Can the Dice be Fair by Dynamics?

A throw of a fair die is commonly considered as a paradigm for chance. The die is usually a cube of a homogeneous material. The symmetry suggests that such a die has the same chance of landing on each of its six faces after a vigorous roll so it is considered to be fair. Generally, a die with a shape of convex polyhedron is fair by symmetry if and only if it is symmetric with respect to all its faces [Diaconis & Keller, 1989]. The polyhedra with this property are called the isohedra. Every isohedron has an even number of faces [Grunbaum, 1960]. The commonly known examples of isohedra are: tetrahedron, octahedron, dodecahedron and icosahedron which are also used as the shapes for dice. Diaconis and Keller [1989] showed that there are not symmetric polyhedra which are fair by continuity. As an example, they considered the dual of n-prism which is a di-pyramid with 2n identical triangular faces from which two tips have been cut with two planes parallel to the base and equidistant from it. If the cuts are close to the tips, the solid has a very small probability of landing on one of two tiny new faces. However, if the cuts are near the base, the probability of landing on them is high. Therefore by continuity, there will be cuts for which new and old faces have equal probability. In [Diaconis & Keller, 1989] it is suggested that the locations of these cuts depend upon the mechanical properties of the die and the table and can be found either experimentally or by the analysis based on the classical mechanics. However, these definitions are not considering the dynamics of the die motion during the throw. This dynamics is described by perfectly deterministic laws of classical mechanics which map initial conditions (position, configuration, momentum and angular momentum) at the beginning of the motion into one of the final configurations defined by the number on the face on which the die lands. From the point of view of the dynamical systems the outcome from the die throw is deterministic, but as the initial condition–final configuration mapping is strongly nonlinear, one can expect deterministic unpredictability due to the sensitive dependence on the initial conditions and fractal boundaries between the basins of different final configurations. The implementation of this idea has been carried out in a few papers which deal mainly with the coin tossing problem [Ford, 1983; Zeng-Yung, 1985; Keller, 1986; Vulovic & Prange, 1986; Kechen, 1990; Balzass et al., 1995; Murray & Teare, 1993; Diaconis et al., 2007]. In these works the consideration of the various simplified models of the coin motion (for a detailed discussion see [Strzalko et al., 2008]) lead to the conclusion that the uncertainties of the outcome are due to the inability of setting