This is interesting, but I have to quibble with this:
If you express this value in any other units, the magic immediately disappears. So, this is no coincidence
Ordinarily, this would be extremely indicative of a coincidence. If you’re looking for a heuristic for non-coincidences, “sticks around when you change units” is the one you want. This is just an unusual case where that heuristic fails.
Actually no, the whole equation boils down to the definition of meter. Or rather, one of the earlier definitions.
Yeah, I read the post. What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general.
For example, the speed of sound is almost exactly 3/4 cubits per millisecond. Why is it such a nice fraction? The magic disappears if you change units… (of course, I just spammed units at wolfram alpha until I found something mildly interesting).
Another bad way to check for non-coincidences is to use a value like g which changes depending on your location.
Pi is the same everywhere in the universe.
g on Earth: 9.8 m/s²
g on Earth's moon: 1.62 m/s²
g on Mars: 3.71 m/s²
g on Jupiter: 24.79 m/s²
g on Pluto: 0.62 m/s²
g on the Sun: 274 m/s²
(Jupiter's estimate for g is at the cloud tops, and the Sun's is for the photosphere, as neither body has a solid surface.)
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.
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!
Sure there is. You don’t expect a paper to explain that the numbers are in decimal and not hexadecimal.
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?
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"
https://en.wikipedia.org/wiki/Pi
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!
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/...
Similarly, g depends on the geometry, and g is a constant 0 for Euclidean space
This whole conversation is painful to read:
1. Your parent was talking about projections from one space to another and getting it confused.
2. Pi is pi and their non-Euclidean pi is still pi (unless you want to argue that a circle drawn on the earth’s surface has a different value of pi).
The problem comes down to projections, then all bets are off.
Yes. That's what makes it a fun fact. Most people never even learn about non-euclidean math, and this is the kind of "wow I never even thought about this" that people should be able to learn about in a comment thread.
Calling it painful to read is downright weird. Pi, the constant, has one value, everywhere. So now let's learn about what pi can also be and how that value is not universal.
It isn't a "fun fact" ... it's plainly incorrect.
π never changes its value. Ever. It is a constant in mathematics, no matter the geometry. However, π can have different ratios in other geometries, but it will still be ~3.14.
It is painful because this statement:
The problem here is *projections*. If you project non-Euclidean space onto Euclidean space, you end up with some seemingly nonsensical things, like straight lines that get projected into arcs. This is your problem. You projected a curved line from non-Euclidean space onto Euclidean space and got an arc, but didn't account for the curvature of your "real non-Euclidean space" and thus ended up with an invalid value for π. If you got something that isn't ~3.14, then you did the math wrong somewhere along the way.
It doesn't have anything to do with projections.
Let's say you live in a non-flat space. You come up with the idea of a "circle" with the usual definition: the set of points on the same plane equidistant from a central point.
You then trace along the circle and measure the length, and compare it to the length of the diameter. It turns out that this ratio changes as a function of the diameter. This truth is inherent to curved spaces themselves, and is not an artifact of choosing to describe the space using a projection.
The definition of pi in this world is no longer the invariant ratio of diameter to circumference. You can still recover pi by taking the limit of this ratio as the diameter length goes to zero. But perhaps mathematicians in this world would (justifiably) not see pi as such a fundamental number.
Now back to GP's example: people living on a sphere (like us) are analogous to inhabitants of a non-Euclidean space. The surface of Earth is analogous to 3-space, and the curvature of Earth is analogous to the curvature of space.
And indeed, if you draw real-life larger and larger circles on our planet, you will find that the ratio of circumference to diameter is smaller than pi. For example, if you start at some point on the Earth (say, the North pole) and trace out a circle 100 miles distant from it, you will find that the circumference of that circle (as measured by walking around that circle) is a little bit _less_ than pi x 100 miles.
Again, we have done no "projection" here. We've limited ourselves to operations that are fairly natural from the perspective of a mathematician living in that space, such as measuring lengths.
The fact that large circle circumferences measure less than pi x diameter on Earth did not change how we developed math, likely because you only notice start to notice this effect with extremely large (relative to us) circles.
But perhaps the inhabitants of a non-Euclidean space that was much more highly curved would notice it much earlier, and it would affect their development of maths, such that the number pi is held in lower regard.
