> 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.
The earths rotation coincides with the phenomenon, so it's likely 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.
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).
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/...
I'm sorry to be the kind of person who feels compelled to make this comment, but you mean a Hertz, not a second.
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.)
An unnecessary complication if you're dropping a brick out of a window.
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
In other words, we assume spherical cows until that approximation no longer works.
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
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.
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
Most people would argue that we don't. They say "the ratio is not Pi" rather than "Pi is a different value".
Heaven forbid people learn something about math that extends beyond the obvious, how dare they!
Sure there is. You don’t expect a paper to explain that the numbers are in decimal and not hexadecimal.
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?
Edit: This picture should make it pretty clear for anyone who is new to this concept: https://en.wikipedia.org/wiki/Non-Euclidean_geometry#/media/...
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.
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.
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.
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.
That said, you could spin this by suggesting that the mathematician living in that non-Euclidean space might also have a different perspective on numbers. If we assume pi is still constant for him, then the numbers he's always known could be shifting in value but maybe that's a stretch.
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.
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).
You’re treating your non-consensus definition not as a hypothetical, but as a fact. Your comment started with “fun fact”.
That's not a rigorous definition, though, because it doesn't tell you what those "..." expand to.
> cos can be defined via its power series or via the exp function (if you use complex numbers).
First of all, that's wrong. It's only true for analytic functions.
Second of all, sin and cos appear in all sorts of contexts that are not primarily geometric (such as harmonic analysis).
Lean's mathlib defines pi precisely the way I've described it - cos is defined via exp, and pi is defined as the unique zero in [0,2]
Which, _again_ is a taylor series. A function having a taylor series does not argue against the geometric nature of cos.
> Second of all, sin and cos appear in all sorts of contexts that are not primarily geometric (such as harmonic analysis).
I'm not sure how you can say that, there is so much material describing the geometric nature and interpretation of fourier/laplace transforms.
Mathlib defines cos x = (exp(ix)+exp(-ix))/2, which is not a definition via Taylor series (even though you can derive the Taylor series of cos quite easily from it). exp is defined as a Taylor series in mathlib, but it might just as easily be defined as the unique solution of a particular IVP, etc. Regardless of this, I have no idea why to you seemingly a definition via Taylor series is not a "true" definition, you could probably crack open half a dozen (rigorous) real or complex analysis texts, they're likely to define sin and cos in some such way (or, alternatively, as the single set of functions satisfying certain axioms), because defining them via geometry and making this rigorous is much harder.
I can't argue whether cos has a "geometric nature" or not, because I don't know what that means. Undoubtedly cos is useful in geometry. It is, however, used in a wide range of other domains that make absolutely no reference to geometry. mathlib's definition of cos doesn't import a single geometry definition or theorem.
Remember that the starting point of this discussion was that somebody was claiming that "pi is different in non-Euclidean geometry", which, no, even in a completely different geometry, the trigonometric functions would be useful and pi is closely related to them.
I am not understanding why you think this is at all relevant.
Its like saying a^2 + b^2 = c^2 is not geometric because it doesnt make reference to triangles. But at the end of the day, everyone understands Pythagorean theorem to be an inherently geometric equation, because geomtry is just equations and numbers.
I realize to some extent this is all subjective, but to me its insane to claim that cos is not an inherently geometric function. Agree to disagree.
But really, I'll refer again to the part of my previous reply where I contextualise why I wrote what I wrote and how that answers the question of whether pi is somehow arbitrary due to the fact that we usually think of space as Euclidean: it's not.
Or after-atmosphere insolation being somewhat on average 1kw/m2.
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
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.
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.
The primary advantage of the SI system is that it has only ONE length unit that you add prefixes to.
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.
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”.
in that sense, oddly enough, the solar constant is not very constant at all
palm : 7,64 cm
span : 12,36 cm
handspan : 20 cm
foot : 32,36 cm
cubit: 52,36 cm
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...
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.
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
My favorite approximation is π·E7 = 31415926.5... , which is a <1% error from the number of seconds in a year.
(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.
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.
Not a long of things go to the fourth power in equations we need to use. Pi^2 directly features in periodicity
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.
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."
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.
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.
"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."
We could imagine removing pi from the pendulum equation, but that would mean putting it back elsewhere, which would be inconvenient.
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.
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.
(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.)
heck, g is not even a constant, it just happens to measure to roughly 9.8 m/s² at most places around here.
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.
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.
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.
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.
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.
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 ?
