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.
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.
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
"This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C / d"
Pi is the "the ratio of circumference to diameter of a circle in Euclidian space". Everyone agrees that. What people are arguing is that when you have a circle in non-Euclidian space, so that the ratio of it's circumference to diameter is different, do we still call this new ratio Pi.
Most people would argue that we don't. They say "the ratio is not Pi" rather than "Pi is a different value".
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?
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?
Sorry for coming off as snotty. It wasn't my intention, I thought my statements were rather matter of fact. It's possible that after having attended two lectures on differential geometry I have forgotten that some of these things like circumference ratios and sum of angles of triangles being different in a curved geometry are not obvious to every one. I'm glad you learned something!
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.
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.
Thanks for the original comment—I picked up something new that I hadn't considered before.
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.
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.
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.
You're analogy is flawed. Pythagoras' theorem is about right triangles. "a^2+b^2=c^2" isn't about triangles, it's not even a theorem, it's simply an equation (typically a diophantine one) that is satisfied by some numbers but not others. Something which typically belongs to number theory. Obviously the two things are closely related (which is the beauty of mathematics - there are a lot of connections between very different fields).
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.
insulation < Latin insula, -ae f "island" (apparently nobody knows where this one comes from)
isolation < French isolation < Italian isolare < isola < Vulgar Latin *isula < Latin insula, -ae f
Spanish aislamiento < aislar < isla < Vulgar Latin *isula < Latin insula, -ae f
Oh and the English island never had an s sound, but is spelled like that because of confusion with isle, which is an unrelated borrowing from Old French (île in modern French, with the diacritic signifying a lost s which was apparently already questionable at the time it was borrowed), ultimately also from Latin insula.
> Oh and the English island never had an s sound, but is spelled like that because of confusion with isle
So the German cognate (I assume) Eiland probably hasn't had one either. Makes sense, since the Nordic Eya / Øy / Ö never felt like they should, and they must be just northern variants that lack the -land (literally the same in English) suffix.
Usefully, the speed of light is extremely close to one foot per nanosecond. This makes reasoning about things like light propagation delays in circuits much easier.
I really wish we had known this back before it was way too late to seriously change our units around. It would mean that our SI length units wouldn't have to have some absolutely ridiculous denominator to derive them from physical constants, and also the term "metric foot" is pretty fun.
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”.
units(1) calls it `solarconstant` or `solarirradiance` but that's the quantity above the atmosphere. the same term is sometimes used for the quantity below the atmosphere: https://en.wikipedia.org/wiki/Solar_constant and of course that depends on exactly how much atmosphere you're below
in that sense, oddly enough, the solar constant is not very constant at all
There is relationship between the metric system and the French royal system. The units used in this system have a fibonacci-like relationship where unit n = unit n-1 + unit n-2.
palm : 7,64 cm
span : 12,36 cm
handspan : 20 cm
foot : 32,36 cm
cubit: 52,36 cm
cubit/foot =~ 1,618 =~ phi, el famoso Golden ratio.
foot/handspan =~ phi too. And so on.
From this it turns out that 1 meter = 1/5 of one handspan = 1/5 x cubit/phi^2
Another way to get at it is to define the cubit as π/6 meters (= 0.52359877559). From this we can tell that
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.
... you all realize that phi is barely a better approximation than 8/5, right? 1.6 vs 1.609 (km in a mile) vs 1.618?
(8/5)/(1 mile/1 km) = 0.9942; (1 mile/1 km)/phi = 0.9946. You're making things way harder on yourself for essentially no improvement in precision, especially when you're just rounding to the nearest whole number.
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.
> 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."
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.
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.
Right, that connection is not a coincidence. The connection the previous commenter drew between the meter, pi, and the circumference of the earth is a coincidence.
> The reason there is pi is probably because a pendulum is defined by its length and follows a circular motion that has the length as its radius.
It’s not quite that easy: For small excursions x the equation of motion boils down to x’’+(g/L)x=0. There is not a π in sight there! But the solution has the form x=cos(√(g/L)t+φ), with a half period T=π√(L/g), thus bringing π back in the picture. So indeed not a coincidence.
It was news to me, but that's what the article says, and it is supported by by Wikipedia, at least. [1]
In addition, I feel the article glosses over the definition of the second. At the time, it was a subdivision of the rotational period of the earth (mostly, with about 1% contribution from the earth's orbital period, resulting in the sidereal and and solar days being slightly different.) Clearly, the Earth's rotational period can (and does) vary independently of the factors (mass and radius) determining the magnitude of g.
The adoption of the current definition of the second in terms of cesium atom transitions looks like a parallel case of finding a standard that could be measured repeatably (with accuracy) and be close to the target unit - though it is, of course, a much more universal measure than is the meridional meter.
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.)
