51–60 of 85 posts
I've considered refactoring my pathfinding code - 2.0 is this magic constant that is floating around all over my code. Inertia has prevented me.
eagerly awaiting π vs τ rap wars..
Earlier quoted context omitted.
The area of the traditional unit circle is π, which has strong ties to the definition of every trigonometric function, and the reason that radians of common fractions of the unit circle are expressed in terms of π is related to the integrals used to derive arc length. Just as an exercise, try setting the area of the unit circle to 2π, and then see how meaningful your radian measurements are. How many radians are in a…
> The area of the traditional unit circle is π The area of the unit circle is 3.14(etc) units squared. The result is a number; it's how you get there that's important. You get there by integrating. The "right" equation for area is not πr^2 or τr^2/2; it's the integral that leads to either of those equations. The 1/2 in the τ version is meaningful because it is an artifact of the integration. The lack of the 1/2 in th…
Was two millennia not long enough? We could take another couple centuries, I guess. I don't think we're ever going to get a new answer for the ratio of the circumference of a circle to the diameter of that circle, though.
> The lack of the 1/2 in the π version shows why it's "wrong"—it's not as meaningful.
Frankly, I don't know what this means. I've read it several dozen times, and each time, it seems increasingly more inane. I can't help but wonder what you would say about the derivative (and antiderivative) of e^x; would you complain that it's not meaningful? Does it need more coefficients? More exponents?
Earlier quoted context omitted.
Why do you refer to the area? Why not the diameter or the circumference? If you care about the area, why not use the unit circle with area 1? A quarter arc of a circle is π/2 radians, regardless of area.
(To answer your question, the unit circle, with radius one, has area π.) A quarter arc of the unit circle is π/2. A quarter arc of the circle with area 2π, though... Here, let's do it in τ for fun. :3 So, the circle with area τ has radius sqrt(2) (from A = πr^2), diameter 2sqrt(2), and circumference τsqrt(2) (from C = Dπ), thus a quarter arc of that circle is τsqrt(2)/4. That's not a pretty number at all to work with…
How much simpler could it get?
Defining the unit circle as a circle with area τ is a contradictory statement and is semantically invalid.
Earlier quoted context omitted.
> The area of the traditional unit circle is π The area of the unit circle is 3.14(etc) units squared. The result is a number; it's how you get there that's important. You get there by integrating. The "right" equation for area is not πr^2 or τr^2/2; it's the integral that leads to either of those equations. The 1/2 in the τ version is meaningful because it is an artifact of the integration. The lack of the 1/2 in th…
> hastily Was two millennia not long enough? We could take another couple centuries, I guess. I don't think we're ever going to get a new answer for the ratio of the circumference of a circle to the diameter of that circle, though. > The lack of the 1/2 in the π version shows why it's "wrong"—it's not as meaningful. Frankly, I don't know what this means. I've read it several dozen times, and each time, it seems incre…
Which shouldn't come as too much of a surprise because the radius is the smallest amount of information that determines what a circle is, as well as the basis for how we define radians.
When you ask hard-core Tauists what the area of a unit circle is, do they actually answer "tau over two"? Or do they just say "pi"? By the way, this whole discussion reminds me of what W.V.O. Quine called "mathematosis".
When you ask hard-core Tauists what the area of a unit circle is, do they actually answer "tau over two"? Or do they just say "pi"? By the way, this whole discussion reminds me of what W.V.O. Quine called "mathematosis".
It's τ/2. The area of a circle is τr²/2. You may be familiar with the idea of x²/2 from calculus: it's an integral, which can be used to compute areas.
Mind you, I don't have a dog in this hunt so I'm not all up to speed on it. All I know for sure is that tau = 2 * pi, so I won't be terribly upset if I see either usage. I generally favor the use of notations which better reveal an underlying concept, but I don't like it when people get all high and mighty about things.
Earlier quoted context omitted.
The problem with using cycles as the primary unit of angle is that the wonderful trigonometric derivative symmetry only occurs when the functions are calibrated for radians. I've never understood the particular argument of "single intermediary constant (tau) rather than two (2,pi)". 2pi is one constant that contains multiple glyphs, just as 1/2 is one constant, just as tau/2 would be one constant. That it is derived…
The two-glyph thing is a matter of description length and parametrization. 10+2 is an expression, whereas 10 is also a constant that is the base of our number system. Thus, 10 carries more meaning than 12, even though both are constant - 10 is potentially the aforementioned parameter, but what is the 2? With sufficient study, 12 becomes a number of important constants as well (such as the number of inches in a foot,…
10+2 is an expression, whereas 10 is also a constant that is the base of our number system.
There is no notion of an "expression" as distinct from a "number" (or "function" if it involves a variable) in any branch of math apart from computer science[1]. In algebraic terms, (12) and (10+2) and (6x2) and (0xC) and (2^4-2) and "twelve" are all literally the same thing. Well, technically they are all equivalent notations for the same abstract concept.
Thus, 10 carries more meaning than 12
Even if I accept this (which I'm not convinced I do), it's beside the point: 10 and 12 are not equal. Unlike with pi and a hypothetical tau, using one where the other is called for would be an error.
[1]There is the notion of the limit, which is subtly different: limits do care how a function behaves at other points. One could make the case that this makes a limit into a sort of expression, but to be honest I think that only obscures the idea.
Earlier quoted context omitted.
The area of the traditional unit circle is π, which has strong ties to the definition of every trigonometric function, and the reason that radians of common fractions of the unit circle are expressed in terms of π is related to the integrals used to derive arc length. Just as an exercise, try setting the area of the unit circle to 2π, and then see how meaningful your radian measurements are. How many radians are in a…
> The area of the traditional unit circle is π The area of the unit circle is 3.14(etc) units squared. The result is a number; it's how you get there that's important. You get there by integrating. The "right" equation for area is not πr^2 or τr^2/2; it's the integral that leads to either of those equations. The 1/2 in the τ version is meaningful because it is an artifact of the integration. The lack of the 1/2 in th…
It was chosen here:
http://arxiv.org/abs/math/0506415
Elegance is found in arguments and proofs, not in results, and so any attempt to look at equations is really missing the point. If you believe Euler's methods might be simplified by using 2pi instead of pi, first consider a look at the methods themselves.