321–330 of 352 posts
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No, Tau is fundamental. Pi only exists because someone mistakenly thought the formula for circumference involved diameter, when in fact it involves radius. ("Quit factoring a 2 out of Tau!" I tell them.)
Any examples of textbooks or papers where the author used tau instead of pi (and the topic was not tau or pi)?
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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.
(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.
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No, Tau is fundamental. Pi only exists because someone mistakenly thought the formula for circumference involved diameter, when in fact it involves radius. ("Quit factoring a 2 out of Tau!" I tell them.)
Eh, you can find plenty of cases where tau is just as awkward as pi is elsewhere. Right off the bat, the area of a circle becomes more awkward with tau, becoming (tau*r^2)/2, and in general, the volume of an n-ball gains weird powers and roots of two in its denominator as n increases if you switch to tau. In general, I don't think you can really claim either one is "more fundamental". It's just a matter of framing.
https://tauday.com/tau-manifesto
And this is the section that addresses your point:
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To quote myself: > cos can be defined via its power series or via the exp function (if you use complex numbers).
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]
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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.
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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.
> 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…
in that sense, oddly enough, the solar constant is not very constant at all
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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
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Eh, you can find plenty of cases where tau is just as awkward as pi is elsewhere. Right off the bat, the area of a circle becomes more awkward with tau, becoming (tau*r^2)/2, and in general, the volume of an n-ball gains weird powers and roots of two in its denominator as n increases if you switch to tau. In general, I don't think you can really claim either one is "more fundamental". It's just a matter of framing.
In case you didn't see this: https://tauday.com/tau-manifesto And this is the section that addresses your point: https://tauday.com/tau-manifesto#sec-circular_area