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The surface shown in my last few posts can be described as a diagonal on the hypersphere. In this 🧵 I'll expand a bit on what I mean by that.

On a sphere we can trace 'loxodrome' curves of constant bearing, diagonal to the longitude/latitude directions, spiralling from one pole of the sphere to the antipodal one.
https://en.wikipedia.org/wiki/Rhumb_line

Stereographically projecting from the top pole onto a horizontal plane gives logarithmic spirals, where one of the poles projects to the point at infinity. However, if instead we turn the sphere on its side before projecting we get a family of beautiful double spiral curves which reveal the true symmetrical nature of the 2 poles.