Modern mathematics is more likely than not going to define pi as twice the unique zero of cos between 0 and 2, and cos can be defined via its power series or via the exp function (if you use complex numbers). None of this involves geometry whatsoever.
What do you mean? Cos is an inherently geometric function.
To quote myself:
Any function can be defined as a taylor series. That is not at all an argument against its geometric nature.
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]
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.
That's not a rigorous definition, though, because it doesn't tell you what those "..." expand to.
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).
I haven’t missed your point. I’ve criticized it.
You’re treating your non-consensus definition not as a hypothetical, but as a fact. Your comment started with “fun fact”.
My physics prof said g is actually a vector field. Because the acceleration has a direction and both magnitude and direction vary from point to point.
Absolutely true on astronomical scales.
An unnecessary complication if you're dropping a brick out of a window.
It's funny how much of physics we do assuming a flat earth.
If you did it "properly" you would calculate the orbit of the brick (assuming earth was a point mass), then find the intersection between that orbit and earth's surface. But for small speeds and distances you can just assume g points down as it would in a flat earth
Your physics Prof is correct of course, and so is GP. "Standard" values for g exist for these bodies, but it also varies everywhere.
This is correct, gravitational constants are a good approximation/simplification since the mass of solar bodies is usually orders of magnitude greater than the other bodies in the problem, and displacement over the course of the problem is usually orders of magnitude smaller than absolute distance between them.
In other words, we assume spherical cows until that approximation no longer works.
I volunteer for the Mars mission as a weight loss tool.
Surviving on Mars will probably involve some mass loss, too.
With the current technology, even getting on a rocket to Mars will involve some weight loss - the mass budget is tight, and each kilogram they can trim off the crew bodies is a kilogram that could be put towards fuel, life support, or scientific equipment.
Or the speed of light being almost a sweet 300 million m/s.
Or after-atmosphere insolation being somewhat on average 1kw/m2.
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.
Even if you made that kind of definition, it wouldn't have been that simple. Would you have used metric seconds or babylonian seconds?
Fun, and poetic too
I use the term "natural foot." It's very useful in simulations.
I always find insolation and insulation to be such an interesting pair of words
I guess the equivelent of "change the units" is "change the language".
French: insolation et isolation
German: Sonneneinstrahlung / Isolierung
Spanish: insolación / aislamiento
Chinese: 日照 / 绝缘
I guess coincidence
insolation < Latin sol, solis m "sun"
insulation < Latin insula, -ae f "island" (apparently nobody knows where this one comes from)
isolation < French isolation < Italian isolare < isola < Vulgar Latin *isula < Latin insula, -ae f
Spanish aislamiento < aislar < isla < Vulgar Latin *isula < Latin insula, -ae f
Oh and the English island never had an s sound, but is spelled like that because of confusion with isle, which is an unrelated borrowing from Old French (île in modern French, with the diacritic signifying a lost s which was apparently already questionable at the time it was borrowed), ultimately also from Latin insula.
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
32 meters is 35 yards, to within about an eighth of an inch. How's that grab you ?
My favorite is 1 mile = phi kilometers with <1% error
I use that approximation, via the Fibonacci sequence, to translate between miles and km. 13 miles ~ 21 km (actually 20.921470).
My favorite approximation is π·E7 = 31415926.5... , which is a <1% error from the number of seconds in a year.
... you all realize that phi is barely a better approximation than 8/5, right? 1.6 vs 1.609 (km in a mile) vs 1.618?
(8/5)/(1 mile/1 km) = 0.9942; (1 mile/1 km)/phi = 0.9946. You're making things way harder on yourself for essentially no improvement in precision, especially when you're just rounding to the nearest whole number.
That tells me you haven't memorized the first ten or twelve terms of the Fibonacci sequence.
1 km = 5 furlong, with < 1% error.
That one’s useful too. If you know a few Fibonacci numbers you can convert miles to kilometres and vice versa with ease.
1, 1, 2, 3, 5, 8, 13, 21, 34, 55 …
21 km is ~13 miles, 13 km is ~8 miles, etc.
A 26 mile marathon? Must be ~42km.
Same for speed limits too; 34 mph is ~55 kmh
I wonder if this is related, but imperial measurements with a 5 in the numerator (and a power of two in the denominator) are generally just under a power of two number of millimeters.
