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Isn't it a lot easier? How can we have day and night at the same time in different places if Earth was flat?
fresh_42 said:Isn't it a lot easier? How can we have day and night at the same time in different places if Earth was flat?
It is interesting how we reach this "breakaway" value of 4 , the least number of points needed to determine whether there is non-trivial curvature. Is there a name for this least value needed?mgkii said:Hi Whitehole
I don't think anyone actually got around to giving you the answer to your specific question. What he means is if you only use 3 cities and make triangles, then you'll never see a problem; you can take a hundred cities and draw a hundred different "planar triangles" (i.e. triangles on a flat piece of paper) and the Earth's curvature will not "show up"; the curvature will simply "get lost" as each triangle will have a small error in each of its angles that makes up for the "lost" curvature as you translate the triangles from the curved Earth onto the flat planar surface of your paper.
However, if you then go back and add a forth city onto each of your planar triangles - extending each triangle into a planar quadrangle, you'll see the curvature problem come into play - if your original triangle was Paris, Berlin, Barcelona (as in the example you used), then you'll not be able to find a single point for the fourth corner that is the correct distance away from all these three cities to Rome.
But how do we then switch from night to day in the dark region?Hornbein said:Big squares blocking the Sun? Gravity bending the Sun's rays so they hit they Earth before they get to the nighttime region?
WWGD said:But how do we then switch from night to day in the dark region?
WWGD said:It is interesting how we reach this "breakaway" value of 4 , the least number of points needed to determine whether there is non-trivial curvature. Is there a name for this least value needed?
I don't know if there is a name, but the theory can be seen as follows:WWGD said:It is interesting how we reach this "breakaway" value of 4 , the least number of points needed to determine whether there is non-trivial curvature. Is there a name for this least value needed?
This is correct. It has already been pointed out earlier in the thread that you need fewer points if include angle measurements.mgkii said:Just throwing a thought on this; I could well be wrong, but I think you might need to think of this from a different starting point. Moving from a 3 dimensional Earth to a 2 dimensional plane by ignoring angles of the triangle is a form of dimensional reduction. You may be able to maintained angles by choosing a different form of reduction, and find a different type of breakaway - i.e. the value of 4 may simply be a feature of the chosen reduction.
Nice. I think you've hit on the weakness in all the comments I've read so far. Namely, what do we mean by 'distance'? How would one know the 'distance'? By definition, the metric applicable to the Earth's surface is the contour length the great circle connecting any two cities. But this is a non-Euclidean geometry, a spherical geometry. (For one thing, the angles of a spherical triangle do not add up to 180 deg, which may play a role in our problem.) Therefore, the inconsistency between the distances defined by the spherical metric and the Euclidean distances is the result of the naive isomorphism between the 2 geometries. Perhaps there is a projection of the sphere onto the plane which preserves the distances between points on the two surfaces and the angles of triangles. I wouldn't know. But I do know what any 6th grader knows - that the Mercator projection is not the right one - all that business about Greenland being bigger than South America and so forth.Dr. Courtney said:To me, this is the most convincing proof not only that the Earth is not flat, but that it rotates as explained in the rotating spheroid model.
For me, it is more convincing, because it can be done in a single room and does not depend on outside sources of data that may be less reliable.
If that's not a slam dunk, the repeating the experiment at several latitudes should be.
Maybe one can somehow generalize the fact that any projection will produce distortions of some sort to make a general argument on the inequality/inequivalence between Euclidean and Spherical.Mark Harder said:Nice. I think you've hit on the weakness in all the comments I've read so far. Namely, what do we mean by 'distance'? How would one know the 'distance'? By definition, the metric applicable to the Earth's surface is the contour length the great circle connecting any two cities. But this is a non-Euclidean geometry, a spherical geometry. (For one thing, the angles of a spherical triangle do not add up to 180 deg, which may play a role in our problem.) Therefore, the inconsistency between the distances defined by the spherical metric and the Euclidean distances is the result of the naive isomorphism between the 2 geometries. Perhaps there is a projection of the sphere onto the plane which preserves the distances between points on the two surfaces and the angles of triangles. I wouldn't know. But I do know what any 6th grader knows - that the Mercator projection is not the right one - all that business about Greenland being bigger than South America and so forth.
