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Hyperbolic Geometry is proving better than Euclidian Geometry at describing human brain function.

Cool to think of how much we have yet to understand.

The following are statements that I can't easily confirm, so first things first, tagging Devac. School me good, in any way, if I'm not 100% correct. Either of you.

- Basically whenever you are projecting something that is 3D onto a flat surface or vice versa you are using non-Euclidean geometry.

I don't think this is technically true(?). This is the case if you were to attempt to translate spherical or cylindrical surfaces into 2D projections at the scale of taking a pocketknife to the surface of a globe or donut or drumstick and attempting to flatten it, but there are relatively neat mathematical ways of doing such a thing, handled by the differential path length treatment.

Non-Euclidean geometry arises when path length and topology (space-time, in this instance) are a function of something, e.g. position, and thus the transformation tensors vary from point to point. Frame dragging, gravitational space-time dilation, and the event horizon are all related concepts.

Edit: I think I'm actually incorrect, and that spherical coordinates *do* seem to violate "Euclideanism", which seems like a dumb formulation anyway. Leaving my original statements, because we can easily reformulate the violations of Euclideanism, like a triangle on a spherical surface with three 90 degree corners, into orthogonal coordinates by imposing a set of conditions, and then we're "Euclidean" once more. This is kinda a waste of time, debating the origins of geometry. Saying that, but acknowledging that I'm spoiled to have been exposed to these ideas casually in my education.

Straight, direct line segments marking the shortest distance between two points are a must for Euclidean metric. Strictly speaking, even Taxicab geometry is non-Euclidean.

Odder is right about both commonality of non-Euclidean geometry and 2D<->3D projection. Projecting sphere onto a plane isn't isometric -- doesn't conserve the distances (and areas) translated between geometries. Stating it and doing some arrow maths should be enough to prove the transform itself is non-Euclidean.