A Rotation by One Turn is 1

by fortmarinus

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.

References:
  1. https://www.michaelhartl.com/ [↩]
  2. https://hexnet.org/files/documents/tau-manifesto.pdf [↩]
  3. https://tauday.com/tau-manifesto [↩]
  4. https://www.imdb.com/title/tt8228288/ [↩]