### A Rotation by One Turn is 1

In Michael Hartl’si Tau Manifestoiiiii, he addresses a criticism of the Tau formulation which points out that the $\tau$-form of Euler’s equation only relates 4 fundamental constants (e, i, circle constant, and 1), while the $\pi$-form relates 5 (despite needing to be re-arranged to do so). In response, he jokingly tacks on a trivial ‘+0’ to the $\tau$-form to put it on par with the $\pi$-form.

 Re-arranged Pi form $e^{i\pi}+1 = 0$ Trivially Modified Tau form $e^{i\tau} = 1 + 0$ Full Pi form $e^{i\pi} = (-1 + 0i)$ Full Tau form $e^{i\tau} = (1 + 0i)$

However, all joking aside, $e^{i\tau}$ is a complex number which evaluates to the complex number (1 + 0i). Thus, it is not trivial to represent the zero, in fact, it is thoroughly correct to do so. Conversely, the cleverly re-arranged $\pi$-form is even uglier than previously thought, as it breaks up its own complex number evaluation (-1 + 0i) and places half of it on either side of the equation.

Both forms properly include a zero, but the modified $\pi$-form breaks the symmetry of the solution, while the $\tau$-form maintains its symmetry without modification.

Now, as a disclaimer on the $\pi-\tau$ debate, I believe that the debate is not about choosing a constant or a symbol that makes equations look prettier; this is utterly secondary and wholly counterproductive. The debate is about the proper definition of the fundamental circle constant. Which, is $\tau$, obviouslyiv.

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