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).
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.
I’m sure I’m being naive, but I was expecting at least one “crazy” and interesting graphic in this article.
I'm a mathematician, and thats crazy. Marvellous.
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