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   Danpiker (danpiker@mathstodon.xyz)'s status on Thursday, 13-Mar-2025 03:40:26 JST Danpiker

   in reply to

   What is the equivalent of these curves in one dimension up?

   Instead of the familiar sphere in 3d space a.k.a. the 2-sphere (which can be parameterised with 2 coordinates, longitude and latitude) we now have the 3-sphere or hypersphere in 4d space.

   One particularly nice way of parameterising the 3-sphere is using Hopf coordinates (https://en.wikipedia.org/wiki/3-sphere#Hopf_coordinates).

   Stereographic projection which maps the 2-sphere to the flat plane can also be generalised to higher dimensions, and it conformally maps the 3-sphere to our familiar Euclidean 3d space.

   This projects Hopf coordinates to a triply orthogonal system composed of spheres, planes and nested tori. At one extreme the torus becomes a horizontal circle and at the other a vertical line (or circle through infinity) through its centre (see toroidal coordinates here https://mathcurve.com/surfaces.gb/tripleorthogonal/tripleorthog.shtml).