things way harder on yourself for essentially no improvement in precision, especially when you're just rounding to the nearest whole number.

1 km = 5 furlong, with < 1% error.

Reminds me of https://xkcd.com/687/

X^2 is a lot more interesting than x*0.0000743 or whatever it is

Ok, then by that thinking, you should find it really interesting that Earth escape velocity is almost exactly ϕ^4 miles per second.

In fact, adding exponents here objectively makes it less interesting, because it increases the search space for coincidences.

What makes the case in the post most interesting to me is that it looks at first glance like it must be a coincidence, and then it turns out not to be.

You would need a compelling argument why x^4 is interesting. Especially something like the golden ratio.

Not a long of things go to the fourth power in equations we need to use. Pi^2 directly features in periodicity

Why is it more interesting? Is it just more interesting because we use such bases, or can it be interesting inherently? That is the question that is being asked, and why some say it's merely a coincidence.

Well every number is the product of another number and some coefficient. If it’s a nice clean number then that implies it could be the result of some scaling unit conversion. But that should be sort of apparent. And it’s not super interesting if true.

If a number is another number squared then that implies some sort of mechanistic relationship. Especially when the number is pi, which suggests there’s a geometric intuition to understanding the definition.

In other bases, it does not actually imply much, even if it were squared. Maybe it really does make sense if it existed in base 10 but I cannot see much if it were part of other bases.

> 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.

Yeah, it was a strange claim, which makes me think that the author may have had his conclusion in mind when writing this. I.e. what he meant to say may have been something more like:

"The relationship vanishes when you change units, which suggests the possibility that the relationship is a function of the unit definitions... and therefore not a coincidence."

Because the cubit is a measure of what a body can reach

I never thought of the cubit this way. It's an interesting idea, but the cubit is the length of a forearm, whereas you can reach around yourself in a circle the length of your extended arm, from finger tip to shoulder.

That would be somewhere between 1.5 to 2 cubits for people whose forearm is about a cubit long.

I think the cubit is mainly a measure of one winding of rope around your forearm. That way you can count the number of windings as you're taking rope from the spool. This is the natural way a lot of us wind up electrical cables, and I'm sure it was natural back in the day when builders didn't have access to precise cubit sticks.

I don't see the connection with the units and sound that you're making. But it is kind of interesting to know that sound travels about 3/4 of a forearm length per millisecond. That's something that's easy to estimate in a physical space.

How does that explain the relationship to the speed of sound?

A bit out of topic, however

https://www.youtube.com/watch?v=0xOGeZt71sg

Note: I'm more inclined to think this is a coincidence given that it establishes a link between the most commented text and the the most commented building in history. However I don't think these kind of relationships based on "magic thought" should be discarded right away just because they are coincidences, and I'd be very interested in an algorithm that automatically finds them.

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.

Pi is related to the circumference of a circle; the meter was originally defined as a portion of the circumference of the Earth, which can be approximated as a circle.

"The meter was originally defined as one ten-millionth of the distance between the North Pole and the equator, along a line that passes through Paris."

But that connection actually is a coincidence. From what I can tell, when they standardized the meter, they were specifically going for something close to half of a toise, which was the unit defined as two pendulum seconds. So they searched about for something that could be measured repeatably and land on something close to a power of ten multiple of their target unit. The relationship to a circle there doesn’t have anything to do with the pi^2 thing.

Not a coincidence. They defined the meter from the second using the pendulum formula, and the pandulum formula has a pi in it, so pi is going to appear somewhere. The reason there is pi is probably because a pendulum is defined by its length and follows a circular motion that has the length as its radius.

We could imagine removing pi from the pendulum equation, but that would mean putting it back elsewhere, which would be inconvenient.

Right, that connection is not a coincidence. The connection the previous commenter drew between the meter, pi, and the circumference of the earth is a coincidence.

> The reason there is pi is probably because a pendulum is defined by its length and follows a circular motion that has the length as its radius.

It’s not quite that easy: For small excursions x the equation of motion boils down to x’’+(g/L)x=0. There is not a π in sight there! But the solution has the form x=cos(√(g/L)t+φ), with a half period T=π√(L/g), thus bringing π back in the picture. So indeed not a coincidence.

It was news to me, but that's what the article says, and it is supported by by Wikipedia, at least. [1]

In addition, I feel the article glosses over the definition of the second. At the time, it was a subdivision of the rotational period of the earth (mostly, with about 1% contribution from the earth's orbital period, resulting in the sidereal and and solar days being slightly different.) Clearly, the Earth's rotational period can (and does) vary independently of the factors (mass and radius) determining the magnitude of g.

