3 ms·

Hardly. It's acknowledging that both choices are definitions. One provides clarity and is related directly to the unit circle, the other is related to the circ

Hardly. It's acknowledging that both choices are definitions. One provides clarity and is related directly to the unit circle, the other is related to the circle with radius 1/2. Which is more intuitive?

scythe 15y ago
>One provides clarity How so? Neither is more intuitive. The unit circle is itself a definition you have grabbed. The notion of defining a circle by its radius comes to us from Euclid: >"Let the following be postulated": >1. "To draw a straight line from any point to any point." >2. "To produce [extend] a finite straight line continuously in a straight line." >3. "To describe a circle with any centre and distance [radius]." >4. "That all right angles are equal to one another." >5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." In fact, pi/2 itself is sitting right there in the fourth axiom, and pi is in the fifth. 2pi is nowhere to be found.
- ianterrell 15y ago
  It's tau/4 that's sitting in the fourth postulate, and tau is right there in the fifth (the sum of all four angles formed by the "straight line falling" on the side of the two straight lines' intersection). Besides, Euclid would have been a tau advocate, as he defined circles with their radius, which is clearly superior to the diameter. I can't help people chose the poorer constant for so long; I can only hope to help correct them.
  - scythe 15y ago
    >tau is right there in the fifth (the sum of all four angles formed by the "straight line falling" on the side of the two straight lines' intersection). ...yes, but that makes the postulate meaningless! You have to look at one side of the line in order for the postulate to have any relevance. >Euclid would have been a tau advocate Oh yeah? Well.. well... Ramanujan would have been a pi advocate! Ha! >which is clearly superior to the diameter. It is expedient in the process of mathematical argumentation. Looking at expedience, though, we see that using a constant 2pi introduces an untoward amount of fractions into just about every mathematical calculation -- see for example here: http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof_using_Fourier_series http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof_... Irrespective of the definition of constants, which is long since forgotten at this point (how much of a pain is it to define a circle, starting from ZFC?), it is kind of disappointing to see you refusing to read the proofs which you claim to be clarifying -- most of them get uglier moving to tau, on a cursory examination of the seminal work Proofs from THE BOOK. Go on, mentally replace every instance of "2pi" with "tau" and "pi" with "tau/2" in, say, this paper: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZe...
    - ianterrell 15y ago
      I can't convince you, but that doesn't make anything you've said above correct.
      - scythe 15y ago
        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. http://jumpmath.org/ http://jumpmath.org/ 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...

scythe 15y ago

>One provides clarity How so? Neither is more intuitive. The unit circle is itself a definition you have grabbed. The notion of defining a circle by its radius comes to us from Euclid: >"Let the following be postulated": >1. "To draw a straight line from any point to any point." >2. "To produce [extend] a finite straight line continuously in a straight line." >3. "To describe a circle with any centre and distance [radius]." >4. "That all right angles are equal to one another." >5. The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles." In fact, pi/2 itself is sitting right there in the fourth axiom, and pi is in the fifth. 2pi is nowhere to be found.

ianterrell 15y ago
It's tau/4 that's sitting in the fourth postulate, and tau is right there in the fifth (the sum of all four angles formed by the "straight line falling" on the side of the two straight lines' intersection). Besides, Euclid would have been a tau advocate, as he defined circles with their radius, which is clearly superior to the diameter. I can't help people chose the poorer constant for so long; I can only hope to help correct them.
- scythe 15y ago
  >tau is right there in the fifth (the sum of all four angles formed by the "straight line falling" on the side of the two straight lines' intersection). ...yes, but that makes the postulate meaningless! You have to look at one side of the line in order for the postulate to have any relevance. >Euclid would have been a tau advocate Oh yeah? Well.. well... Ramanujan would have been a pi advocate! Ha! >which is clearly superior to the diameter. It is expedient in the process of mathematical argumentation. Looking at expedience, though, we see that using a constant 2pi introduces an untoward amount of fractions into just about every mathematical calculation -- see for example here: http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof_using_Fourier_series http://en.wikipedia.org/wiki/Basel_problem#A_rigorous_proof_... Irrespective of the definition of constants, which is long since forgotten at this point (how much of a pain is it to define a circle, starting from ZFC?), it is kind of disappointing to see you refusing to read the proofs which you claim to be clarifying -- most of them get uglier moving to tau, on a cursory examination of the seminal work Proofs from THE BOOK. Go on, mentally replace every instance of "2pi" with "tau" and "pi" with "tau/2" in, say, this paper: http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZeta.pdf http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Zeta/EZe...
  - ianterrell 15y ago
    I can't convince you, but that doesn't make anything you've said above correct.
    - scythe 15y ago
      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. http://jumpmath.org/ http://jumpmath.org/ 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...