Actually no, the whole equation boils down to the definition of meter. Or rather, one of the earlier definitions.
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).
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.
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.
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 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
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.
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
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.
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.
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.
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.)
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.
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/...
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".
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.
In other words, we assume spherical cows until that approximation no longer works.
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
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."
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...
heck, g is not even a constant, it just happens to measure to roughly 9.8 m/s² at most places around here.
"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."
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.
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.
(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.)