Today I Learned (TIL)

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TIL Colorado isn’t a rectangle and actually has 697 sides, mostly due to poor measurement tools (www.atlasobscura.com)

submitted 3 months ago by [email protected] to c/[email protected]

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[–] [email protected] 4 points 3 months ago (1 children)

So it doesn't fit on a sphere at all?

[–] kogasa 1 points 3 months ago (1 children)

Good point. Four equal angles, then, although they will each have to be greater than 90 degrees.

[–] [email protected] 1 points 3 months ago* (last edited 3 months ago) (1 children)

I don't think that would work for just 4 lines? I think you have to have arcs, not straight lines

[–] kogasa 3 points 3 months ago

It's possible to have an equiangular quadrilateral, i.e. whose sides are geodesics (the analogue of "straight line" on a sphere). The Gauss-Bonnet theorem implies their total interior angle is greater than 2pi, so four right angles can't work.

Here's an interactive demo of quadrilaterals on the sphere: https://geogebra.org/m/q83rUj8r

Notice that each side is a segment of a great circle, i.e. a circle that divides the sphere in half. That's what it means for a path to be a geodesic on the sphere.