I think we're agreeing at this point and esp. with that quote. Nobody here is saying that the planet isn't roughly spherical without any reference to scale. And there are certainly tools and techniques to detect the real shape of the planet. That has been true for a very long time. But none of us use spherical trigonometry to calculate distances, for example. We behave as if the earth is flat. The question is why is there a round/flat Earth problem and specifically why does it seem flat to us & our day to day behaviour assumes it is flat. And of course this was the observation of ancient people too. And I'm delighted to see the Wolfram article actually using our example to explain manifolds and esp to mention the flat/round Earth problem. This is not just a heuristic and this explanation actually played a roll in the development of manifolds.durangopipe wrote: ↑Tue Nov 14, 2017 8:47 pmI'll let the comment stand. The earth is not flat, even locally. In this case the manifold is a heuristic, not a physical reality.
"Possibly what's confusing you is that the statement refers to the surface of the sphere, not the space the sphere is sitting in. If we ignore relativistic effects (and the fact that the Earth isn't quite a sphere), yes the sphere is in a Euclidean 3-dimensional space. In that space all the convenient Euclidean things apply. But the statement didn't refer to the sphere, or the space it's sitting in: it referred to the surface of the sphere. The surface of the sphere is a 2-dimensional space, not a 3-dimensional one, and points on it only need 2 coordinates (latitude and longitude, for example).
To help see the difference, consider a straight line between London and New York. In the 3-dimensional Euclidean space in which the Earth is embedded, that straight line goes through the Earth. But if we're only considering the surface of the Earth, that line doesn't exist. The straight line (shortest distance between the two points) on the surface lies along the great circle. Now consider drawing the lines from both New York and London to, say, Capetown, to make a triangle. Yes, if you draw the lines through the Earth you will get a nice Euclidean triangle with angles that add up to 180 degrees. But those lines don't exist in the space you are considering: you can only draw lines on the surface of the Earth. The angles of the triangle drawn on the surface of the Earth add up to more than 180 degrees, so the space must be non-Euclidean.
Edit: The bit after the "but" just seems to be saying that for most purposes you can treat a smallish bit of the Earth as if it were flat. You probably don't need to worry about the curvature of the Earth when looking at a street-map of your town."
I think the explanation is interesting and that is why I shared it (along with a bit of humour). And I hope that my original explanation has sufficient fidelity to the authoritative source so that it is useful.Wolfram wrote:To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly "flat" on small scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, as first codified by Poincaré.