Regarding 1/2 tau r^2

An enlightening discussion about pi and tau.

Regarding 1/2 tau r^2

by Chris Park » Sat Jul 09, 2011 4:59 pm

One of the prevalent arguments for tau that might impress the compulsive pattern-spotter in each of us was the resemblance of the following in Article 3.2, Table 3 of The Tau Manifesto:

[tex]y = \frac{1}{2}g t^2
U = \frac{1}{2}k x^2
T = \frac{1}{2}m v^2
A = \frac{1}{2}\tau r^2[/tex]

In short, I think this pattern is misleading. You might look at those equations and conclude that nature intended a symmetry between them, and that tau completes the set. I would argue that the similarity here is not a reflection of symmetry in nature but of symmetry imposed by the people who chose modern curriculum. It's easier to remember equations that look like one another.

Just take a look: why is an equation for energy stored in a spring being compared to a universal geometrical constant? The former is very specific and arbitrary, the latter is very general and natural.

The only reason we teach students in high school physics about springs is because they resemble elements of other mechanisms found in nature. You can approximate the force interaction between a pair of molecules with a spring model. You can approximate the deformation of an elastic solid with linear-regime bending. The key word is "approximate", though, because there is no such thing as a truly linear field strength in nature. It's a good educational tool, but there's nothing universal or natural about it. One of the many valid reasons we use it in teaching is because it resembles 1/2 mv^2.

Which brings me to another point: 1/2 mv^2 is also a simplification. Kinetic energy is only represented this way when mass is held constant (dm/dt = 0). When Newton wrote his Second Law, it almost certainly didn't read F = ma, but probably resembled dp/dt = F (or p_dot = F).

This reminds us that if we want to look at these equations in a way that comments on nature, we should look at their differential equations.

[tex]\frac{dy}{dt} = v
\frac{dU}{dx} = F
\frac{dT}{dv} = p
\frac{dA}{dr} = C[/tex]

Much more Maxwellian, right?

In this differential form, even the spring equation that I just finished bashing manages to take on a broader meaning about work done in a space-dependent field (not necessarily a linear one). This form also reminds us that the bottom line is not that nature divides by two and squares, but that the antiderivative of the product of two meaningful variables gives us a third meaningful variable

When g and k are held constant, we may rewrite the above as

[tex]\frac{dy}{dt} = v = gt
\frac{dU}{dx} = F = kx
\frac{dT}{dv} = p = mv
\frac{dA}{dr} = C = ?r[/tex]

Note that the first two equations now depend on a constant acceleration and a constant spring stiffness. p, however, is well defined whether or not m or v change in time or space. That's a departure from the pattern.

So before we jump to the conclusion that the fourth equation has to fit the pattern ('C = tau r'), we should ask ourselves why we didn't choose to write, say, v = 2Gt where g = 2G, or F = 2Kx where k = 2K, or p = 2Mv where m = 2M.

We didn't choose those forms because g, not G, is a measurable physical quantity. g is the acceleration of an object in freefall while G is nothing in particular. m, v and x are also physical variables. These variables are worth preserving within their respective equations, so there is no reason to insert 2s into their equations.

I would argue that k is not a directly measurable variable. It can be thought of as the strength of a potential field, and it can only be expressed in terms of other SI units (Newtons per meter). It is also not a natural variable like p, which follows its own conservation law. This is a man-made variable (it was not discovered, but invented) - another departure from the pattern.

So the pattern has been broken twice, and the choice to do so in both cases is seen to be a matter of prudence. Therefore we must ask "Why pick C = 2 pi r over C = tau r or vice versa?" Clearly r is measurable while neither pi nor tau are directly measurable quantities (you have to express them in terms of C, D and/or r).

So overall, I don't think you can count this silly pattern as evidence for either case. The similarity is striking at first, however, if you give the matter a little thought, you realize that it's not really such an amazing coincidence of nature that these formula take the "1/2 x y^2" form. Not that it's arbitrary, either - ending up with this form just means that you started with a product of two variables ("x y") and took the antiderivative with respect to only one of them. And there's nothing profound about a first-derivative relationship, really. I think that what we see in the tau manifesto is a collection of arbitrary equations that share this common calculus-based operator.