The adoption of the current definition of the second in terms of cesium atom transitions looks like a parallel case of finding a standard that could be measured repeatably (with accuracy) and be close to the target unit - though it is, of course, a much more universal measure than is the meridional meter.

[1] https://en.wikipedia.org/wiki/History_of_the_metre

Can you explain what you’re taking issue with in the post, then? Because it specifically lays out how the historical relationship between the meter and the second does in fact involve pi^2 and the force of gravity on earth.

(Granted, from what I can tell, it’s waving away a few details. It was the toise which was based on the seconds pendulum, and then the meter was later defined to roughly fit half a toise.)

pi is always just pi, but g may be defined in terms of the meter.

sure, that's the entire point.

heck, g is not even a constant, it just happens to measure to roughly 9.8 m/s² at most places around here.

It does, and the formula in the post explains the connection

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.

> since reality is definable as that which is independent of measurement.

In terms of quantum mechanics, would that mean the wave function is real until it collapses due to measurement? Or am I misunderstanding your use of measurement there?

Something about that is sticking in my mind in an odd way, but I can't put my finger on exactly what it is - which is intriguing.

Measurement can change what is measured, but it doesnt change it from illusion to reality.

I cannot measure santa clause into existence. But I can change the temperature of some water by measuring it with a very hot thermometer.

That measurement changes what is measured is the norm in almost all cases, except in classical physics which describes highly simplified highly controlled experiments. The only 'unusual' thing about QM is its a case in physics where measurement necessarily changes the system, but this is extremely common in every other area. It is more unusual that in classical physics, measurement doesn't change the system.

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.

The comma differentiates. The comma indicates a short pause and a certain intonation in speech (the period means a longer pause and a different intonation). If you say that sentence with and without a pause/comma, you'll see (hear) that the sentence is correct. Reading unambiguously is also hard.

The problem with that is that writers are not consistent with comma usage either, particularly when it comes to informal writing, where prescriptive rules are out the window anyway. And I would argue that it would be a bit of a norm violation even in informal writing to introduce this new point at the end of a paragraph rather than starting a new one, which makes me think that that was not the author's intent.

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.

Only because you're using metric seconds instead of "imperial seconds" (the time it takes for a 1 foot long pendulum to complete a full oscillation).

Sure, if you change either of the units you can always change the other one to fix the equation again.

But does it work when you use the right Imperial technique?

If I come up with my own measuring unit, let's call it the sneezle (whatever the actual length I assign to it) I will be able to also define a duration unit (say, the snifflebeat) based on the time it takes for a pendulum one sneezle long to complete a full oscillation, and vice versa I can define the sneezle by adjusting the length of a pendulum so that it oscillates in two snifflebeats. Here are the maths:

T = 2π√(l/g)

T/2π = √(l/g)

(T/2π)^2=l/g

g = l/(T/2π)^2

g = l/(T^2/4π^2) = 4π^2xl/T^2

Now substitue T with 2 and l with 1 an you get

g = 4π^2x1/2^2 = π^2

It doesn't matter what the pair of units assigned to T and l are. However, they'll be interrelated.

There is nothing arbitrary, and no coincidences behind g =~ π^2. It just requires to do some history of metrology and some basic maths/physics.

If you want to discuss coincidences, may I suggest you to comment on this remark I made and which hasn't received any attention yet ?

https://news.ycombinator.com/item?id=41209612

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

A more natural way to say it is that equality requires that the unit of length is the length of an arbitrary pendulum and the unit of time is the half-period of the same pendulum.

The pendulum is a device that relates pi to gravity.

Sounds universal. Get a different value on the Moon? Of course... pi squares differently on the moon :)

The arbitrary length pendulum with a period of 2 seconds which is your unit of length, (or 1 Catholic meter) is much shorter on the moon. In local Catholic meters gravity would be pi squared Catholic meters / second. As it would on any planet.

I think I understand. Distance scales with Planet Size? That actually makes a lot of sense.

The equation holds in imperial units as well. The length of the 2 second pendulum needs to be in feet AND the value of g in ft/sec2.

π^2 ≈ 32 to you?

Solving the equation for pi we get:

PI = sqrt(g/L)

g = 9.81. L=1

g = 32.174. L=3.174

Either way works to approximately pi. There is a particular length where it works out exactly to pi which is about 3.2 feet, or about 1 meter. My point was that equations like that remain true regardless of units.

The reason pi squared is approximately g is that the L required for a pendulum of 2 seconds period is approximately 1 meter.

Replace s in your calculation with imperial s instead of metric s and it isn't imperial feet per metric seconds.

Imperial seconds were very, very close to metric seconds.

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?

It’s more precisely the difference between “rationalized” and “unrationalized” units.

