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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