Abel's Lemniscate Division Theorem

Deliberately paired with Gauss's heptadecagon result as the atlas's own second constructibility mark: in 1827, Niels Henrik Abel extended Gauss's constructibility criterion for regular polygons to a completely different curve, the lemniscate of Bernoulli (a figure-eight shape), showing that it too can be divided into n equal-arc-length pieces by compass and straightedge exactly when n satisfies an arithmetic condition directly analogous to Gauss's Fermat-prime criterion. The result is a striking early instance of a pattern, an arithmetic condition governing a geometric constructibility question, recurring in a genuinely different setting, and it grew out of Abel's wider work founding the theory of elliptic functions, of which the lemniscate's arc length is an early example.