Areas of sectors - an argument for tau

An enlightening discussion about pi and tau.

Areas of sectors - an argument for tau

by SpikedMath » Mon Jul 04, 2011 9:24 pm

I know in the Pi Manifesto it is stressed that when looking at areas it is a clear win for [tex]\pi[/tex]. But when looking at areas of sectors a bit more closely, it seems more natural to think of it in terms of [tex]\tau[/tex]. In particular, the area of a circular sector with central angle [tex]\theta[/tex] is:

[tex]A = \frac{1}{2}\theta r^2[/tex]

When [tex]\theta=2\pi[/tex], the 2's cancel giving the area of a circle. The area of a circle is just a special case of the above formula and so it seems there should be a [tex]1/2[/tex] in it.
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Re: Areas of sectors - an argument for tau

by metric man » Sat Dec 24, 2011 12:34 am

Hi everyone,

I would like to say two things: First, that I agree with Spiked Math that areas of sectors provide an argument for [tex]\tau[/tex].

But I would even go beyond what Spiked Math said when writing: "I know in the Pi Manifesto it is stressed that when looking at areas it is a clear win for [tex]\pi[/tex]."

In my opinion, "tauists" are being too generous with our adversaries when they make such a concession regarding the area of a circle. I believe that when advocates for [tex]\pi[/tex] make such a claim, then advocates for [tex]\tau[/tex] are letting off the pro-[tex]\pi[/tex] people too easily.

First, let me say that I recently saw someone else make essentially the same argument that I am about to make, albeit with different words. I wish that I could give proper attribution, but I cannot remember where I saw the argument -- perhaps it was in this very forum.

In any event, pro-[tex]\pi[/tex] advocates will point out that the following formula looks so tidy:

[tex]\pi = \frac{A}{{{r^2}}}[/tex]

So it does. But in writing that formula, those who advocate [tex]\p[/tex] fail to have the courage of their convictions. They claim that diameter is the fundamental circle constant, not the radius. When it suits them, however, as it does here in order to make a succinct formula, then they are quick to abandon diameter and utilize radius in its place.

I say that the advocates of [tex]\p[/tex] do not have the right to refer to radius in the context of explaining any of the parameters of a circle. To be consistent with their own argument, they are logically bound to always refer to diameter and never to radius when discussing circles. And hence, the formula they should use is the much more cluttered version of the circle's diameter, as such:

[tex]\pi = \frac{4A}{{{d^2}}}[/tex]

In short, if pro-[tex]\pi[/tex] people don't like radius, if they don't recognize radius to be the fundamental circle constant, then they shouldn't use it.

Thanks to all of you who read my commentary.
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Re: Areas of sectors - an argument for tau

by josephlindenberg » Sat Dec 24, 2011 8:04 pm

metric man wrote:First, let me say that I recently saw someone else make essentially the same argument that I am about to make, albeit with different words. I wish that I could give proper attribution, but I cannot remember where I saw the argument -- perhaps it was in this very forum.

It was probably Bob Palais' website where you saw that:
http://www.math.utah.edu/~palais/pi.html wrote:"It seems to me that you can't have it both ways on area [tex]A= \pi r^2[/tex] and circumference [tex]C= \pi d[/tex]. If you believe diameter is fundamental, then it should be [tex]A= {\pi d^2}[/tex]/[tex]4[/tex]."
PI is the SEMICIRCLE constant, not the circle constant . . . http://sites.google.com/site/taubeforeitwascool
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Re: Areas of sectors - an argument for tau

by josephlindenberg » Sun Dec 25, 2011 3:38 am

For the Wikipedia article on tau, this is how I summarized why radius is more fundamental:
  • A circle is defined as all points in a plane a certain distance — the radius — away from a center point.
  • Standard circle formulas use radius:
    [tex]\;r^2 = (x-a)^2 + (y-b)^2[/tex]
    [tex]\;x = a\,+\,r\,cos\,t \;\;y = b\,+\,r\,sin\,t[/tex]
  • The unit circle — note the word unit — has a radius of 1, not a diameter of 1.
  • Angles are measured in radians.
PI is the SEMICIRCLE constant, not the circle constant . . . http://sites.google.com/site/taubeforeitwascool
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Re: Areas of sectors - an argument for tau

by metric man » Sun Dec 25, 2011 1:00 pm

I agree with all four of josephlindenberg's excellent points.

