Building a monostable tetrahedron

In this paper we describe what we believe to be the first working model of a monostable tetrahedron, which, if supported on a horizontal plane, would only rest in a stable position on one of its faces. If positioned on any other face, it would tip over its edges to arrive at the stable face. The sequence in which faces are visited is called the tipping sequence of the monostable tetrahedron. We prove that for any convex body K and any desired center of mass O in the interior of K , the ratio of the maximum and minimum density associated with any density distribution producing O as the center of mass of K will achieve its minimum if K is binary , i.e. it is composed by two homogeneous parts, separated by a plane passing through O . If O is not the center of mass of the homogeneous body then this plane is unique. We define 6 combinatorial classes of binary monostable tetrahedra based on their geometry and on their respective tipping sequences. Combining this theory with results on monostable tetrahedra from an earlier paper and with a numerical search algorithm we identify local minima for the ratio of maximal and minimal density in 5 out of the 6 combinatorial classes and we design a binary tetrahedron belonging to the lowest minimum out of these 5. Here, the ratio of densities of the heavy and light parts is approximately 907, and we believe the structure to be close to the global optimum in a degenerate combinatorial class, belonging simultaneously to two combinatorial classes. We also demonstrate a working physical model of this tetrahedron which we dubbed Bille .