When Is a Curve an Octahedron?

When Is a Curve an Octahedron?

We consider complex curves of genus 0 and answer the above riddle. Namely, the lemniscate of Bernoulli, which has obvious fourfold symmetry, actually has the octahedral group as its symmetry group, and may in fact be characterized by this symmetry.

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Veröffentlicht in:The American mathematical monthly 2010-12, Vol.117 (10), p.889-902
Hauptverfasser: Langer, Joel C., Singer, David A.
Format: Artikel
Sprache:eng
Schlagworte:
Algebra
Coordinate systems
Curvature
Curves
Geometric planes
Linear transformations
Mathematical cusps
Octahedrons
Symmetry
Tangents
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Beschreibung
Zusammenfassung:We consider complex curves of genus 0 and answer the above riddle. Namely, the lemniscate of Bernoulli, which has obvious fourfold symmetry, actually has the octahedral group as its symmetry group, and may in fact be characterized by this symmetry.
ISSN:0002-9890
1930-0972
DOI:10.4169/000298910X523371