4 ms·

I don't agree, I thought what he said was very interesting. It never occurred to me that pi might vary, and over a non-flat space I can see what they're saying.

I don't agree, I thought what he said was very interesting. It never occurred to me that pi might vary, and over a non-flat space I can see what they're saying. I think it's intrinsically interesting simply because it breaks one of my preconceptions, that pi is a constant. Talking about it being 'not very useful' just seems far too casually dismissive.

mr_mitm 2y ago
Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry. Clearly everyone means euclidean space unless specified otherwise. Any other interpretation will only lead to problems, which is why it's not useful. There is really no ambiguity about this in mathematics. Mathematicians still use the pi symbol as a constant when they compute the circumference of a circle in a given geometry as a function of the radius.
- TheRealPomax 2y ago
  There is no "clearly" in Math. The fact that pi is a constant while at the same time not being "the same constant" in all spaces, and not even being "a single value, even if we alias it as the symbol pi" is what makes it a fun fact. Heaven forbid people learn something about math that extends beyond the obvious, how dare they!
  - travisjungroth 2y ago
    > There is no "clearly" in Math. Sure there is. You don’t expect a paper to explain that the numbers are in decimal and not hexadecimal.
    - TheRealPomax 2y ago
      I don't know which ones you read so I can't comment on that, but the ones I read most definitely specify which fields of math they apply to, and which axioms are assumed true before doing the work, because the math is meaningless without that? Papers on non-Euclidean spaces always call that out, because it changes which steps can be assumed safe in a proof, and which need a hell of a lot of motivation. And of course, that said: yeah, there are papers for proofs about things normally associated with decimal numbers that explicit call out that the numbers they're going to be using are actually in a different base, and you're just going to have to follow along. https://en.wikipedia.org/wiki/Conway_base_13_function https://en.wikipedia.org/wiki/Conway_base_13_function is probably the most famous example?
- 8note 2y ago
  Similarly, g depends on the geometry, and g is a constant 0 for Euclidean space
- _a_a_a_ 2y ago
  Ur being rather snotty about this. I've just realised something important which is so obvious to you that you consider it trivial, but it's not. I realised something important today, you might just want to feel pleased for me, and a bit pleased that the world is a little less ignorant today...? Or not?
  - mr_mitm 2y ago
    Sorry for coming off as snotty. It wasn't my intention, I thought my statements were rather matter of fact. It's possible that after having attended two lectures on differential geometry I have forgotten that some of these things like circumference ratios and sum of angles of triangles being different in a curved geometry are not obvious to every one. I'm glad you learned something! Edit: This picture should make it pretty clear for anyone who is new to this concept: https://en.wikipedia.org/wiki/Non-Euclidean_geometry#/media/File:Comparison_of_geometries.svg https://en.wikipedia.org/wiki/Non-Euclidean_geometry#/media/...
    - _a_a_a_ 2y ago
      No probs, and yes I certainly learnt something important, thanks.
- jcelerier 2y ago
  > Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry. Can you share some place where Pi isn't defined exclusively as being "the ratio of circumference to diameter of a circle"? I have never heard any other definition in my life, and couldn't find any other through the first few Google results
  - _a_a_a_ 2y ago
    "This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C / d" https://en.wikipedia.org/wiki/Pi https://en.wikipedia.org/wiki/Pi
  - rjmunro 2y ago
    Pi is the "the ratio of circumference to diameter of a circle in Euclidian space". Everyone agrees that. What people are arguing is that when you have a circle in non-Euclidian space, so that the ratio of it's circumference to diameter is different, do we still call this new ratio Pi. Most people would argue that we don't. They say "the ratio is not Pi" rather than "Pi is a different value".

mr_mitm 2y ago

Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry. Clearly everyone means euclidean space unless specified otherwise. Any other interpretation will only lead to problems, which is why it's not useful. There is really no ambiguity about this in mathematics. Mathematicians still use the pi symbol as a constant when they compute the circumference of a circle in a given geometry as a function of the radius.

TheRealPomax 2y ago
There is no "clearly" in Math. The fact that pi is a constant while at the same time not being "the same constant" in all spaces, and not even being "a single value, even if we alias it as the symbol pi" is what makes it a fun fact. Heaven forbid people learn something about math that extends beyond the obvious, how dare they!
- travisjungroth 2y ago
  > There is no "clearly" in Math. Sure there is. You don’t expect a paper to explain that the numbers are in decimal and not hexadecimal.
  - TheRealPomax 2y ago
    I don't know which ones you read so I can't comment on that, but the ones I read most definitely specify which fields of math they apply to, and which axioms are assumed true before doing the work, because the math is meaningless without that? Papers on non-Euclidean spaces always call that out, because it changes which steps can be assumed safe in a proof, and which need a hell of a lot of motivation. And of course, that said: yeah, there are papers for proofs about things normally associated with decimal numbers that explicit call out that the numbers they're going to be using are actually in a different base, and you're just going to have to follow along. https://en.wikipedia.org/wiki/Conway_base_13_function https://en.wikipedia.org/wiki/Conway_base_13_function is probably the most famous example?
8note 2y ago
Similarly, g depends on the geometry, and g is a constant 0 for Euclidean space
_a_a_a_ 2y ago
Ur being rather snotty about this. I've just realised something important which is so obvious to you that you consider it trivial, but it's not. I realised something important today, you might just want to feel pleased for me, and a bit pleased that the world is a little less ignorant today...? Or not?
- mr_mitm 2y ago
  Sorry for coming off as snotty. It wasn't my intention, I thought my statements were rather matter of fact. It's possible that after having attended two lectures on differential geometry I have forgotten that some of these things like circumference ratios and sum of angles of triangles being different in a curved geometry are not obvious to every one. I'm glad you learned something! Edit: This picture should make it pretty clear for anyone who is new to this concept: https://en.wikipedia.org/wiki/Non-Euclidean_geometry#/media/File:Comparison_of_geometries.svg https://en.wikipedia.org/wiki/Non-Euclidean_geometry#/media/...
  - _a_a_a_ 2y ago
    No probs, and yes I certainly learnt something important, thanks.
jcelerier 2y ago
> Pi doesn't vary. The ratio of circumference to diameter of a circle may vary depending on the geometry. Can you share some place where Pi isn't defined exclusively as being "the ratio of circumference to diameter of a circle"? I have never heard any other definition in my life, and couldn't find any other through the first few Google results
- _a_a_a_ 2y ago
  "This definition of π implicitly makes use of flat (Euclidean) geometry; although the notion of a circle can be extended to any curve (non-Euclidean) geometry, these new circles will no longer satisfy the formula π = C / d" https://en.wikipedia.org/wiki/Pi https://en.wikipedia.org/wiki/Pi
- rjmunro 2y ago
  Pi is the "the ratio of circumference to diameter of a circle in Euclidian space". Everyone agrees that. What people are arguing is that when you have a circle in non-Euclidian space, so that the ratio of it's circumference to diameter is different, do we still call this new ratio Pi. Most people would argue that we don't. They say "the ratio is not Pi" rather than "Pi is a different value".