After all, length*width = area, and speed*time = distance, but that doesn't mean we should insist that apple*orange = banana.
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Re: Regarding 1/2 tau r^2

by SpikedMath » Sun Jul 10, 2011 1:03 am

Very good points! I was wondering if the pattern occurred naturally or if some equations were chosen to fit the pattern in physics, but just didn't have the necessary background to discuss it in detail like you did.
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Re: Regarding 1/2 tau r^2

by Eon » Sun Jul 10, 2011 4:06 am

Formulas like these might only seem simpler with tau or pi depending on whether they coincidentally have a factor 1/2 or 2 in them which then apparently cancels out nicely and makes them simpler to write.

But we should not be tempted to try and minimize the number of symbols needed to represent a formula, rather to look again at the original definition and choose what is simpler there. I like having a symbol for "one revolution" in radians.

I thus disagree with the author of the tau manifesto on the point of the mentioned formulas, but ultimately agree with him/her that tau is nicer to work with.
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Re: Regarding 1/2 tau r^2

by xander » Sun Jul 10, 2011 11:26 am

[tex]\int x\,dx = \frac{1}{2}x^2+C[/tex]

If we are defining things in terms of derivatives, it seems logical to integrate both sides. Doing so gives us a term that looks like [tex]\frac{1}{2}x^2[/tex], times some constants. Thus it seems appropriate to me that there should be tons of identities that look like [tex]k\left(\frac{1}{2}x^2\right)[/tex] in mathematics and physics---as you say, it falls out of the integration. There is a similarity in the forms of the equations not because of any deep natural relationship, but because of a mathematical similarity in how they are derived.

That being said, it seems to me that your argument basically comes down to "There is no reason choose [tex]\tau[/tex] over [tex]\pi[/tex] just to make some equations look similar to each other." The counter to that is that there is no compelling reason to choose [tex]\pi[/tex] over [tex]\tau[/tex] on the same grounds. It doesn't matter which constant we choose---so why not choose the constant that is best in other circumstances (whether it be [tex]\pi[/tex] or [tex]\tau[/tex]).

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Re: Regarding 1/2 tau r^2

by DMAshura » Wed Jul 13, 2011 11:42 am

I agree with what Xander says in that sometimes pi is more convenient and that sometimes tau is more convenient. However, since I also agree that the 1/2 comes from the process of integration, I feel this indicates the tau was "originally there", and the pi is a simplification, making tau the more fundamental constant.
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Re: Regarding 1/2 tau r^2

by Chris Park » Thu Jul 14, 2011 12:38 am

xander wrote:...it seems to me that your argument basically comes down to "There is no reason choose [tex]\tau[/tex] over [tex]\pi[/tex] just to make some equations look similar to each other." The counter to that is that there is no compelling reason to choose [tex]\pi[/tex] over [tex]\tau[/tex] on the same grounds.


Actually I think we agree here: I don't think the pattern supports either tau nor pi. I think it's insubstantial to the argument. All smoke and mirrors. For this reason, I think it should be discounted from the Tau Manifesto.

I think the only form of those equations that describes nature is the differential form -- anything that's already been integrated assumes something about the shape of the field in space/time. In all four of the cases above, the field has been stipulated as being linear everywhere in space, which definitely isn't "natural" at all but is a simplification/approximation of nature used for educational purposes. So of course when you decide to make a whole bunch of different fields linear and then integrate them you're going to get similar-looking results, but they're only artificially similar.
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A much stronger, simpler argument in favor of 1/2 tau r^2

by josephlindenberg » Thu Jul 14, 2011 8:37 am

A much stronger, simpler argument in favor of [tex]\frac{1}{2}\tau r^2[/tex] is that you get to memorize two math equations for the price of one.

For a SECTOR OF A CIRCLE of angle θ
AREA OF THE SECTOR = [tex]\frac{1}{2}\theta r^2[/tex]


A full circle is just the special case where the sector has angle TAU
AREA OF THE WHOLE CIRCLE = AREA OF A SECTOR OF ANGLE TAU = [tex]\frac{1}{2}\tau r^2[/tex]


Math students only have to remember the one equation!
And it's a more general, more powerful equation!

