On equilibria of tetrahedra

摘要: The curious mechanical properties of tetrahedral solids were first studied in 1967 when Aladár Heppes constructed a homogeneous tetrahedron [8] that could rest stably on only two of its faces. John Horton Conway then showed [3] that for a homogeneous tetrahedron, this number cannot be improved; ie, it cannot be monostable.A three-dimensional weighted convex polyhedron with center of mass O supported by a fixed horizontal plane has three sorts of equilibria: stable (on a face), unstable (on a vertex), and saddle (on an edge). These correspond to local minima, maxima, and saddle points of the radial height function describing the boundary of as a distance measured from O. The global study of such equilibrium points, today associated with Morse theory, goes back (on smooth surfaces interpreted in a Cartesian coordinate system) to Arthur Cayley [1] in 1859. James Clerk Maxwell [9] noted a few years …