Comment by mistercow
1 year ago
Yeah, I read the post. What I’m saying is “this relationship vanishes when you change units, so it must not be a coincidence” is a bad way to check for non-coincidences in general.
For example, the speed of sound is almost exactly 3/4 cubits per millisecond. Why is it such a nice fraction? The magic disappears if you change units… (of course, I just spammed units at wolfram alpha until I found something mildly interesting).
Alpha brainwaves are almost exactly 10hz, in humans and mice. The typical walking frequency (for humans) is almost exactly 2hz (2 steps per second). And the best selling popular music rhythm is 2hz (120bpm) [1].
Perhaps seconds were originally defined by the duration of a human pace (i.e. 2 steps). These are determined by the oscillations of central pattern generators in the spinal cord. One might suspect that these are further harmonically linked to alpha wave generators. In any case, 120bpm music would resonate and entrain intrinsic walking pattern generators—this resonance appears to make us more likely to move and dance.
Or it’s just a coincidence.
[1] https://www.frontiersin.org/journals/neurorobotics/articles/...
Well, a second is also a pretty good approximate resting heart rate (60 bpm)
> 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.
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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.)
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.
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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.
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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.
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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.
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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.
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Or the speed of light being almost a sweet 300 million m/s.
Or after-atmosphere insolation being somewhat on average 1kw/m2.
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.
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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.
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> 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.
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...
32 meters is 35 yards, to within about an eighth of an inch. How's that grab you ?
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.
My favorite is 1 mile = phi kilometers 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 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.
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1 km = 5 furlong, with < 1% error.
Reminds me of https://xkcd.com/687/
X^2 is a lot more interesting than x*0.0000743 or whatever it is
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.
You would need a compelling argument why x^4 is interesting. Especially something like the golden ratio.
Not a long of things go to the fourth power in equations we need to use. Pi^2 directly features in periodicity
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.
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> 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."
Because the cubit is a measure of what a body can reach
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.
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.