Any time they want the area of a whole circle, they just replace one greek letter (θ) with another (TAU).

(SpikedMath pointed out this formula in another thread here. And I think we should all thank him by using it against him.)
PI is the SEMICIRCLE constant, not the circle constant . . . http://sites.google.com/site/taubeforeitwascool
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Re: Regarding 1/2 tau r^2

by josephlindenberg » Fri Oct 07, 2011 11:20 pm

I've fleshed out the other argument for [tex]\frac{1}{2}\tau r^2[/tex] at the beginning of http://sites.google.com/site/taubeforeitwascool

Take a look and see if it convinces you.


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Re: Regarding 1/2 tau r^2

by second-grade-math » Thu Oct 20, 2011 5:21 pm

Really liked the tau before it was cool link. In search for sector of a circle formula I stumbled on this site and literally to this post with a detail explanations of pi and tau in different math and physics equations. But I teach to my students, the symmetry and use it to memorize more than one equations.
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Re: Regarding 1/2 tau r^2

by Kodegadulo » Fri Jun 15, 2012 9:18 pm

Chris Park wrote:So overall, I don't think you can count this silly pattern as evidence for either case. The similarity is striking at first, however, if you give the matter a little thought, you realize that it's not really such an amazing coincidence of nature that these formula take the "1/2 x y^2" form. Not that it's arbitrary, either - ending up with this form just means that you started with a product of two variables ("x y") and took the antiderivative with respect to only one of them. And there's nothing profound about a first-derivative relationship, really. I think that what we see in the tau manifesto is a collection of arbitrary equations that share this common calculus-based operator.

The choice of [tex]k[/tex] for [tex]A = k r^2[/tex] is not at all arbitrary. Do you think this is the only integration step there is to this? Have you ever heard of n-spheres? Let's take a look at all n-spheres, for all dimensions [tex]n \in \mathbb{N}[/tex], shall we?

Here's how you get the n-dimensional "area" ([tex]A_n[/tex]) and n-dimensional "volume" ([tex]V_n[/tex]) for every n-sphere, where an n-sphere is defined as a set of points a given distance [tex]r[/tex] from a given center point, in n-dimensional space. I'll do it as a recursive algorithm because it's easier to understand that way.

First, keep in mind that the way this works is modulo 2: All the even-dimensioned n-spheres are built on each other, and all the odd-dimensioned n-spheres are built on each other. They don't mix. But other than having two different starting points, dimension 0 versus dimension 1, the algorithm for building the higher dimensions is exactly the same.

So let's start with the base cases. First, 0 dimensions. Anybody read Flatland? Zero dimensions is "Pointland". Nothing but a single point, with no dimensions. That's the 0-sphere. Does it have anything analogous to an "area"? Not really. No surface. So let

[tex]A_0 = 0[/tex]

Does it have anything analogous to a "volume"? Well, yes, sort of. It's got itself, a single point. So let

[tex]V_0 = 1[/tex]

That means, all the even powers are going to be built starting from a simple coefficient of 1.

Now let's go to 1 dimension. Welcome to "Lineland". Here at least we've got points and length, so we can pick a center point and a radius. What is the set of points a given distance from a given point, on a line? Well, there's one point to the right that's 1 radius away, and one to the left that's 1 radius away. 2 points, the vertices delimiting a line segment. The line segment is our 1-sphere, and its vertices constitute its "surface". So let

[tex]A_1 = 2[/tex]

We can actually stop right here, we've got all we need to start with. But notice the 2: All the odd powers are going to be built starting from that simple coefficient of 2. That's pretty much the only difference between the odd and even powers, the fundamental difference between a 1-point "volume" in Pointland and a 2-point "area" in Lineland.

Now, here's the program: If you have the "area" of your n-sphere, [tex]A_n[/tex], you get its "volume" by multiplying by the radius, [tex]r,[/tex] and dividing by the dimension, [tex]n[/tex]. In other words:

(1) [tex]V_n = A_n (\frac{r}{n})[/tex]

This represents a classic step of integral calculus, raising the function of [tex]r[/tex] by a power, and pulling out fractional coefficient based on the new power, which for each n-sphere winds up being its current dimension [tex]n[/tex].