There is a physical notion of distance which implements the mathematical notion, and abstracts from any knowledge of the geometry (metric). Then it is valid to ask whether how a set of such measurements can give you information about the geometry.Mark Harder said:Nice. I think you've hit on the weakness in all the comments I've read so far. Namely, what do we mean by 'distance'? How would one know the 'distance'? By definition, the metric applicable to the Earth's surface is the contour length the great circle connecting any two cities. But this is a non-Euclidean geometry, a spherical geometry. (For one thing, the angles of a spherical triangle do not add up to 180 deg, which may play a role in our problem.) Therefore, the inconsistency between the distances defined by the spherical metric and the Euclidean distances is the result of the naive isomorphism between the 2 geometries. Perhaps there is a projection of the sphere onto the plane which preserves the distances between points on the two surfaces and the angles of triangles. I wouldn't know. But I do know what any 6th grader knows - that the Mercator projection is not the right one - all that business about Greenland being bigger than South America and so forth.
spamanon said:If the Earth is flat, then why has nobody ever taken a picture of the edge?
I don't get it. Are there really people out there who honestly believe the Earth is flat?
Hi mgkii, thanks for your explanation. You've got a point in your first statement, haha. Their explanation kinda pointed out what you are saying and I know what they mean but I wanted to understand what Zee is saying. I know this topic can be explained in a lot of ways but your explanation exactly answered my question. Thanks for clearing it up!mgkii said:Hi Whitehole
I don't think anyone actually got around to giving you the answer to your specific question. What he means is if you only use 3 cities and make triangles, then you'll never see a problem; you can take a hundred cities and draw a hundred different "planar triangles" (i.e. triangles on a flat piece of paper) and the Earth's curvature will not "show up"; the curvature will simply "get lost" as each triangle will have a small error in each of its angles that makes up for the "lost" curvature as you translate the triangles from the curved Earth onto the flat planar surface of your paper.
However, if you then go back and add a forth city onto each of your planar triangles - extending each triangle into a planar quadrangle, you'll see the curvature problem come into play - if your original triangle was Paris, Berlin, Barcelona (as in the example you used), then you'll not be able to find a single point for the fourth corner that is the correct distance away from all these three cities to Rome.
No, following a straightest possible path locally, where difference from flatness cannot be discerned, is completely well defined and can, in principle, be carried out over any distance. All definitions of intrinsic curvature assume you can perform a variety of operations based on local asymptotic flatness without any apriori knowledge of curvature. For example, in principle, if radar distance between 5 objects in some some non-coplanar configuration were done on earth, and all the pairwise distnances were put in the order 4 Cayley-Menger determinant, the result would not vanish, indicating curvature of our universe. Of course, the precision required for this is unattainable in practice for a laboratory sized device (about one part in 10^30).Buzz Bloom said:I would like to suggest a thought that might well be considered to be only a nit.
The use the distances between six pairs of cities (using four cities) to PROVE that the Earth is not flat ignores that the practical measurement of these six distances seems to require prior knowledge (e.g., GPS technology) that the Earth is not flat.
Regards,
Buzz
Hi Paul:PAllen said:No, following a straightest possible path locally, where difference from flatness cannot be discerned, is completely well defined and can, in principle, be carried out over any distance.
Practical measurements, probably not. I would say the biggest practical difficulty on Earth (rather than a mathematical surface) is lack of local flatness. Even building roads, you always find the need to go around and up and down because of obstacles. But if you could build roads that are everywhere locally flat and straight, then measure them with an odometer, you could get enough precision. The thing is, we don't really live in a 2-manifold; the geometry problem is really answering questions about measurements within a 2-manifold.Buzz Bloom said:Hi Paul:
My admitted nit was intended to raise a question about practicality, rather than what might be doable in principle. If the four cities were too close together, then the distance measurements would perhaps be consistent with flatness. If the cities were far enough apart for the measurements to reveal the non-flatness, then would radar technology be a practical means of measurement?
I am curious whether any current technology not dependent on prior knowledge of non-flatness would be practical to make the six measurements to show non-flatness.
Regards,
Buzz
1oldman2 said:This thread seems to be taking on a life of its own, or am I imagining that?. Its interesting to see serious people discussing such an off the wall fringe notion. I sense a little entertainment value here.![]()
ehm, ehm, on a word ...Hornbein said:Well, it would be pretty sad if we couldn't prove the Earth was flat,
Oops. I'd also like to know who that South African flat Earth prime minister was. That was pre-Internet.fresh_42 said:ehm, ehm, on a word ...
I think the main point was a slam dunk right from the go, but I have enjoyed this thread immensely. Thus the bonus entertainment value.Hornbein said:Well, it would be pretty sad if we couldn't prove the Earth was flat, wouldn't it? My fave is applying general relativity to the flat Earth hypothesis.