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.
Heaven forbid people learn something about math that extends beyond the obvious, how dare they!
Sure there is. You don’t expect a paper to explain that the numbers are in decimal and not hexadecimal.
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 is probably the most famous example?
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/...
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
Most people would argue that we don't. They say "the ratio is not Pi" rather than "Pi is a different value".
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.
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.
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.
Let's say you live in a non-flat space. You come up with the idea of a "circle" with the usual definition: the set of points on the same plane equidistant from a central point.
You then trace along the circle and measure the length, and compare it to the length of the diameter. It turns out that this ratio changes as a function of the diameter. This truth is inherent to curved spaces themselves, and is not an artifact of choosing to describe the space using a projection.
The definition of pi in this world is no longer the invariant ratio of diameter to circumference. You can still recover pi by taking the limit of this ratio as the diameter length goes to zero. But perhaps mathematicians in this world would (justifiably) not see pi as such a fundamental number.
Now back to GP's example: people living on a sphere (like us) are analogous to inhabitants of a non-Euclidean space. The surface of Earth is analogous to 3-space, and the curvature of Earth is analogous to the curvature of space.
And indeed, if you draw real-life larger and larger circles on our planet, you will find that the ratio of circumference to diameter is smaller than pi. For example, if you start at some point on the Earth (say, the North pole) and trace out a circle 100 miles distant from it, you will find that the circumference of that circle (as measured by walking around that circle) is a little bit _less_ than pi x 100 miles.
Again, we have done no "projection" here. We've limited ourselves to operations that are fairly natural from the perspective of a mathematician living in that space, such as measuring lengths.
The fact that large circle circumferences measure less than pi x diameter on Earth did not change how we developed math, likely because you only notice start to notice this effect with extremely large (relative to us) circles.
But perhaps the inhabitants of a non-Euclidean space that was much more highly curved would notice it much earlier, and it would affect their development of maths, such that the number pi is held in lower regard.
That said, you could spin this by suggesting that the mathematician living in that non-Euclidean space might also have a different perspective on numbers. If we assume pi is still constant for him, then the numbers he's always known could be shifting in value but maybe that's a stretch.