*[Corrected, thanks – T.]*

All this can be perhaps best understood in a modern context via the Clifford algebra which I write as P(Cl(2,0,1))*: that is, the Clifford algebra with degenerate metric (2,0,1), projectivized (so it acts on projective space P2 and not vector space R3). The ‘*’ means that 1-vectors represent lines, not points (dual construction). The even subalgebra of this Clifford algebra (when normalized called the Spin group) is then isomorphic to the planar quaternions described above, just like the even subalgebra of the nondegenerate (“elliptic”) Clifford algebra P(Cl(3,0))* gives the ordinary quaternions. The indirect isometries are also present (in both cases, naturally) via the so-called Pin group, generated by “sandwiches” gxg~ with a 1-vector g (lines).

More about P(Cl(2,0,1))* and similar results for 3D can be found in the above-quoted paper:

Charles Gunn

On the Homogeneous Model Of Euclidean Geometry

AGACSE (2011)

https://arxiv.org/abs/1101.4542

Good question! I don’t see how the grid example in R^2 extends easily to the S^2 case, so in principle one might be able to do better. On the other hand, I cannot think of a technique which would give better results in the S^2 setting than the R^2 case; typically in mathematics, the flat case is easier than the curved case, so the results tend to be stronger in the former. If the sharp lower bound for R^2 is ever established rigorously, this may shed more light on what is going on at S^2.

]]>In your Remark 3, is it obvious that a log factor is required for the distances problem on the sphere? For example, must the lattice grid example for R^2 carry over to S^2? Or might it be possible to remove the log and obtain N distances?

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