Oh come on. You appealed to elegance and failed to show any. I could just as easily define a circle as the shape which encloses the most area for a given perimeter. If you think this is confusing, consider that it is the same as defining it as the shape which a small water droplet forms on a piece of glass.
The thing is that most of us knew what a circle was before we knew what a radius was. You've probably been encountering circles since before you could speak, and the term was certainly in your vocabulary long before you ever took a course in geometry. Appealing to the definition of a circle as elegant is weird when you consider the intrinsic inelegance of trying to formally define an intuitive concept. It makes more sense to measure it, which could be why Archimedes, Liu Hui, and Brahmagupta all ended up studying the same number.
Euclid's formalization of geometry was a landmark achievement in mathematics and possibly the most important single technique of antiquity. However, it was superseded multiple times before set theory became the foundation of essentially all of modern mathematics. Today, a circle is not an axiom but a construct itself derived from the distance formula and the definition of R^2 (a collection of points all the same distance...).
What got me involved in this argument is the assertion that it would make life easier for students learning mathematics. I, like most HN'ers, regard with serious concern the deterioration of mathematics education in the United States, but, also like most HN'ers, am not apt to solve my problems with snake oil. As a student myself, I regularly got pi/3 confused with pi/6, as the latter was a third of a right angle. Since angles and their respective sines and cosines were always diagrammed in class as portions of a right angle, I slipped up a few times between pi and pi/2.
This doesn't mean anything, though, other than a vagary of the way I used to think at the ripe old age of ten. Students of mathematics quite often have their own individual approaches and understandings of the concepts as presented, and this switch of constants is not really likely to make things any easier. This is why I kept pressuring you (unreasonably I do admit) to demonstrate that some essential proof or argument is simplified by using tau.
It is more annoying, though, when good, practical, tested, and effective solutions to educational problems go ignored in favor of something that geeks find interesting.
This is an example of a far more effective use of our collective time than the definition of any fundamental constant, be it pi (perhaps tau/2), e (perhaps 1/d, where d is the decay constant), i (perhaps -i), gamma (perhaps log(gamma-prime), since e^gamma appears as often as gamma), etc...
Really I've just tired of it today, and also had to cook dinner. :)
> You appealed to elegance and failed to show any.
You trot out Euler and then don't find it inelegant that his proof uses not a unit circle but a circle with a diameter of 1? Even as you also trot out Euclid who defines circles with radii? Fixing even just that is elegant.
And if you do find a few cases where π is super convenient (probably because you only care about half the rotation of something), feel free to substitute half tau. :)
The forest really is there, in addition to the trees.
But I really am done. Feel free to leave your last (I'm sure to be exceptionally) clever rebuttal for posterity.
>then don't find it inelegant that his proof uses not a unit circle but a circle with a diameter of 1?
I don't know, do you find it elegant? It's like a goddamn footnote, that's the whole point!
For reference, you trotted out Euclid when you defined the circle. I only pointed out whom you referenced.
>And if you do find a few cases where π is super convenient (probably because you only care about half the rotation of something), feel free to substitute half tau. :)
yawn
It does go without saying that you won't read this post, doesn't it? You didn't read the previous one.