You need a factor 4pi in either Gauss’ law or Coulomb’s law (because they are related by the area 4pi*r^2 of a sphere), and different unit systems picked different ones.

It’s more akin to how you need a factor 2pi in either the forward or backward Fourier transform and different fields picked different conventions.

> It’s more akin to how you need a factor 2pi in either the forward or backward Fourier transform and different fields picked different conventions.

Some fields even use the unitary transform -- they split the difference and just throw in a 1/sqrt(2pi) in both directions.

https://en.wikipedia.org/wiki/Fourier_transform#Angular_freq...

It is more involving [1]

[1] https://phys.libretexts.org/Bookshelves/Electricity_and_Magn...

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.

I initially had an objection due to a misconception I was carrying.

I see now that the pendulum formula is a pure relation between time and distance/length. It will apply regardless of the units used. For example if we measure time in fortnite and length and furlongs the formula will be the same. The gravitational acceleration of course will be in those units: furlongs per fortnights squared. Needless to say that will not be 9.81.

Now the meter unit was chosen in relation to the second unit by the length of a pendulum that produced an integer period. So that choice/relation caused the gravitational acceleration g to take on such a value that its square root cancels out the π on the outside of the root.

I was confused for a moment thinking that the definition of the kilogram would somehow be mixed up in this but of course it isn't. g doesn't incorporate mass; and of course pendulum swings are dependent only on length and not mass.

There are all sorts of situations in which certain units either give us a nice constant inside the formula or eliminate it is entirely.

For instance Ohm's law, V = IR. It's no coincidence that the constant there is 1. If we change resistance to some other unit without changing how we measure voltage and current we get V = cIR.

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.

Truth feels like a coincidence when 1 small thing can make anything wrong.

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.

Another one would be philosophy: there’s nothing, something and everything. Or logic: ∃, ¬∃ and ∀. Just rambling here, but seems like universal concepts across fields.

Chiming in from theoretical linguistics: it is impossible for natural languages to "count", i.e. make reference to numbers other than 0, 1 or infinity.

As an example, there are languages where prenominal genitives are impossible (0).

Then, there are languages, such as German, where only one prenominal genitive is possible (1):

> Annas Haus

> *Annas Hunds Haus

Finally, there are languages, such as English, where an infinite number of prenominal genitives are possible (infinity).

> Anna's house

> Anna's dog's house

> Anna's mother's dog's house

> Anna's mother's sister's ... dog's house

But there are no languages where only two or three prenominal genitives are possible.

This property is taken to be part of Universal Grammar, i.e. the genetic/biological/mental system that makes human language possible.

As a German speaker, the claim "more than one prenominal genitive is impossible" seems interesting but perhaps not entirely accurate. As a non-linguist I probably misunderstand the meaning of prenominal genitive and might be missing your argument's point. For the overarching discussion we should note though the great variety of refering to "more than one element" in German.

> Anna's mother's sister's dog's house

> Das Haus des Hundes der Schwester der Mutter von Anna.

It's difficult to parse though. We only get to know about Anna at the end of the sentence. Consequently, we avoid such sentences or use workarounds.

> Von Annas Mutters Schwester dem Hund sein Haus.

If you ever use such a construction German speakers will correct your bad language but they will perfectly understand the sentence's meaning.

A quick search brought me to this presentation: https://www.linguistik.hu-berlin.de/de/staff/amyp/downloads/...

The take-away seems to be: "German does not work like English" (slide 8) so be careful when comparing PreGen in German and English.

He already got rid of "three" and just needed a little help to get rid of "two." Since we already have 0 and (almost) everything else can just be "one more" than something else, we only need 0 and one more ... or 1.

One and Done!

Very much like lambda calculus

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”.

All numbers are imaginary.

sqrt(-1) should have been called w, for "weird".

I use w for the complex roots of 1 though, when rewriting FFT notes.

You mean the legendary “i”. There is no n.

And -0 and NaN

And int_max

That's a implementation detail.

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.

Definitely not serious :)

But from the fact that an Earth year happens to be roughly pi*10^7 seconds long, it follows that in 10^7 seconds Earth travels about two radians, or one orbital diameter, and equivalently that the diameter of Earth's orbit is roughly 10^7 seconds times Earth's orbital speed.

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.

Me too.

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.

> Meter, second, and kilogram were all chosen to be approximately the scale of a human,

“Approximately the scale of a human” leaves so much wiggle room that I don’t see how one can defend that claim.

> and the combined multiplicative units like Pascal

You don’t explicitly claim it, but I wouldn’t say the Pascal is “Approximately the scale of a human”. Atmospheric air pressure is about 10⁵ pascal, human blood pressure about 10⁴ pascal, and humans can very roughly produce about that pressure by blowing.