The reason is fun, and as far as I know, historically unintentional. To convert from 5/(2^n) inches to mm, we multiply by 25.4 mm/in. So we get 5*25.4/(2^n) mm, or 127/(2^n) mm. This is just under (2^7)/(2^n) mm, which simplifies to 2^(7 - n) mm.
This is actually super handy if you're a maker in North America, and you want to use metric in CAD, but source local hardware. Stock up on 5/16" and 5/8" bolts, and just slap 8 mm and 16 mm holes in your designs, and your bolts will fit with just a little bit of slop.
So the error is 1.6%. Acceptable for everyday hardware I guess.
X^2 is a lot more interesting than x*0.0000743 or whatever it is
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.
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.
Alpha brainwaves are almost exactly 10hz, in humans and mice. The typical walking frequency (for humans) is almost exactly 2hz (2 steps per second). And the best selling popular music rhythm is 2hz (120bpm) [1].
Perhaps seconds were originally defined by the duration of a human pace (i.e. 2 steps). These are determined by the oscillations of central pattern generators in the spinal cord. One might suspect that these are further harmonically linked to alpha wave generators. In any case, 120bpm music would resonate and entrain intrinsic walking pattern generators—this resonance appears to make us more likely to move and dance.
Or it’s just a coincidence.
[1] https://www.frontiersin.org/journals/neurorobotics/articles/...
Well, a second is also a pretty good approximate resting heart rate (60 bpm)
I'm sorry to be the kind of person who feels compelled to make this comment, but you mean a Hertz, not a second.
You're right. Thanks for being that guy!
Youre right, you should be sorry. Dont be that guy.
Because the cubit is a measure of what a body can reach
How does that explain the relationship to the speed of sound?
A bit out of topic, however
https://www.youtube.com/watch?v=0xOGeZt71sg
Note: I'm more inclined to think this is a coincidence given that it establishes a link between the most commented text and the the most commented building in history. However I don't think these kind of relationships based on "magic thought" should be discarded right away just because they are coincidences, and I'd be very interested in an algorithm that automatically finds them.
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.
Reminds me of https://xkcd.com/687/
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."
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.
cubit/foot =~ 1,618 =~ phi, el famoso Golden ratio. foot/handspan =~ phi too. And so on.From this it turns out that 1 meter = 1/5 of one handspan = 1/5 x cubit/phi^2
Another way to get at it is to define the cubit as π/6 meters (= 0.52359877559). From this we can tell that
1cbt = π/6m
π meters = 6 cubits
Source: https://martouf.ch/crac/index.php?title=Quine_des_b%C3%A2tis...
It does not? Pi has nothing to do with our arbitrary unit system.
Pi is related to the circumference of a circle; the meter was originally defined as a portion of the circumference of the Earth, which can be approximated as a circle.
"The meter was originally defined as one ten-millionth of the distance between the North Pole and the equator, along a line that passes through Paris."
But that connection actually is a coincidence. From what I can tell, when they standardized the meter, they were specifically going for something close to half of a toise, which was the unit defined as two pendulum seconds. So they searched about for something that could be measured repeatably and land on something close to a power of ten multiple of their target unit. The relationship to a circle there doesn’t have anything to do with the pi^2 thing.
Not a coincidence. They defined the meter from the second using the pendulum formula, and the pandulum formula has a pi in it, so pi is going to appear somewhere. The reason there is pi is probably because a pendulum is defined by its length and follows a circular motion that has the length as its radius.
We could imagine removing pi from the pendulum equation, but that would mean putting it back elsewhere, which would be inconvenient.
Right, that connection is not a coincidence. The connection the previous commenter drew between the meter, pi, and the circumference of the earth is a coincidence.
It’s not quite that easy: For small excursions x the equation of motion boils down to x’’+(g/L)x=0. There is not a π in sight there! But the solution has the form x=cos(√(g/L)t+φ), with a half period T=π√(L/g), thus bringing π back in the picture. So indeed not a coincidence.
It was news to me, but that's what the article says, and it is supported by by Wikipedia, at least. [1]
In addition, I feel the article glosses over the definition of the second. At the time, it was a subdivision of the rotational period of the earth (mostly, with about 1% contribution from the earth's orbital period, resulting in the sidereal and and solar days being slightly different.) Clearly, the Earth's rotational period can (and does) vary independently of the factors (mass and radius) determining the magnitude of g.
