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.