The polyhedron that features so prominently in Melencolia I occupies a privileged place in the histories of both art and geometry. Art historians and mathematicians have puzzled over the precise shape, debating whether it is a truncated triangular trapezohedron or rhombohedron, and possibly a precursor to the study of quasicrystals. In these histories the polyhedron is elusive, withdrawn, and indifferent—it exudes melancholy. This chapter considers the evolution of the polyhedron as a melancholic object, and later appropriations by artists such as Alberto Giacometti, Anselm Kiefer, and, most recently, Fabien Mérelle.