[HN Gopher] Mathematicians marvel at 'crazy' cuts through four d...
       ___________________________________________________________________
        
       Mathematicians marvel at 'crazy' cuts through four dimensions
        
       Author : nsoonhui
       Score  : 69 points
       Date   : 2024-04-23 13:21 UTC (9 hours ago)
        
 (HTM) web link (www.quantamagazine.org)
 (TXT) w3m dump (www.quantamagazine.org)
        
       | Sniffnoy wrote:
       | I wish this article talked more about how this plays out in the
       | 3-dimensional analogue (knot theory) for comparison. I have very
       | little picture of how I would expect this to work, so it would be
       | helpful to start with a case I can picture, and then say how this
       | is similar and how it's different.
        
         | VyseofArcadia wrote:
         | I wouldn't call knot theory the three dimensional analog.
         | 
         | They spend a lot of time talking about analogs you can
         | visualize, like the sphere and the torus.
        
       | Jun8 wrote:
       | 4D is really interesting and not because it's higher than 3D.
       | There are many mathematical aspects that are unique to it! Check
       | out the second answer to the MATH SE question to see one:
       | https://math.stackexchange.com/questions/3344266/are-there-m...
       | 
       | Also the number of regular polytopes is highest (6) in 4D
       | (https://oeis.org/A060296).
        
       | teleforce wrote:
       | Kudos to the Quanta Magazine writer and staff, such a well
       | written article for layman understanding.
       | 
       | This make me wonder on the connection between 2, 3 and 4
       | dimensions, and Hamilton found out the hard that you need 4
       | numbering system or quaternion, in order to properly represent 3
       | dimensions [1].
       | 
       | This article hinting a direct connection between two and four
       | dimensions, and the interplays between the two but not 3
       | dimensions and it seems to me that the 3 dimensions exist only as
       | a curious transition.
       | 
       | Recently someone come up with the equivalent of complex number
       | analytic signal (an indispensable tool in modern engineering) in
       | the quartenion space and called it quaternion embedding, or
       | probably the better name should be quaternion analytic signal
       | [2].
       | 
       | It is great to see the synergy between Topology and Group Theory
       | as the article mentioned in solving some of the the former's list
       | of problems and their new found solutions. It looks like the
       | topology group is rapidly cleaning their house as the article
       | aptly put it, and at this rate (after 30 years of winter hiatus)
       | we will probably see the results spilled over to the applied math
       | fields for examples physics and engineering applications.
       | 
       | [1] Quaternion:
       | 
       | https://en.wikipedia.org/wiki/Quaternion
       | 
       | [2] Polarization spectrogram of bivariate signals:
       | 
       | https://ieeexplore.ieee.org/abstract/document/7952905
        
         | empath-nirvana wrote:
         | > This make me wonder on the connection between 2, 3 and 4
         | dimensions, and Hamilton found out the hard that you need 4
         | numbering system or quaternion, in order to properly represent
         | 3 dimensions [1].
         | 
         | Sort of. Quarternions have some nice properties as
         | representations of _rotations_ in 3d space. 3d space works just
         | fine with 3 dimensions.
        
       | codeulike wrote:
       | I'm a mathematician, and thats crazy. Marvellous.
        
       ___________________________________________________________________
       (page generated 2024-04-23 23:00 UTC)