Timeline for Center of mass from the abstract point of view, or could the ancient Greeks invent modern analysis?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Nov 19, 2015 at 21:02 | comment | added | Alexandre Eremenko | Center of mass is not really a characteristic of a set, it is a characteristic of a measure. (This is implicitly contained in the answer of Deane Yang). | |
| Dec 23, 2012 at 20:35 | comment | added | Robert Israel | @Deane Yang: Yes, but there's a difference between a limiting process and a "completed infinity". | |
| Dec 22, 2012 at 22:02 | comment | added | Deane Yang | Although countable sets might be "out of the ballpark" for the ancient Greeks, didn't they understand the concept of approximating curved shapes by polygons? My impression is that they understood how to compute the area of a circle and the volume of a ball by approximation. | |
| Dec 22, 2012 at 19:05 | history | edited | fedja | CC BY-SA 3.0 |
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| Dec 22, 2012 at 18:41 | comment | added | Tom Leinster | No one seems to have mentioned this yet, so just in case anyone's not aware: there's a whole lot of classical work on characterizing means in the one-dimensional case. For instance, you can find a lot on this in Hardy, Littlewood and Pólya's book Inequalities, in among stuff on power means. | |
| Dec 22, 2012 at 2:33 | answer | added | Deane Yang | timeline score: 11 | |
| Dec 21, 2012 at 22:53 | history | edited | José Hdz. Stgo. | CC BY-SA 3.0 |
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| Dec 21, 2012 at 14:42 | comment | added | Suvrit | I think already allowing for an element that is not in $E$ to be "the center of mass" of $E$ is a fairly advanced concept...but it seems essential to have if we follow your definition of center, which feels closer to "centroid" than to "median"....... | |
| Dec 21, 2012 at 11:16 | answer | added | Liviu Nicolaescu | timeline score: 8 | |
| Dec 21, 2012 at 11:01 | comment | added | Liviu Nicolaescu | If we, as physicists do, would allow points to have (positive) masses, then things would be much easier, and it would justify the terminology center of mass. One can think of a point of mass $20$ situated at coordinate $x_0$ on as a cluster of $20$ points clouding the location $x_0$. | |
| Dec 21, 2012 at 10:10 | comment | added | Robert Israel | Countable sets are way out of the ballpark for the ancient Greeks: most (well, at least the followers of Aristotle) didn't believe in a "completed infinity." | |
| Dec 21, 2012 at 6:12 | comment | added | Will Sawin | Without an approximation argument, it seems hard to believe that you can get the center of mass of the region bounded by an arbitrary smooth curve. Say, take one whose coordinates are interesting transcendental numbers - how are you going to compute those using this method? | |
| Dec 21, 2012 at 6:08 | comment | added | Will Sawin | We can compute the center of mass of a finite set (with measure 1 on each element), as long as no three points are colinear, by induction. A finite set of $k$ elements can be divided into two smaller sets, which we can compute the center of mass of, giving us a line the center of mass of the whole set must be on. As long as we get two different lines, we can find the real center of mass. This only fails if the entire set is colinear, which by assumption implies it consists of two elements, so we can find the center of mass by symmetry and axiom 2. | |
| Dec 21, 2012 at 6:04 | comment | added | Will Sawin | Via axiom $3$, we can find the center of mass of a square as the unique point of symmetry - the center. Via axiom 4, this allows us to compute the center of mass of any object that's a square grid, as long as we know the definition of the center of mass of a finite set. If we had an approximation principle, we could approximate a nice round shape by square grids, but obviously we didn't. | |
| Dec 21, 2012 at 4:53 | comment | added | Marty | "I assume that the ancient Greeks had an idea of a complete normed space (${\mathbb R}$ and ${\mathbb R}^2$ would be enough for our purposes for quite a while), a set, a linear transformation, and the center of mass." Really? Really??! | |
| Dec 21, 2012 at 4:26 | comment | added | Alexandre Eremenko | Could you please explain these ponts: a) What is the center of mass of a SET? (I know what is the center of mass of a MEASURE in R or in R^2). What is the center of mass of a COUNTABLE set? What is the center of mass of the sequence 1/n in R? b) You say "normed space". What does the norm have to do with your definition? | |
| Dec 21, 2012 at 4:03 | history | asked | fedja | CC BY-SA 3.0 |