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Although previous research has found several facts concerning chord lengthsof regular polytopes, none of these investigations has considered whether anyof these facts define relationships that might generalize to the chord lengthsof all regular polytopes. Consequently, this paper explores whether fourfindings of previous studies-- viz., the four facts relating to the sums andproducts of squared chord lengths of regular polygons inscribed in unitcircles-- can be generalized to all regular ($n$-dimensional) polytopes(inscribed in unit $n$-spheres). We show that (a) one of these four factsactually does generalize to all regular polytopes (of dimension $n \geq 2$),(b) one generalizes to all regular polytopes except most simplices, (c) onegeneralizes only to the family of crosspolytopes, and (d) one generalizes onlyto the crosspolytopes and 24-cell. We also discover several corollaries (due toreciprocation) and some theorems specific to the three-dimensional regularpolytopes along the way.

Sums and Products of Regular Polytopes' Squared Chord Lengths