The adoption of the current definition of the second in terms of cesium atom transitions looks like a parallel case of finding a standard that could be measured repeatably (with accuracy) and be close to the target unit - though it is, of course, a much more universal measure than is the meridional meter.
[1] https://en.wikipedia.org/wiki/History_of_the_metre
pi is always just pi, but g may be defined in terms of the meter.
sure, that's the entire point.
heck, g is not even a constant, it just happens to measure to roughly 9.8 m/s² at most places around here.
It does, and the formula in the post explains the connection
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.)
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
Doesn't the relationship hold if we change units? It seems like it must.
When I worked with electric water pumps I loved that power can be easily calculates from electrical, mechanical, and fluid measurements in the same way if you use the right units. VoltsAmps, torquerad/sec, pressure*flow_rate all give watts.
No, the equality requires the length of a 2 second period pendulum be g / pi^2. Change your definition of length - that no longer holds true.
g in imperial units is 32 after all. g has units; pi does not
The equation holds in imperial units as well. The length of the 2 second pendulum needs to be in feet AND the value of g in ft/sec2.
π^2 ≈ 32 to you?
Replace s in your calculation with imperial s instead of metric s and it isn't imperial feet per metric seconds.
Imperial seconds were very, very close to metric seconds.
Solving the equation for pi we get:
PI = sqrt(g/L)
g = 9.81. L=1
or
g = 32.174. L=3.174
Either way works to approximately pi. There is a particular length where it works out exactly to pi which is about 3.2 feet, or about 1 meter. My point was that equations like that remain true regardless of units.
The reason pi squared is approximately g is that the L required for a pendulum of 2 seconds period is approximately 1 meter.
A more natural way to say it is that equality requires that the unit of length is the length of an arbitrary pendulum and the unit of time is the half-period of the same pendulum.
The pendulum is a device that relates pi to gravity.
Sounds universal. Get a different value on the Moon? Of course... pi squares differently on the moon :)
The arbitrary length pendulum with a period of 2 seconds which is your unit of length, (or 1 Catholic meter) is much shorter on the moon. In local Catholic meters gravity would be pi squared Catholic meters / second. As it would on any planet.
Nope, it completely vanishes in other units. If you do all your distance measurements in feet, for example, the value of pi is still about 3.14 but the acceleration due to gravity at the earth's surface is about 32 feet s^(-2). If you do your distance measurements in furlongs and your time measurements in hours then the acceleration due to gravity becomes about 630,000 furlongs per hour squared and pi (of course) doesn't change.
Only because you're using metric seconds instead of "imperial seconds" (the time it takes for a 1 foot long pendulum to complete a full oscillation).
Sure, if you change either of the units you can always change the other one to fix the equation again.
But does it work when you use the right Imperial technique?
If I come up with my own measuring unit, let's call it the sneezle (whatever the actual length I assign to it) I will be able to also define a duration unit (say, the snifflebeat) based on the time it takes for a pendulum one sneezle long to complete a full oscillation, and vice versa I can define the sneezle by adjusting the length of a pendulum so that it oscillates in two snifflebeats. Here are the maths:
T = 2π√(l/g)
T/2π = √(l/g)
(T/2π)^2=l/g
g = l/(T/2π)^2
g = l/(T^2/4π^2) = 4π^2xl/T^2
Now substitue T with 2 and l with 1 an you get
g = 4π^2x1/2^2 = π^2
It doesn't matter what the pair of units assigned to T and l are. However, they'll be interrelated.
There is nothing arbitrary, and no coincidences behind g =~ π^2. It just requires to do some history of metrology and some basic maths/physics.
If you want to discuss coincidences, may I suggest you to comment on this remark I made and which hasn't received any attention yet ?
https://news.ycombinator.com/item?id=41209612
This is not quite the same situation, as you are calculating a value having a dimension (that of power, or energy per second) three different ways using a single consistent system of units, and getting a result demonstrating / conforming to the conservation of energy. If you were to perform one of these calculations in British imperial units (such as from pressure in stones per square hand and rate of flow in slugs per fortnight) you would get a different numerical value (I think!) that nonetheless represents the same power expressed in different units. The article, however, is discussing a dimensionless ratio between a dimensionless constant and a physical measurement that is specific to one particular planet.