So let's do it for dimension n=1. What's the "volume" of a line with a given radius? That would be its length, and that's twice the radius, right? Let's check:

[tex]V_1 = A_1 (\frac{r}{1}) = 2r[/tex]

Yep, that works.

Now, notice that so far we've only dealt with points, lengths, and dimension numbers. Nothing so far about either [tex]\tau[/tex] or [tex]\pi[/tex]. We go from an "area" to its associated "volume" without involving either. Got it?

But now for the next step: You can get the "area" of a higher n-sphere, [tex]A_n[/tex], from the "volume" from two dimensions down, [tex]V_{n-2}[/tex], by multiplying that by a circumference. I'll leave it as an exercise for the reader how to integrate around a rotation to get a circumference like this. But what we get is:

(2) [tex]A_n = V_{n-2} C[/tex]

where

[tex]C = \tau r = 2 \pi r[/tex]

So now we have a choice of doing this with [tex]\tau[/tex] or [tex]\pi[/tex]. How about we use [tex]\tau[/tex].

So let's go from Pointland to Flatland itself, 2 dimensions. At last we get some "circularity". The 2-sphere is none other than the circle, and the "area" of its "surface" is simply its circumference:

[tex]A_2 = V_0 C = (1)(\tau r) = \tau r[/tex]

And its "volume" is, amusingly enough, its area. We get that by going back to step (1) where [tex]n=2[/tex]:

[tex]V_2 = A_2 (\frac{r}{2}) = (\tau r)(\frac{r}{2}) = \frac{1}{2}\tau r^2[/tex]

Okay, now let's go back to Lineland and use that to jump to Spaceland, 3 dimensions. Familiar territory for you and me. The 3-sphere is none other than the sphere itself. To get its surface area, which for once actually is an area, we go back to step (2) and multiply the 1-volume by a circumference to get the 3-area:

[tex]A_3 = V_1 C = (2r)(\tau r) = 2\tau r^2[/tex]

Notice that 2 coefficient we inherit from the two vertices of the line segment. Okay, now apply step (1) again to get the 3-volume, the actual volume of the actual sphere:

[tex]V_3 = A_3 (\frac{r}{3}) = (2\tau r^2)(\frac{r}{3}) = \frac{2}{3}\tau r^3[/tex]

Next stop, 4-space. Land of the tesseract. We're on the hunt for the 4-sphere. To get its "surface area", which to us really feels like a volume, we go back to Flatland and slap on another circumference:

[tex]A_4 = V_2 C = (\frac{1}{2} \tau r^2)(\tau r) = \frac{1}{2} \tau^2 r^3[/tex]

And we throw on another radius and divide by the dimension, 4, to get the hypervolume:

[tex]V_4 = A_4 (\frac{r}{4}) = (\frac{1}{2} \tau^2 r^3)(\frac{r}{4}) = \frac{1}{2\cdot 4} \tau^2 r^4[/tex]

So let's see where this takes us as we fly up the dimensions:

[tex]\begin{tabular}{l l l l l l ll l l l}
n = 0 & & A_0 = 0 & & V_0 = 1 & & n = 1 & & A_1 = 2 & & V_1 = 2r \\
n = 2 & & A_2 = \tau r & & V_2 = \frac{1}{2}\tau r^2 & & n = 3 & & A_3 = 2\tau r^2 & & V_3 = \frac{2}{3} \tau r^3 \\
n = 4 & & A_4 = \frac{1}{2}\tau^2 r^3 & & V_4 = \frac{1}{2\cdot 4}\tau^2 r^4 & & n = 5 & & A_5 = \frac{2}{3} \tau^2 r^4 & & V_5 =\frac{2}{3\cdot 5} \tau^2 r^5 \\
n = 6 & & A_6 = \frac{1}{2\cdot 4}\tau^3 r^5 & & V_6 =\frac{1}{2\cdot 4\cdot 6}\tau^3 r^6 & & n = 7 & & A_7 = \frac{2}{3\cdot 5} \tau^3 r^6 & & V_7 = \frac{2}{3\cdot 5\cdot 7} \tau^3 r^7 \\
n = 8 & & A_8 = \frac{1}{2\cdot 4\cdot 6}\tau^4 r^7 & & V_8 = \frac{1}{2\cdot 4\cdot 6\cdot 8}\tau^4 r^8 & & n = 9 & & A_9 = \frac{2}{3\cdot 5\cdot 7} \tau^4 r^8 & & V_9 = \frac{2}{3\cdot 5\cdot 7\cdot 9} \tau^4 r^9 \\
n = 10 & & A_{10} = \frac{1}{2\cdot 4\cdot 6\cdot 8}\tau^5 r^9 & & V_{10} = \frac{1}{2\cdot 4\cdot 6\cdot 8\cdot 10}\tau^5 r^{10} & & n = 11 & & A_{11} = \frac{2}{3\cdot 5\cdot 7\cdot 9} \tau^5 r^{10} & & V_{11} = \frac{2}{3\cdot 5\cdot 7\cdot 9\cdot 11} \tau^5 r^{11} \\
\end{tabular}[/tex]