I'll try to see these units you linked thanks
g in imperial units is 32 after all. g has units; pi does not
The pendulum is a device that relates pi to gravity.
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.
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.
All measurement metrics are “fake” - nothing is truly universal, and can easily be correlated with another human made measure eg Pi.
There are just too many physical quantities we find significant, and too many ways to mix numbers together to make expressions that look notable.
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.
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...
[1] https://phys.libretexts.org/Bookshelves/Electricity_and_Magn...
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 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.
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.
Not sure why everyone is surprised.
Ah, and a year is pi*10e9 seconds (IIRC)
Hysterical, especially for the fact that he quotes 'two' and 'three' in the sentence itself.
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.
> 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.
One and Done!
Don’t believe those mathematicians when they tell you that `i` is “imaginary”.
sqrt(-1) should have been called w, for "weird".
- Posted by the quantum gang
In guess it's a good thing I left physics after my PhD.
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.
On the other hand, 10 e9 = 10 * 10^9.
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.
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.
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 ...
`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!
“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.
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."
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.
https://www.simonandschuster.com/books/The-Measure-of-All-Th...
The site completely breaks when I visit it. After some investigation, I found out that if I enable Stylus (a CSS injection extension) with any rules (even my global ones), the site becomes unusable. Since it's built using the React framework, it doesn't just glitch; it completely breaks.
After submitting a ticket and getting a quick response from the Stylus dev, it turns out that this website (and any site built with caseme.io) will throw an error and break if it detects any node injected into `<html>`.
I highly doubt this bit of strategy would have worked with sellers of said fabric. They may have not had formal measurements but they weren't stupid either.
So if I understand correctly: the meter was defined using gravity and π as inputs (distance a pendulum travels in 1 cycle), so of course g and π would be connected.
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 think what you meant to say was that he wouldn't be speaking like this if people were born with 3 fingers.
Is standardization the sans-serif of civilization?
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.
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?
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.
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.
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.
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.
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.
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.
Also my reading of TFA is that the pendulum definition was in fact a redefinition that didn't catch on.
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?
Some of these are test vectors for math systems.
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.
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.
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.
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.
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.
Pretty sure it was done back in Sumer first.
>“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.
(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.
Interestingly, the Sumerians did not seem to employ this method, they would count 6 instances of counting to 10.
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.
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.
Second, we'd be having exactly the same conversation if it happened to be g = 2pi^2, or 4pi^2, or any other reasonably artificial number.
Third, if you do the math, the mass of the earth and gravitational constant do conspire.
g = G * M_earth / R_earth^2
M_earth is approximately (4/3) * pi * R_earth^3 * Density_earth
G is approximately (2/3) * 10^-10 m^3.kg^-1.s^-2
We can eliminate our human units and rewrite it as
G = 2/3 * 10^8 / ( Density_water * Earth_orbital_period^2)
Put this all together and you get
g = 8/9 * pi * 10^8 * (Density_earth / Density_water) * ( R_earth / Earth_orbital_period^2 )
the ratio of the densities of earth and water is a dimensionless number that is independent of our units of measurement, and is approximately 5.5. With a little rearrangement we get
g = 88/9 * (pi/2) * 10^8 * R_earth / Earth_orbital_period^2
That 88/9 happens to be equal to pi^2 to within 1% error. This comes purely from nature.
1 m/s^2 is defined to be (pi/2) * 10^8 * R_earth / Earth_orbital_period^2 and thus we get the nice and neat g = pi^2 in metric units, but getting (pi^3)/2 * 10^8 in natural units is just as remarkable.
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.
Can you?
432: yes it's a super fun and interesting number 432 cycles per second: seconds are not in fact special
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?
But also 2Pi is fundamental, who defines a ratio of something to 2 of something (radius)?
If it seems awkward to you, it's only because of a lifetime of seeing it done in terms of pi.
I can also make arguments that pi/2 would have been a better constant from a teaching perspective. The pi/2 version of the Euler identity, for example, would give you all the tools you need to link complex multiplication to rotation.
But at the end of the day, the choice of multiple used for the constant is a convention. None is going to be ideal in every case, and no math fundamentally changes because of a particular choice. Trying to argue for a change in convention at this point is just silly.
https://tauday.com/tau-manifesto
And this is the section that addresses your point:
- Speed of light: 299,792,458 m/s
- Great Pyramid of Giza: 29.9792458°N
"[ErrorBoundary]: There was an error: {}"
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?
why not?
https://en.wikipedia.org/wiki/Mathematical_coincidence
See also this blog: https://martouf.ch/tag/coudee-royale-egyptienne/
One french royal cubit ≈ one egyptian cubit ≈ about π/6 meters. One royal span ≈ 1/5 meter = 20cm.
