Comment by withinboredom | Hacker News Reader

Comment by withinboredom | Hacker News Reader

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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.

It doesn't have anything to do with projections.

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.

Thanks for the original comment—I picked up something new that I hadn't considered before.

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.

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