There you have it, the first dozen n-spheres. Powers of [tex]\tau[/tex] and [tex]r[/tex] keep going up in a regular pattern. There's a double-factorial process rolling out those skyrocketing denominators in the coefficients. But the numerators are fixed: 1 for the even dimensions, 2 for the odds, carried through all the way down from Pointland and Lineland. Overall, a fairly tidy system. And even though the even and odd dimensions are operating in parallel, you can still see how unified it is.

So let met ask you, in the face of those double-factorial denominators, what do you hope to stuff into that [tex]\tau[/tex] constant to cancel any of that out? What is so special about the [tex]1/2[/tex] in [tex]V_2[/tex] that it should be canceled out by the [tex]2[/tex] in [tex]2\pi[/tex]? But not, let's say the [tex]1/3[/tex] integrated into [tex]V_3[/tex]? Or the [tex]1/4[/tex] integrated into [tex]V_4[/tex]? The [tex]1/5[/tex] in [tex]V_5[/tex]?

There is no justification for using anything but [tex]\tau[/tex] here!
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Re: Regarding 1/2 tau r^2

by dimension10 » Sun Jul 01, 2012 11:15 pm

Chris Park wrote:One of the prevalent arguments for tau that might impress the compulsive pattern-spotter in each of us was the resemblance of the following in Article 3.2, Table 3 of The Tau Manifesto:

[tex]y = \frac{1}{2}g t^2
U = \frac{1}{2}k x^2
T = \frac{1}{2}m v^2
A = \frac{1}{2}\tau r^2[/tex]

In short, I think this pattern is misleading. You might look at those equations and conclude that nature intended a symmetry between them, and that tau completes the set. I would argue that the similarity here is not a reflection of symmetry in nature but of symmetry imposed by the people who chose modern curriculum. It's easier to remember equations that look like one another.

Just take a look: why is an equation for energy stored in a spring being compared to a universal geometrical constant? The former is very specific and arbitrary, the latter is very general and natural.

The only reason we teach students in high school physics about springs is because they resemble elements of other mechanisms found in nature. You can approximate the force interaction between a pair of molecules with a spring model. You can approximate the deformation of an elastic solid with linear-regime bending. The key word is "approximate", though, because there is no such thing as a truly linear field strength in nature. It's a good educational tool, but there's nothing universal or natural about it. One of the many valid reasons we use it in teaching is because it resembles 1/2 mv^2.

Which brings me to another point: 1/2 mv^2 is also a simplification. Kinetic energy is only represented this way when mass is held constant (dm/dt = 0). When Newton wrote his Second Law, it almost certainly didn't read F = ma, but probably resembled dp/dt = F (or p_dot = F).

This reminds us that if we want to look at these equations in a way that comments on nature, we should look at their differential equations.

[tex]\frac{dy}{dt} = v
\frac{dU}{dx} = F
\frac{dT}{dv} = p
\frac{dA}{dr} = C[/tex]

Much more Maxwellian, right?

