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Culture War Roundup for the week of September 19, 2022

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Isn’t the “imprecision” of that rectangle definition meant to generalize to other geometries? My theory is pretty rusty, but I’d think you could have a space where two sets of parallel lines don’t have symmetrical angles. Maybe one pair right angles, one pair not?

this relation cannot be expressed with mere definitions

I’m quite confident that it can. Geometry has a lot of mere definitions, and stringing a few of them gets you from any of the other rectangles definitions to the four-angles one. In the case of the single-right-angle version, adding Euclid’s fifth axiom confirms that the opposite angles must be identical, and either the 360° definition of a quadrilateral or the rules of supplementary angles should give us the final proof.

This is really important because theoretical math is all about reducing definitions.

My theory is pretty rusty, but I’d think you could have a space where two sets of parallel lines don’t have symmetrical angles. Maybe one pair right angles, one pair not?

What do you mean by "a space"? I'm in an ambivalent state where I can't tell if you're missing an obvious point, or talking theory at a level I'm entirely missing. Leaning towards the latter.

A space in the mathematical sense: a set of locations plus some rules which describe their relations. Our vanilla geometry is defined over Euclidean spaces, but by screwing around with one or more of the relation rules, we get other ones. The classic example is swapping out the parallel postulate to get "elliptic" and "hyperbolic" geometries. The former includes geometry on the surface of a sphere, which gives us triangles with two right angles and other shenanigans. I looked to [Wikipedia](https://en.wikipedia.org/wiki/Space_(mathematics)) for a better definition but only left more confused.

This was on my mind from trying to grasp quaternion basics for work. They're 4-dimensional, which I'm pretty sure is still handled by a Euclidean space, but also have some nice properties regarding spatial rotations.

In a space that has more than two dimensions, skew lines never meet but also are not parallel.

Actually you might be right the "at least" is basically there to apply to hyperbolic space or something like that. Of course, high schooler me had no idea about that stuff.