Since space is curved, it's entirely legit to say the earth is flat. Well, except for mountains and trees.

'':-) Well, yes... Earth's surface is the points of the space where a perpendicular vector can be defined and the (local) curvature is 'zero' (ie. measure curvature and substract space-time curvature...) That can be a broad definition of a plane... :-) But Euclides would die if it happened to be around and read this ;-)''

But this does not follow at all. Since the surface of my apple here is curved,
the circle I draw on it is a straight line? Come on. A simple experiment will
convince all you infidels. Start at the north pole and walk due south,
until you reach the equator. Then, turn 90 degrees to the right and walk
around a quarter of the equator. Then, turn again 90 degrees right and walk
and walk... until you are again on the north pole. Now take a flat piece of
paper and try to draw the route you just walked...

''Oh, that's easy, just declare that the second direction you walked occurs on a plane not represented by the paper.  Then, according to the paper, you walked due south, turned 90 degrees, moved your legs for a bit, turned 90 degrees and walked back the way you come.  Not that hard, is it?''
-BethanyAndresBeck

A plane not represented by the paper? That's not possible if the earth is flat.
If the earth is flat, its surface is completely represented by a flat sheet
of paper. And vice versa. After all, they are both flat.

''Well, actually, a flat sheet of paper is not a 'flat' surface in Earths geometry... because it's not actually flat, in the potential way... Consider the center of the sheet of paper to be in such a way that the sheet's director vector is parallel to the radial vector from the center of gravitation. Then, if lenth of paper is L, and the length of position vector from the center of Earth to the paper is R, then the deviation at the paper borders from 'flatness' would be L^2/(4R) (taking L << R). Thus, the curvature of such a flat sheet of paper can indeed be measured (well, gedanken experiment). The deviation from the 'perfect' earth curvature of a flat object would be 0, as the definition of flatness comes from the curvature of stace-time at the place, with can be measured from Earth's surface.''

''Regarding the not-euclidean triangle apparent paradox, give a sheet of paper big enough to walk about 20000 km towards any direction from a certain point, and I'll show you it has to bee exactly like you do when walking on Earth's surface itself... Just think where are you going to put the paper on to be able to draw over it... if space-time is curved, whatever you put on it will follow the curvature... :-)''

Perhaps it would be simpler to consider it like this:  Consider (under your normal preconceptions about the geometry of the earth and pieces of paper), holding a piece of paper onto the planet, making it completely flat.  In this situation, the center of the paper forms a tangent to the Earth's curvature at that point, and the corners of the paper are elevated (relative to the Earth's surface).

Now, how do you tell the difference between that situation and one where the Earth is flat, and the piece of paper is curved up at the corners?

It's all about relative curvature.

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Everyone who attends to these "flat universe" Wiki pages needs to go to this web page: http://www.brunothebandit.com/d/20000224.html

''A pretty poor attempt at cryptotheism I think. Learn a little QuantumTheory - no way you can measure the position and velocity of just one atom at one time, much less all of them at all times.''