In this differential form, even the spring equation that I just finished bashing manages to take on a broader meaning about work done in a space-dependent field (not necessarily a linear one). This form also reminds us that the bottom line is not that nature divides by two and squares, but that the antiderivative of the product of two meaningful variables gives us a third meaningful variable

When g and k are held constant, we may rewrite the above as

[tex]\frac{dy}{dt} = v = gt
\frac{dU}{dx} = F = kx
\frac{dT}{dv} = p = mv
\frac{dA}{dr} = C = ?r[/tex]

Note that the first two equations now depend on a constant acceleration and a constant spring stiffness. p, however, is well defined whether or not m or v change in time or space. That's a departure from the pattern.

So before we jump to the conclusion that the fourth equation has to fit the pattern ('C = tau r'), we should ask ourselves why we didn't choose to write, say, v = 2Gt where g = 2G, or F = 2Kx where k = 2K, or p = 2Mv where m = 2M.

We didn't choose those forms because g, not G, is a measurable physical quantity. g is the acceleration of an object in freefall while G is nothing in particular. m, v and x are also physical variables. These variables are worth preserving within their respective equations, so there is no reason to insert 2s into their equations.

I would argue that k is not a directly measurable variable. It can be thought of as the strength of a potential field, and it can only be expressed in terms of other SI units (Newtons per meter). It is also not a natural variable like p, which follows its own conservation law. This is a man-made variable (it was not discovered, but invented) - another departure from the pattern.

So the pattern has been broken twice, and the choice to do so in both cases is seen to be a matter of prudence. Therefore we must ask "Why pick C = 2 pi r over C = tau r or vice versa?" Clearly r is measurable while neither pi nor tau are directly measurable quantities (you have to express them in terms of C, D and/or r).

So overall, I don't think you can count this silly pattern as evidence for either case. The similarity is striking at first, however, if you give the matter a little thought, you realize that it's not really such an amazing coincidence of nature that these formula take the "1/2 x y^2" form. Not that it's arbitrary, either - ending up with this form just means that you started with a product of two variables ("x y") and took the antiderivative with respect to only one of them. And there's nothing profound about a first-derivative relationship, really. I think that what we see in the tau manifesto is a collection of arbitrary equations that share this common calculus-based operator.

After all, length*width = area, and speed*time = distance, but that doesn't mean we should insist that apple*orange = banana.


Great point!
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Re: Regarding 1/2 tau r^2

by 1=0 » Sat Apr 01, 2017 11:08 pm

Why is it that the piists are focusing so much on the Tau Manifesto's argument regarding the pattern 1/2kx^2 and calling it contrived yet not addressing the entire argument? The Tau Manifesto showed exactly where that pattern comes from, integration of a linear function. This integration gives us a factor of 1/2 in 2D areas and 1/3 in 3D volumes. All of the other formulas in the Tau Manifesto also come from integrating a linear function. When people say that the pattern is contrived just because the other formulas have nothing to do with circles, all they're doing is ignoring where the pattern actually comes from; no one is claiming that those other formulas have anything to do with circles, but they are derived in the same way as the formula for area of a circle. As Xander said above, "There is a similarity in the forms of the equations not because of any deep natural relationship, but because of a mathematical similarity in how they are derived."
Also, it really doesn't make any sense intuitively to say circumference (a distance) is just the derivative (not a distance) of area, but it makes very much intuitive sense to say that the area of a circle is an integral, another area. The Tau Manifesto also showed why the area is the integral of the circumference, but can anyone find an intuitive way to demonstrate that circumference is the derivative of area without first proving that area is the integral of the circumference?
It is also important that circumference is one-dimensional while area is 2D. Doesn't it make a lot more sense to start with one dimension, and then go to two dimensions via integration, and so on rather than to start with two dimensions and then go backwards to one dimension?

I do think the Tau Manifesto could improve its area of a circle argument by including the part about summing the areas of infinitely thin "skinny triangles." I think that argument is the most convincing because of how clearly it demonstrates the factor of 1/2 and how it comes from triangles, which have the formula 1/2 bh for area, and because it doesn't require calculus, so anyone can understand it. It's a shame that that argument wasn't in the manifesto, but luckily it is in "Tau Before It Was Cool," which is one of the reasons I think everyone looking into the pi-tau debate should read that site.
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