The author had a nice try with pi "showing up" in places, but just a few things I've seen so far (after looking for only a few minutes):

1. "Hey I have more functions than you!" isn't exactly a good argument. This author is doing himself exactly what the article is accusing "tauists" of doing --- conveniently asking questions that have pi as their answer.

2. I would argue that a lot of the functions in which pi is more convenient are much more complex and found in higher areas of mathematics, and those in which tau is found are much more elementary --- basically that pi is based off tau, rather than the other way around.

3. Some of those functions have the pi in there because it was a simplification from things cancelling out with tau ... like the formula for an ellipse. Sure, pi a b is more compact, but where does it come from? Integrating a polar form from 0 to 2pi ... or 0 to tau.

4. In the section on Euler's identity, if the author wants to say the result with pi is "stronger" he should say WHY it is stronger.

5. As for the 1/2 in the area formula for a circle ... it actually follows directly from Archimedes' original calculation of the area of a circle, which is that it has the same area as a triangle with its base equal to the circumference and its height equal to the radius --- in short, 1/2 C r.

6. The trigonometric functions having multiples of pi for their periods takes away again from the MEANING of those periods. When you say that sin(x) has a period of 2pi and tan(x) has a period of pi, big deal. You've said they're numbers. When you say the periods are tau and 1/2 tau, you realize something ... sin(x) takes 1 full circle (1 tau) to repeat itself, and tan(x) takes 1/2 a circle (1/2 tau) to do the same.

7. The argument for pi being convenient for area isn't bad, but I would argue measures of length (1-dimensional) are more fundamental than measures of area (2-dimensional). All other dimensions are extensions of 1 dimension, not 2. In grade school you learn that if you double the length in a figure, you quadruple the area, and multiply the volume by 8. Should we be teaching that if you double the area, you multiply the length by √(2) and the volume by 2√(2)?

Here's the thing ... I don't advocate COMPLETELY getting rid of pi like the author insinuates, but I do believe that tau is the more fundamental constant. As far as I'm concerned, tau should be taught as the more fundamental constant when students are learning it, and pi would be learned for convenience in calculating various other formulas.