Louis Poinsot describes the two remaining Kepler–Poinsot polyhedra

Background. Kepler recognized the small and great stellated dodecahedra as regular around 1619. Poinsot rediscovered these figures in 1809 by assembling star pentagons around each vertex. The Kepler–Poinsot polyhedra are obtained by extending regular icosahedra and dodecahedra through stellation. These polyhedra exist in dual pairs and cover their circumscribed spheres more than once.

Publication. Poinsot's memoir on polygons and polyhedra was published in the Journal de l'École Polytechnique, volume 9, pages 16–48, in 1810; the passages give the title and journal citation but not the memoir's content. The following year, Arthur Cayley gave the Kepler–Poinsot polyhedra the names by which they are generally known today. By 1809 Poinsot had an excellent reputation from published works on geometry, mechanics and statics.

Mathematical Significance. In 1810 Cauchy proved, using Poinsot's definition of regular, that the enumeration of regular star polyhedra is complete. Arthur Cayley gave a modified form of Euler's formula, using density of the vertex figures and faces, holding for both convex polyhedra and the Kepler–Poinsot polyhedra; the passages date neither this contribution nor Cayley's naming precisely. Poinsot became assistant professor of analysis and mechanics at the École Polytechnique on 1 November 1809.