Comment by mr_mitm
1 year ago
In non-euclidean spaces, your definition of pi wouldn't even be a value. It's not well defined because the ratio of circumference to diameter of a circle is dependent on the size of the circle and the curvature inside the circle.
It's probably true that it's only well defined in euclidean space. Your relaxed definition, which I have never seen before, is not very useful.
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.
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.
> 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
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Similarly, g depends on the geometry, and g is a constant 0 for Euclidean space
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!
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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?
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This whole conversation is painful to read:
1. Your parent was talking about projections from one space to another and getting it confused.
2. Pi is pi and their non-Euclidean pi is still pi (unless you want to argue that a circle drawn on the earth’s surface has a different value of pi).
The problem comes down to projections, then all bets are off.
Yes. That's what makes it a fun fact. Most people never even learn about non-euclidean math, and this is the kind of "wow I never even thought about this" that people should be able to learn about in a comment thread.
Calling it painful to read is downright weird. Pi, the constant, has one value, everywhere. So now let's learn about what pi can also be and how that value is not universal.
It isn't a "fun fact" ... it's plainly incorrect.
π never changes its value. Ever. It is a constant in mathematics, no matter the geometry. However, π can have different ratios in other geometries, but it will still be ~3.14.
It is painful because this statement:
> draw a circle on a sphere. That circle has a curved diameter that is bigger than if you drew it on a flat sheet of paper. The ratio of the circle circumference to its diameter is less than 3.1415etc, so is that a different pi? You bet it is: that's the pi associated with that particular non-Euclidean, closed 2D plane.
The problem here is *projections*. If you project non-Euclidean space onto Euclidean space, you end up with some seemingly nonsensical things, like straight lines that get projected into arcs. This is your problem. You projected a curved line from non-Euclidean space onto Euclidean space and got an arc, but didn't account for the curvature of your "real non-Euclidean space" and thus ended up with an invalid value for π. If you got something that isn't ~3.14, then you did the math wrong somewhere along the way.
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