Which is why I have always kinda hated kilogram. Such an ugly unit for it having prefix. Grav should have been correct answer, but instead we ended up with something that is too small... That is in reality gram. For grams we could simply have milligravs or decigravs for 100g equivalents... Not that hard considering decilitres are used and decimetres are kinda tried in schools.

Pascal is pretty small and m^3 is pretty large though. That's why we still often use millibar and liter. They just happen to largely cancel out.

A human is ~1 meter, 1e2 kilograms and 1e9 seconds.

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.

> why is this interesting?

Physicists need some precise definitions of units, and this is hard. Harder than most people expect. You can't do physics properly using your current king's foot size. This, more than the actual computations, was Huygens' valuable insight.

So you need a universal constant to serve as a standard, and it turns out very few things are in our world. One of them is the ratio of the perimeter of a circle over its diameter. So it's no wonder that this ratio comes up under various forms in our standard units, more often than chance would predict.

This is interesting because students of physics need to understand the complexity and importance of coming up with a standard set of measurement units, based on universal constants.

This is also interesting because the reason we need standard units is that we need science to be reproducible. If all I care about is to understand the world on my own then using the size of my own foot will do just fine as a unit.

Accessorily it's also useful to address the nonsense belief that such coincidences prove the existence of god or the perfection of nature.

None of this will come as radically insightful to you, but there are a lot of people in this world for whom this is not the case.

I'm also not a fan of over the top language, but this seems to be the norm of our attention-seeking times.

>This is not physics, just some number coincidence in the metric unit system

It's not a coincidence. The meter was (historically) intentionally defined as how long a pendulum is that swings in 2 seconds. When you do that, g = pi^2.

>The fact the author calls this a "wonderful coincidence"

The author doesn't call it a wonderful coincidence. The author asks the question of whether it's a wonderful coincidence or not, and comes to the conclusion: no.

Consider this to be an article about physics, not history. The article can be boiled down to one sentence: “the meter was originally defined to make pi^2=g”. This was a fun fact I didn’t know.

Too late to edit, but I meant to say “consider this to be an article about history, not physics”.

> 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.

Probably because 12 is a much better base than 10. 12 can be divided by 2, 3, 4 and 6 and still results in whole numbers, which helps a lot when doing rounding and fraccional numbers. The only reason we use base 10 is because is much easier to count with our fingers.

There's also a method of counting where you touch your thumb to the sections of your fingers on the same hand, which naturally lends itself to base 12. This can be extended by keeping track of how many times you've counted to 12 on the other hand, which lends itself to either base 60 or base 144.

Interestingly, the Sumerians did not seem to employ this method, they would count 6 instances of counting to 10.

Because it traverses the distance twice would be my guess. If you show someone a pendulum going through 3 periods and asked a group of people a generic question like “how many times did it move” without clarifying what you meant I would bet maybe half the people would say 6 as long as everyone counted correctly.

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.

As far as I can tell, the most recent experiment to measure the mass of the Earth by 1790, when they decided on the definition of the meter, was the 1772 Schiehallion experiment, which gave a value 20% below the actual value. So if pi^2 were to somehow fall out of that it would likely be so far off as to be unrecognizable.

But even that doesn’t matter, because the mass of the Earth didn’t play a direct role in the definition of the meter. If you take out the whole thing about the meter’s definition targeting half a toise, then all you have is “related to the circumference of the Earth”, and it would be a monumental coincidence if the mass of the earth and gravitational constant just conspired to somehow drop an unadulterated pi^2 out of the math.

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.

Well, defining seconds via "caesium frequency" is perhaps more brutal.

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.

Actually, no: that Tau-centric area formula you gave derives naturally from taking the integral. Your example actually fits the expectation you have from what you learned in Calculus I. You should _expect_ that 1/2 scaling to be there.

If it seems awkward to you, it's only because of a lifetime of seeing it done in terms of pi.

In case you didn't see this:

https://tauday.com/tau-manifesto

And this is the section that addresses your point:

https://tauday.com/tau-manifesto#sec-circular_area

It's not really about being awkward (that's a tell not the motivation), it's about basing on a radius or diameter: which is more fundamental? Or the arc length of a unit circle or half a circle, which isn't an arbitray formula it's the definition.

Any examples of textbooks or papers where the author used tau instead of pi (and the topic was not tau or pi)?

There might be, I don't know. But as you're already aware, we are culturally stuck with pi and so that's how it is taught.

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?

"[ErrorBoundary]: There was an error: {}"

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

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?

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).

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.

You are correct but the point is the way the meter is calculated, g in meters per second should come to pi squared.

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