Changing units in Electrodynamics for instance comes with unexpected factors in formulas though, indeed containing π. (CGS <-> SI)
Isn't that just the change between rad/s and Hz?
It’s more precisely the difference between “rationalized” and “unrationalized” units.
You need a factor 4pi in either Gauss’ law or Coulomb’s law (because they are related by the area 4pi*r^2 of a sphere), and different unit systems picked different ones.
It’s more akin to how you need a factor 2pi in either the forward or backward Fourier transform and different fields picked different conventions.
Some fields even use the unitary transform -- they split the difference and just throw in a 1/sqrt(2pi) in both directions.
https://en.wikipedia.org/wiki/Fourier_transform#Angular_freq...
It is more involving [1]
[1] https://phys.libretexts.org/Bookshelves/Electricity_and_Magn...
I agree. But if you remove the "so", there is no contradiction. It is possible the author used "so" not to mean "in other words", but simply as a relatively meaningless discourse marker.
Huh, interesting point. Writing unambiguously is ridiculously hard.
The comma differentiates. The comma indicates a short pause and a certain intonation in speech (the period means a longer pause and a different intonation). If you say that sentence with and without a pause/comma, you'll see (hear) that the sentence is correct. Reading unambiguously is also hard.
The problem with that is that writers are not consistent with comma usage either, particularly when it comes to informal writing, where prescriptive rules are out the window anyway. And I would argue that it would be a bit of a norm violation even in informal writing to introduce this new point at the end of a paragraph rather than starting a new one, which makes me think that that was not the author's intent.
I'm surprised at the number of people disagreeing with your quibble. I had the exact same thought as you!
If pi^2 were _exactly_ g, and the "magic" disappeared in different units, THEN we could say "so this is no coincidence" and we could conclude that it has to be related to the units themselves.
But since pi^2 is only roughly equal to g, and the magic disappears in different units, I would likely attribute it to coincidence if I hadn't read the article.
It would be useful if people carried around some card with all the information that they understood on it, since opinions are largely symptoms of this.
In almost all cases any apparent phenomenon specific to one system of measurement is clearly a coincidence, since reality is definable as that which is independent of measurement.
In terms of quantum mechanics, would that mean the wave function is real until it collapses due to measurement? Or am I misunderstanding your use of measurement there?
Something about that is sticking in my mind in an odd way, but I can't put my finger on exactly what it is - which is intriguing.
Measurement can change what is measured, but it doesnt change it from illusion to reality.
I cannot measure santa clause into existence. But I can change the temperature of some water by measuring it with a very hot thermometer.
That measurement changes what is measured is the norm in almost all cases, except in classical physics which describes highly simplified highly controlled experiments. The only 'unusual' thing about QM is its a case in physics where measurement necessarily changes the system, but this is extremely common in every other area. It is more unusual that in classical physics, measurement doesn't change the system.
Not necessarily. One of the things I was taught when studying astronomy is that if you observe periodicity that is similar to a year or a day, that's probably not a coincidence, you probably failed to account for the earth's orbit or rotation.
This is a good example, but actually this is exactly what GP was referring to. It is a coincidence that the thing you're observing is periodic with earth's rotation. Observing a similar thing from a satellite (allegorically the same as "changing bases") would remove the interesting periodicity.
The earths rotation coincides with the phenomenon, so it's likely a coincidence.
In the example case, the earth's rotation is producing the apparent observation: it's the cause, not a separate phenomenon that happens to coincide, or that might be indicative of a deeper relationship. For something to be a coincidence, it must be otherwise unconnected causally, which is not the case if the reason you found a ~24 hour period is that you forgot to account for the earth's rotation.
It’s really the best and only way to find non-coincidences involving the definition of units, though. All such non-coincidences will have this property
All coincidences involving the definition of units will also have this property. Once you’ve narrowed to that specific domain, invariance to change of units is completely uninformative.
Reading this gave me a chill. Please take my temperature and compare it to the norm temperature of humanity.
It is not unusual case. The heuristic you want is working. It's nothing more than a coincidence.
What are you disputing about the explanation given in the post? As far as I can tell, it’s basically accurate (although the pendulum unit was called the toise, and the meter seems to have targeted half a toise). If you accept that account, it’s not a coincidence.
