Apollo of the Belvedere with parabolic curve
Apollo of the Belvedere with parabolic curve. Gypsum.
Felix Klein wanted to explain the artistic beauty of a face by mathematical relations. To experiment with this idea, he drew the parabolic curves (where the Gaussian curvature vanishes) on a copy of the Apollo of the Belvedere; this was considered the pinnacle of classical beauty.
Unfortunately, these curves did not display any particular structure, and no general law describing their form could be found.
Showcase of this model is Case number 43
Hilbert, D.; Cohn-Vossen(1932). Anschauliche Geometrie, Springer-Verlag, Berlin, p. 174f. Online version