Timeline for Definition of area
Current License: CC BY-SA 3.0
29 events
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| Nov 15, 2015 at 3:13 | comment | added | ARi | Is the concept of a dimension necessary to define that of the area : If yes then does such a definition automatically follow from it? | |
| Nov 14, 2015 at 20:50 | review | Close votes | |||
| Nov 14, 2015 at 22:56 | |||||
| Nov 14, 2015 at 19:34 | history | edited | Sam Nead | CC BY-SA 3.0 |
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| Mar 20, 2015 at 17:39 | history | edited | Ricardo Andrade |
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| Jan 30, 2013 at 6:34 | comment | added | Daniel Litt | My feeling is that if this is aimed at undergraduate students, you might have to bite the bullet and work with a grid...it's boring but easy. | |
| Jan 30, 2013 at 5:15 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Jan 27, 2013 at 23:02 | comment | added | Anton Petrunin | @roy, "right" only means "no cheating", it does not mean "no mistake" and it does not mean I like it. | |
| Jan 27, 2013 at 20:26 | comment | added | roy smith | Since you say Millman and Parker do it "right", I assume you are interested in the approach they take. As mentioned elsewhere here, that is due to Hilbert, and is clearly explained in Hartshorne's Geometry: Euclid and Beyond, sections 20,22,23. As Millman and Parker make clear, the theories of area and similarity are essentially equivalent, but doing similarity first avoids assuming Euclid's appropriate common notion for area. Finally it seemed to me that M-P, make an error in the proof of their key Thm.10.2.5, i.e. that the sentence on p.262 beginning "The crucial point.." is false. | |
| Jan 27, 2013 at 17:42 | comment | added | Anton Petrunin | @Misha, I want to check, maybe I miss something simple. If that is the case I will think about suitable audience. So far I got no answers which were not described in the question. Box counting def looks best so far. | |
| Jan 27, 2013 at 17:31 | comment | added | Misha | Anton: It would help if you were to describe your target audience. For instance, Liviu's or Andy's answers would be suitable for participants of the MASS program at Penn State; for Russian undergrads or US grad students, Lebesgue integral might be most appropriate; but I would not recommend any of these for the Penn State undergrads, unless you are willing to hire an armed guard for a year or so. | |
| Jan 27, 2013 at 17:29 | comment | added | Deane Yang | Anton, am I right in believing that the key difficulty is proving that area is invariant under, say, rigid motions or, more generally, affine motions with determinant 1? The arguments involving decomposition into triangles are designed to handle this more eaisly but the ones involving square grids would appear to require more work to prove this invariance. | |
| Jan 27, 2013 at 16:02 | comment | added | user21349 | What's wrong with Euclid's approach? Area is a primitive, so he doesn't define it. It has the properties given in common notions 1-5. If area were non-unique, it would contradict common notion 1, and the system would be inconsistent. Existence isn't an issue, because he isn't trying to construct a map onto the reals. Non-measurable sets don't arise, because the approach is constructive, and the methods of construction are incapable of creating non-measurable sets. Calculus and measure theory are topics for courses in calculus and measure theory. | |
| Jan 27, 2013 at 13:58 | comment | added | Gerald Edgar | Is there some reason you do not want to do this with Lebesgue measure? | |
| Jan 27, 2013 at 11:26 | answer | added | Liviu Nicolaescu | timeline score: 8 | |
| Jan 27, 2013 at 11:20 | answer | added | Loïc Teyssier | timeline score: 7 | |
| Jan 27, 2013 at 6:49 | comment | added | alvarezpaiva | Maybe it's just the reference: Chapter IV of Hilbert's "Foundations of Geometry" treats precisely on this topic in about 10 pages. If you add Supplement III that gives another 7 pages. Boltyankii's little book "Equivalent and Equidecomposable figures" is also nice. I don't think any math student would be bored with it. Of course this equidecomposability approach does not work in dimensions higher than 2. That was the subject of Hilbert's third problem. | |
| Jan 27, 2013 at 0:59 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Jan 27, 2013 at 0:24 | comment | added | Anton Petrunin | @Deane, I modified the question again, so it should be more clear what I want. | |
| Jan 27, 2013 at 0:23 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Jan 27, 2013 at 0:02 | comment | added | Deane Yang | Anton, thanks. Do I understand correctly that you don't like the decomposition proofs described in the answers so far, because it's a lot of work to show that the answer is independent of the decomposition? So what you really want is a more direct and geometric way to see that area is a well-defined invariant for some family of sets? | |
| Jan 26, 2013 at 20:42 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Jan 26, 2013 at 20:19 | comment | added | Deane Yang | Could you say more about what you mean by "geometrically attractive"? | |
| Jan 26, 2013 at 20:10 | answer | added | Andy Putman | timeline score: 11 | |
| Jan 26, 2013 at 18:42 | answer | added | Mariano Suárez-Álvarez | timeline score: 13 | |
| Jan 26, 2013 at 18:40 | answer | added | Daniel Litt | timeline score: 20 | |
| Jan 26, 2013 at 18:27 | history | edited | Anton Petrunin | CC BY-SA 3.0 |
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| Jan 26, 2013 at 18:23 | comment | added | alvarezpaiva | Can you be a bit more specific on the type of spaces and structures you are considering ? | |
| Jan 26, 2013 at 18:18 | comment | added | Andy Putman | How general a notion of area are you looking for? Eg do you want something very general like Lebesgue measure, or do you just want to be able to find the areas of simple figures in the plane (and if so, how simple? For instance, would subspaces with polygonal boundary be enough?) | |
| Jan 26, 2013 at 18:14 | history | asked | Anton Petrunin | CC BY-SA 3.0 |