When Is a Curve an Octahedron?

@article{Langer2010WhenIA,
  title={When Is a Curve an Octahedron?},
  author={Joel Langer and David A. Singer},
  journal={The American Mathematical Monthly},
  year={2010},
  volume={117},
  pages={889-902}
}
We consider complex curves of genus zero and answer the above riddle. Namely, the lemniscate of Bernoulli, which has obvious four-fold symmetry, actually has the octahedral group as its symmetry group, and may in fact be characterized by this symmetry.