[HN Gopher] Mathematicians cross the line to get to the point
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       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.
        
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