[HN Gopher] Mathematicians cross the line to get to the point ___________________________________________________________________ Mathematicians cross the line to get to the point Author : nsoonhui Score : 30 points Date : 2023-09-26 05:17 UTC (17 hours ago) (HTM) web link (www.quantamagazine.org) (TXT) w3m dump (www.quantamagazine.org) | 1letterunixname wrote: | This maybe a tangent, but damn those union scabs set on angling | to make a point on the surface by blocking intersections. It | parallels the obtuse congruence of last year. Where will it end? | scubbo wrote: | Hopefully, the limit does not exist. | subroutine wrote: | > Keep repeating this process, and from a certain perspective, | you'll have nothing left: The resulting set will cover so little | of the original line segment that its length will be zero. But it | is, in both an intuitive and a mathematical sense, "bigger" than | just a single point. Its Hausdorff dimension is about 0.6. | | I don't follow. Can someone clarify what "it" is with a dimension | of 0.6? Any point? Or all the points remaining after you remove | specifically 1/3 from each remaining line segment a number (an | infinite?) of times? Would it be different if we removed 1/4 | repeatedly? Would it be different if the line was longer than 0 | -> 1? | Sniffnoy wrote: | The _set_ is what has the dimension. And the set under | discussion (the Cantor set) is the set remaining after | repeating this process infinitely many times. That is to say, | every time you repeat the process, the set gets smaller; so, if | you take the intersection of all the finite stages, then you | get what 's left after repeating the process infinitely many | times. That remaining set is the Cantor set being discussed. | | > Would it be different if we removed 1/4 repeatedly? | | That would result in what's known as a "fat Cantor set". If I'm | not mistaken, it would have Hausdorff dimension 1, rather than | than something intermediate like the usual Cantor set. | | > Would it be different if the line was longer than 0 -> 1? | | No, the length of the starting line segment is not material | here. ___________________________________________________________________ (page generated 2023-09-26 23:00 UTC)