Incidentally, could somebody please explain to me how to quote another posting. Thank you.
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Re: Areas of sectors - an argument for tau

by bmonk » Sun Dec 25, 2011 9:49 pm

metric man wrote:I agree with all four of josephlindenberg's excellent points.

Incidentally, could somebody please explain to me how to quote another posting. Thank you.

when signed in, the upper right corner has a box with four dots and a right arrow. Click that, and the quote will open.
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Re: Areas of sectors - an argument for tau

by metric man » Mon Dec 26, 2011 9:29 am

Thank you, bmonk.

bmonk wrote:when signed in, the upper right corner has a box with four dots and a right arrow. Click that, and the quote will open.
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Re: Areas of sectors - an argument for tau

by 1=0 » Thu Mar 23, 2017 5:51 pm

metric man wrote:Hi everyone,

I would like to say two things: First, that I agree with Spiked Math that areas of sectors provide an argument for [tex]\tau[/tex].

But I would even go beyond what Spiked Math said when writing: "I know in the Pi Manifesto it is stressed that when looking at areas it is a clear win for [tex]\pi[/tex]."

In my opinion, "tauists" are being too generous with our adversaries when they make such a concession regarding the area of a circle. I believe that when advocates for [tex]\pi[/tex] make such a claim, then advocates for [tex]\tau[/tex] are letting off the pro-[tex]\pi[/tex] people too easily.

First, let me say that I recently saw someone else make essentially the same argument that I am about to make, albeit with different words. I wish that I could give proper attribution, but I cannot remember where I saw the argument -- perhaps it was in this very forum.

In any event, pro-[tex]\pi[/tex] advocates will point out that the following formula looks so tidy:

[tex]\pi = \frac{A}{{{r^2}}}[/tex]

So it does. But in writing that formula, those who advocate [tex]\p[/tex] fail to have the courage of their convictions. They claim that diameter is the fundamental circle constant, not the radius. When it suits them, however, as it does here in order to make a succinct formula, then they are quick to abandon diameter and utilize radius in its place.

I say that the advocates of [tex]\p[/tex] do not have the right to refer to radius in the context of explaining any of the parameters of a circle. To be consistent with their own argument, they are logically bound to always refer to diameter and never to radius when discussing circles. And hence, the formula they should use is the much more cluttered version of the circle's diameter, as such:

[tex]\pi = \frac{4A}{{{d^2}}}[/tex]

In short, if pro-[tex]\pi[/tex] people don't like radius, if they don't recognize radius to be the fundamental circle constant, then they shouldn't use it.

Thanks to all of you who read my commentary.

I agree, I have noticed that the "piists" seem to not be able to decide on a consistent argument. They start out by trying to say that diameter is more important than the radius, despite the obvious falsehood of that argument, but then they will point out the area formula, in terms of the radius, as a win for pi! Is that not incredibly hypocritical? The piists seem to think that whichever suits pi must be more fundamental. If they want to make a real argument for pi being right, they can't say that diameter is more fundamental when looking at the circumference but radius is more fundamental when looking at area; they have to choose. It gets even worse though. Since radian angle measure uses both the radius and circumference, they will instead claim that a semicircle is more fundamental than a circle (except that they'll still say area of a circle is a win for pi even though, by their own logic, they should be looking at the area of a semicircle). Maybe even worse than that, in some places, there's neither a semicircle, nor an area, nor a diameter; there's a full circle and its radius, and tau. But then from some other calculation, a factor of 1/2 comes in and gets multiplied by tau. Then piists will still claim it shows pi to be more fundamental despite the factor of 1/2 coming from a completely different place than any of those other things (basically what Kodegadulo is talking about here: viewtopic.php?f=30&t=369).

I should point out though, that Spiked Math is neither a tauist nor a piists, so he's not exactly our "adversary," which I think he has shown by making this post.

Another thing that should be mentioned about the area of sectors is that the formulas τr and 1/2τr^2 are exactly the same as θr and 1/2θr^2 except that theta is replaced with tau. This is because they are the special case of these formulas when θ=τ because the circle is a tau radian arc/sector. With tau, you don't even have to memorize the sector formulas because they're the same as the full circle formulas you already know. With pi, you have to memorize an extra set of formulas, and you can end up getting the two sets mixed up. So even though area is simpler with pi, this makes it more convenient with tau, in addition to already being more intuitive and natural. So I think even area of a circle is a clear win for tau, despite being simpler with pi.
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