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However, like the lower dimensional case, we can also apply a 4-dimensional rotation to the 3-sphere before projecting, so that this circle and straight line become a symmetrical pair of linked circles. My earlier posts show the resulting triply orthogonal system.

https://mathstodon.xyz/@Danpiker/113625825121929332

https://mathstodon.xyz/@Danpiker/113636827127103066

To get an equivalent in 3d to the projected loxodrome *curve* in 2d, we now want to take a diagonal *surface* in this coordinate system.
In 3 dimensions this could mean either a diagonal of 2 directions or a diagonal of all 3 directions.

Taking the diagonal of only 2 directions gives the surface shown here:

https://mathstodon.xyz/@Danpiker/111727832885683368

https://skfb.ly/oPFYM

Notice how this wraps around one circle like a smoke ring, while at the other circle it becomes a tapering tube nesting into itself like an Ouroboros.