Regarding "Catholic meter": its definition depends on time measurement. How did they ensure that "seconds" of different clocks were equal?
A wonderful coincidence or an expected connection: why π² ≈ g
221–230 of 352 posts
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#222What an amazing post! Such an interesting investigation. These kinds of write-ups make me realize how truly far we are from AGI. Sure, it can write amazing code, poems, songs, but can it draw interesting conclusions from first principles? I asked both ChatGPT and Claude, the same question, and both failed at pointing out the connection the author states. This is not to deride feats of AI today, and I am sure it will…
The only thing with more bollocks in it than this article is your aibro comment.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#223Earlier quoted context omitted.
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.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#224Earlier quoted context omitted.
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?
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.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#225Another "wonderful coincidence" is that the conversion between miles and kilometers involves this constant of conversion : kilometers = miles * 1.609344. Let's call 1.609344 the "km" constant. As it happens, km is very close the the Golden Ratio (sqrt(5)+1)/2 = 1.618033989... (call this "gr"). In fact they only differ by about 1/2 of one percent (100 * (gr/km - 1) = 0.54%)! As the author of the original article says,…
Do miles connect to feet and feet vs. gravity connect to the Golden ratio? 6 feet is like 2 pi?
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#226> this can’t just be a simple coincidence.
uhhhhhh yes it can?
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#227Earlier quoted context omitted.
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 intere…
Or the speed of light being almost a sweet 300 million m/s. Or after-atmosphere insolation being somewhat on average 1kw/m2.
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Re: A wonderful coincidence or an expected connection: why π² ≈ g
#228Earlier quoted context omitted.
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.
I use the term "natural foot." It's very useful in simulations.
Re: A wonderful coincidence or an expected connection: why π² ≈ g
#229Earlier quoted context omitted.
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.