Timeline for Triangle area on surfaces of constant curvature
Current License: CC BY-SA 3.0
14 events
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| May 16, 2013 at 23:37 | comment | added | Anton Petrunin | @Alexandre, Euclid (and Kiselev) did not prove the existence, essentially they add the existence as an axiom, but they did not say that it is an "axiom". This axiom follows from the rest of axioms, but it takes 20 pages at least. Instead of unit square you have to use other normalization (which essentially defines curvature). A rigourous way to introduce area given in "Elementary Geometry From An Advanced Standpoint" by Moise 35 pages Euclidean plane onlyy,and in "Geometry: A Metric Approach with Models" by Millman and Parker 40 pages neutral plane and contains a gap. | |
| May 16, 2013 at 20:41 | comment | added | Alexandre Eremenko | Anton, sorry I looked at 119953 and I don't understand your objection. In elementary geometry we deal with areas of polygons. The area is defined by a) finite additivity, b) monotonicity, invariance with respect to motion, c) the area of the unit square is 1. From this it is easy to derive that the area of a polygon exists and is unique. And I believe Euclid did it rigorously. Kiselev (who wrote the common Russian high school geometry text) did it rigorously, and I studied this in 8-th grade. What's wrong with all this? | |
| May 16, 2013 at 18:33 | comment | added | Anton Petrunin | @BS, check this question mathoverflow.net/questions/119953/definition-of-area I would be very happy if you know a better answer. | |
| May 16, 2013 at 9:46 | comment | added | BS. | What about the scissors congruence definition af area of simple polygons ? | |
| May 15, 2013 at 4:13 | comment | added | Anton Petrunin | @Alexandre, I do not see what exactly you disagree with. A rigorous intro to area from the axioms takes 20-40 pages, and once it is done the formula is already proved. So these sort of "proofs" confuse poorly educated students and they prove nothing to those who know what area is. | |
| May 14, 2013 at 20:47 | comment | added | Alexandre Eremenko | Anton: I disagree with what you say. The area of a TRIANGLE is an elementary notion. (The theory of areas of triangles in Euclid is completely rigorous, by all modern standards.) And the formula has a really elementary proof. | |
| May 13, 2013 at 17:45 | comment | added | Anton Petrunin | P.S. surface of constant curvature κ are spheres plane or Lobachevsky plane. All these things are "elementary" for me. | |
| May 13, 2013 at 17:02 | comment | added | Anton Petrunin | @Kofi, you ask for an elementary derivation. For me "Riemannian metric" and "integral" are not elementary and the geometry as it was used to be covered in the school (but not any more) is elementary. | |
| May 13, 2013 at 7:12 | comment | added | Matthias Ludewig | I don't understand what you mean. I have a Riemannian metric of constant curvature on a Surface. This gives me curvature and a volume density, so I know both what curvature and area is and I can in theory calculate what the area of a geodesic triangle is. Where do I need an axiom? | |
| May 13, 2013 at 2:27 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| May 13, 2013 at 2:17 | comment | added | Anton Petrunin | @Sergei, yes sure, all I wanted to say is that if one knows what is area and curvature then there is nothing to prove. | |
| May 12, 2013 at 22:38 | comment | added | Sergei Ivanov | You would need some normalization axiom, in order to distinguish between proportional measures. Do you have a specific one in mind? | |
| May 12, 2013 at 22:20 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| May 12, 2013 at 18:16 | history | answered | Anton Petrunin | CC BY-SA 3.0 |