I initially had an objection due to a misconception I was carrying.
I see now that the pendulum formula is a pure relation between time and distance/length. It will apply regardless of the units used. For example if we measure time in fortnite and length and furlongs the formula will be the same. The gravitational acceleration of course will be in those units: furlongs per fortnights squared. Needless to say that will not be 9.81.
Now the meter unit was chosen in relation to the second unit by the length of a pendulum that produced an integer period. So that choice/relation caused the gravitational acceleration g to take on such a value that its square root cancels out the π on the outside of the root.
I was confused for a moment thinking that the definition of the kilogram would somehow be mixed up in this but of course it isn't. g doesn't incorporate mass; and of course pendulum swings are dependent only on length and not mass.
There are all sorts of situations in which certain units either give us a nice constant inside the formula or eliminate it is entirely.
For instance Ohm's law, V = IR. It's no coincidence that the constant there is 1. If we change resistance to some other unit without changing how we measure voltage and current we get V = cIR.
What? The entire point is that it’s no coincidence in this unit set. Saying that changing units indicates a coincidence is like saying that if we see Trump suddenly driving a Tesla after Elon stated throwing money at him, that must be just a coincidence because if we change the car model to a ford then there would be nothing odd about it.
That analogy is so bizarre that I have no idea how to respond to it.
Truth feels like a coincidence when 1 small thing can make anything wrong.
I don’t agree with this. You could literally redefine any unit (as we have done so multiple times in the past) and end up with zero coincidences.
All measurement metrics are “fake” - nothing is truly universal, and can easily be correlated with another human made measure eg Pi.
I seriously doubt you could define any system of units that has zero coincidences, even with significant computational effort. Some things in the real world are just going to happen to line up close to round numbers, or important mathematical constants, or powers or roots of mathematical constants, and then you’ll have some coincidences.
There are just too many physical quantities we find significant, and too many ways to mix numbers together to make expressions that look notable.
I don't have this heuristic drilled into me, so I saw the point immediately. To be frank, I suspected the general direction of the answer after reading the headline, and this general direction, probably, can be expressed the best by pointing at the sensitivity of the approx. equation from the headline to the choice of units.
So, I think, the reaction to this quote says more about the person reacting, then about this quote. If the person tends to look answers in a physics (a popular approach for techies), then this quote feels wrong. If the person thinks of physics as of an artificial creation filled with conventions and seeking answers in humans who created physics (it is rarer for techies and closer to a perspective of humanities and social sciences), then this quote is the answer, lacking just some details.
No, like objectively, a dimensionless number lining up with a meaningful constant is more likely to be because of some underlying mathematical connection, and a dimensioned number lining up is more likely to be a coincidence. There are only a handful of ways for a unit’s heritage to have a connection to a local physical phenomenon like the post describes, and that’s what it takes to have a unit-dependent non-coincidence. That’s not dependent on your perspective.
The thing that’s interesting in this case is that the meter’s connection to g is obscured by history, whereas most of the time a unit’s heritage is well known. Nobody is going to be surprised by constants coming out of amps, ohms, and volts, for example, because we know that those units are defined to have a clean relationship.
Your quibble seems nitpicky and unwarranted. What the author is saying is that the relationship becomes evident if we consider the units of m/s^2 for gravity. They just didn't quite say it like that.
Obviously it’s nitpicky. That’s what a quibble is. But I don’t think it’s unwarranted. How you reason your way to a conclusion is at least as important a lesson as the conclusion itself. And in this case, the part I quoted is a bad lesson.
I think what the author want to convey is that the metric system was designed based on the assumption that pi^2 = g. The assumption pi^2 = g is one of the source of the metric system (at least for the relationship between meter and second). The deviation was due to the size of earth being incorrectly measured by French in the original expedition.)
you can rule that heuristic out immediately because pi is unitless, surely?
The "magic" doesn't disappear in "any" other units.
Period = 2π√(length/g)
So the "magic" holds in any units where the unit of time is the period of a pendulum with unit length.
I think you are right and i had the exact same thought. I think people are misunderstanding you.
Agreed. Irrespective of how the story is later developed, "So, this is no coincidence", is a baffling thing to put immediately after apparently demonstrating a coincidence!