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The orbit space of an octupole, a traceless symmetric third-rank tensor of$SO(3)$, is shown to be a three-dimensional body with three cusps and twocuspidal edges. It is demonstrated that for a unique choice of orbit spacecoordinates its boundary turns into a tangent developable surface of degree sixwhich we identify with one in the classification by Chasles and Cayley. Theclose relation of the octupole's orbit space to the moduli space of a binarysextic form is described and relevant work by Clebsch and Bolza from thenineteenth century is recalled. Upon complexification, the octupole's orbitspace yields the geometry of the moduli space of hyperelliptic curves of genustwo. Its boundary is found to be a non-orientable tangent developable surface.As preamble, the orbit space of a set of three vectors is shown to be boundedby Cayley's nodal cubic surface and the boundary of the orbit space for avector plus a quadrupole is found to correspond to Cremona's first class in hisclassification of quartic ruled surfaces.

Tangent Developable Orbit Space of an Octupole