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原始链接: https://news.ycombinator.com/item?id=41208988

Pi (τau) 是一个数学常数，其近似值为 3。14159。它将圆的周长与直径之比联系起来，无论其在欧几里得空间中的位置如何。虽然该值保持恒定，但在非欧几里得空间中，周长与直径的比率可能会变化，从而导致根据几何形状产生不同的 pi（或 tau）值。例如，在球体上画一个圆会导致直径弯曲，从而使周长与直径之比小于 3。14159，这表明与该特定非欧几里得闭合二维平面相关的唯一 pi（或 tau）。此外，pi 在其特定环境（即欧几里得几何）中被认为是通用常数，但实际上，它在不同的几何中会有所不同。因此，圆周率在宇宙中的各处并不相同，因为它取决于所使用的基础几何形状。此外，虽然 pi 由于与圆相关而广为人知，但它在三角学和微积分等各个数学领域中发挥着至关重要的作用，其属性揭示了与 e 和 phi（黄金比例）等其他数学常数的有趣关系。 Lastly, despite common misconceptions, pi is neither "irrational" nor "transcendent," as there exist algebraic expressions representing approximated values for pi。 Instead, pi belongs to the class of Liouville numbers, which are real numbers that cannot be rational but have bounded partial sums of convergent series representations。

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.

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)

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

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...

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.)

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

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.

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.

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.

Yes. That's what makes it a fun fact. Most people never even learn about non-euclidean math, and this is the kind of "wow I never even thought about this" that people should be able to learn about in a comment thread.

Calling it painful to read is downright weird. Pi, the constant, has one value, everywhere. So now let's learn about what pi can also be and how that value is not universal.

It isn't a "fun fact" ... it's plainly incorrect.

π never changes its value. Ever. It is a constant in mathematics, no matter the geometry. However, π can have different ratios in other geometries, but it will still be ~3.14.

It is painful because this statement:

> 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.

The problem here is *projections*. If you project non-Euclidean space onto Euclidean space, you end up with some seemingly nonsensical things, like straight lines that get projected into arcs. This is your problem. You projected a curved line from non-Euclidean space onto Euclidean space and got an arc, but didn't account for the curvature of your "real non-Euclidean space" and thus ended up with an invalid value for π. If you got something that isn't ~3.14, then you did the math wrong somewhere along the way.

It doesn't have anything to do with projections.

Let's say you live in a non-flat space. You come up with the idea of a "circle" with the usual definition: the set of points on the same plane equidistant from a central point.

You then trace along the circle and measure the length, and compare it to the length of the diameter. It turns out that this ratio changes as a function of the diameter. This truth is inherent to curved spaces themselves, and is not an artifact of choosing to describe the space using a projection.

The definition of pi in this world is no longer the invariant ratio of diameter to circumference. You can still recover pi by taking the limit of this ratio as the diameter length goes to zero. But perhaps mathematicians in this world would (justifiably) not see pi as such a fundamental number.

Now back to GP's example: people living on a sphere (like us) are analogous to inhabitants of a non-Euclidean space. The surface of Earth is analogous to 3-space, and the curvature of Earth is analogous to the curvature of space.

And indeed, if you draw real-life larger and larger circles on our planet, you will find that the ratio of circumference to diameter is smaller than pi. For example, if you start at some point on the Earth (say, the North pole) and trace out a circle 100 miles distant from it, you will find that the circumference of that circle (as measured by walking around that circle) is a little bit _less_ than pi x 100 miles.

Again, we have done no "projection" here. We've limited ourselves to operations that are fairly natural from the perspective of a mathematician living in that space, such as measuring lengths.

The fact that large circle circumferences measure less than pi x diameter on Earth did not change how we developed math, likely because you only notice start to notice this effect with extremely large (relative to us) circles.

But perhaps the inhabitants of a non-Euclidean space that was much more highly curved would notice it much earlier, and it would affect their development of maths, such that the number pi is held in lower regard.

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.

Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry. Clearly everyone means euclidean space unless specified otherwise. Any other interpretation will only lead to problems, which is why it's not useful. There is really no ambiguity about this in mathematics. Mathematicians still use the pi symbol as a constant when they compute the circumference of a circle in a given geometry as a function of the radius.

> Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry.

Can you share some place where Pi isn't defined exclusively as being "the ratio of circumference to diameter of a circle"? I have never heard any other definition in my life, and couldn't find any other through the first few Google results

Ur being rather snotty about this. I've just realised something important which is so obvious to you that you consider it trivial, but it's not. I realised something important today, you might just want to feel pleased for me, and a bit pleased that the world is a little less ignorant today...? Or not?

There is no "clearly" in Math. The fact that pi is a constant while at the same time not being "the same constant" in all spaces, and not even being "a single value, even if we alias it as the symbol pi" is what makes it a fun fact.

Heaven forbid people learn something about math that extends beyond the obvious, how dare they!

I don't know which ones you read so I can't comment on that, but the ones I read most definitely specify which fields of math they apply to, and which axioms are assumed true before doing the work, because the math is meaningless without that?

Papers on non-Euclidean spaces always call that out, because it changes which steps can be assumed safe in a proof, and which need a hell of a lot of motivation.

And of course, that said: yeah, there are papers for proofs about things normally associated with decimal numbers that explicit call out that the numbers they're going to be using are actually in a different base, and you're just going to have to follow along. https://en.wikipedia.org/wiki/Conway_base_13_function is probably the most famous example?

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.

There are many equivalent definitions and many of them do not refer to geometry at all. If you don't want to go through cos you can always define pi as sqrt(6 * sum from 1 to inf 1/n^2).

You can also define it as `3.14159...` just because. Obviously, a π definition entirely divorced from geometry becomes irrelevant to it - instead, in geometry, you'd still use π = circumference/diameter, or π = whatever(cos), and those values would happen to be the same as a non-geometric π, but only if the geometric π is the one from Euclidean geometry.

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 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.

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.

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.

I’m saying that the single SI length unit could have been defined precisely as the light nanosecond, or “metric foot”, had people known that that length fit closely to an existing unit back around 1790.

There would still be one unit with prefixes added, but that unit would have a really clean correspondence to physics rather than a hacky conversion factor.

But you have to go back that far in time for it to work, because it’s a fraction of a percent off of the current standard foot. They were happy to make those kinds of changes (as in the case of defining the meter to be ~0.51 toises) back when all of the existing measurements were pretty imprecise to begin with.

Of course, that’s why it could never have worked out this way. By the time we could measure a light nanosecond, we were already committed to defining units very closely to their existing usage.

> 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”.

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.

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.

> 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."

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.

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.

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.

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

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.

> 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.)

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.

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.

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.

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.)

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.

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

Solving the equation for pi we get:

PI = sqrt(g/L)

g = 9.81. L=1

or

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.

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.

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.

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.

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.

> 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.

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.

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.

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.

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.

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.

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!

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.

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.

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!

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.

> 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.

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.

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?

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 ``.

[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.

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

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.

> 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.

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.

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?

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.

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.

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.

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.

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.

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.

>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.

> 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 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.

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.

Regarding "Catholic meter": its definition depends on time measurement. How did they ensure that "seconds" of different clocks were equal?

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.

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.

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.

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.

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.

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.

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?

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.

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.

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.

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