MEANS:Animals: Turtles

FROM: J. Nonlinear Sci. Vol. 16: pp. 255–281 (2006)

DOI: 10.1007/s00332-005-0691-8:

Summary.By appealing to the Poincar´e-Hopf Theorem on topological invariants, we introduce a global classification scheme for homogeneous, convex bodies based on the number and type of their equilibria.We show that beyond trivially empty classes all other classes are non-empty in the case of three-dimensional bodies; in particular we prove the existence of a body with just one stable and one unstable equilibrium. In the case of two-dimensional bodies the situation is radically different: the class with one stable and one unstable equilibrium is empty (Domokos, Papadopoulos, Ruina,J. Elasticity36 [1994], 59–66). We also show that the latter result is equivalent to the classical Four-Vertex Theorem in differential geometry.We illustrate the introduced equivalence classes by various types of dice and statistical experimental results concerning pebbles on the seacoast.

This shape has also been shown to be remarkably similar to specific species of tortoises. See: “Mono-monostatic Bodies *do *Exis” in: [4] P.L. Várkonyi, G. Domokos: Mono-monostatic bodies: the answer to Arnold's question Mathematical Intelligencer 28 (4) pp34-38 (2006)