I'm wondering whether some of these coincidences could be explained by the anthropic principle, which deals with these quasi-equalities, for instance:
>An excited state of the 12C nucleus exists a little (0.3193 MeV) above the energy level of 8Be + 4He. This is necessary because the ground state of 12C is 7.3367 MeV below the energy of 8Be + 4He; a 8Be nucleus and a 4He nucleus cannot reasonably fuse directly into a ground-state 12C nucleus. However, 8Be and 4He use the kinetic energy of their collision to fuse into the excited 12C (kinetic energy supplies the additional 0.3193 MeV necessary to reach the excited state), which can then transition to its stable ground state. According to one calculation, the energy level of this excited state must be between about 7.3 MeV and 7.9 MeV to produce sufficient carbon for life to exist, and must be further "fine-tuned" to between 7.596 MeV and 7.716 MeV in order to produce the abundant level of 12C observed in nature.
Source: https://en.wikipedia.org/wiki/Triple-alpha_process#Improbabi...
The idea goes like this:
1. A more fundamental aspect under the anthropic principle which underpins the existence of complex life and intelligent observers is the quasi-alignment of values such as the fundamental constants in physics within a short margin.
2. If you consider the universe to be the product of a random sampling process over these constants (either real or virtual, it occurred many times or just once), and given the fact we exist, which implies an abundance of coincidences, the maths seem to tell us that we should expect to observe superfluous coincidences that are non-functional for the appearance of complex life, rather than the strictly minimal set of functional coincidences necessary for its emergence.
3. This implies that coincidences and pattern seeking are not just features (or bugs) of our complex minds but are present in the universe latently since it is not just fine-tuned for the emergence of complex life but for the presence of coincidences such as these https://medium.com/@sahil50/a-large-numbers-coincidence-299c....
4. It may be even testable by running computer experiments relying on genetic programming/symbolic regression to see whether there is something special about the value of physical constants in our universe when compared to the value they would have in other universes. I think such experiments should factor the fact that not all equations with the same mathematical complexity (number of operands and operators) have the same cognitive complexity. Indeed, if you look at the big equation in the link above, you'll remark that it can be further compressed into a/b = c/d (where a is the photon redshift radius for instance). So I guess you'd also have to throw into the mix Kolmogorov algorithmic complexity to assess this aspect (which is in fact used in some cognitive theories of relevance to tackle this kind of stuff to the tune of "simpler to describe than to generate")
Thoughts ?
uhhhhhh yes it can?
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
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.
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 mean, one meter is defined as 1/10^7 of the distance between the equator and the poles which leads to a round number in base 10.
A unit system is not just something that matches objective reality but something that has some cognitive ergonomy.
Beautifully stated!
And that's one reason why I like the US units of measurement better than SI. I mean, the divide-by-ten thing is nice and all. But _within a project_ how often are you converting between units of the same measurement (e.g, meters to centimeters)? You pick the right "size" unit for your work and then tend to stay there. So you don't get much benefit from the easy conversion in practice.
But if you're doing real hands-on work, you often need to divide by 2, 3, 4, and so on. So, for example, having a foot easily divisible by those numbers works well. And even the silly fractional stuff make sense when you're subdividing while working and measuring.
Of course it all finally breaks down when you get to super high precision (and that's probably why machinists go back to thousands of an inch and no longer fractions).
I think there's a little bit of academic snobbery with the SI units (though, it is a good idea for cross-country collaboration), but for everyday hand-on work the US system works really well. I always love the meme: There are two kinds of countries in the world, those who use the metric system and those who've gone to the moon.
I'm an AMO physicist by training and my choice of units are the "Atomic Units" where hbar, mass of the electron, charge of the electron, and permittivity are all 1. That makes writing many of the formulae really simple. Which is what you say: it has cognitive ergonomy (and makes all of the floating point calculations around the same magnitude). Then when we're all done we convert back to SI for reporting.
"I only ran the first half of the program, but it didn't seem to give the correct answer, so it's obviously broken."
"I only read the first half of the proof, but the answer wasn't contained there, so I'm forced to conclude the proof is worthless."
You simply gave up before encountering the mathematical reason the relationship exists, why the units are different, and so on. You just ran with your incorrect initial assumption.