General comments and conclusions
Updated: April 15, 2002
Center of mass height
In considering the dynamics of non-isohedral dice, it is
natural to consider the height of the center of mass. In a loaded die,
for example, the center of mass is lower for a certain face, and the
probability of that face can be expected to be higher. Another way of
thinking about this is that the probability of each final state depends
on the energy of that state. This motivates the Boltzmann factor approach
suggested by Eugene Levin. ["Experiments with loaded dice" American Journal
of Physics, volume 51, 1983, pages 149-152].
In the case of the cylindrical die, it is interesting to
consider, as George Hart
[www.georgehart.com] has suggested [private communication, 2002] the situation where the height of the center of mass is the same for all three faces. In this situation, the probabilities
should not have anything to do with the final state energy since it is the
same in each case.
This case of equal center of mass height occurs when the
diameter of the die is equal to its height. In that case, the arctangent of
height/diameter equals 45 degrees. From the experimental data obtained,
the 15.50 mm thick glass table gives a probability of landing on an end
of 38%, while for the 2.40 mm thick glass table the probability is
24%. Another way of stating this is that for 15.50 mm glass, the
probabilities of the three faces are 19%, 19% and 62%. For 2.40 mm
glass, the probabilities are 12%, 12% and 76%. Initially, one might
guess that equal center of mass heights should mean that the probabilities
are all the same. However, the edge (curved) face has a much larger surface
area. Assuming diameter and height are both 2 units, the area of each end
face is 3.1416, while the area of the edge face is 12.5664. The total area
of the die is 18.8496 units. So, each end has an area which is 16.67% of
the total, while the edge has an area which is 66.67% of the total.
These numbers do not exactly equal, but are close to the actual probabilities. So this suggests that there is some sense to the apparently naive notion
of an "area rule" where the probability of a face is proportional to its area.
Alternatively, the solid angle subtended by each end is 1.840 steradians,
while the solid angle subtended by the curved side is 8.886 steradians.
This gives probabilities of 14.6%, 14.6% and 70.7%.
Energy dissipation
The experimental data shows clearly different probabilities
for the two glass surfaces. Subjectively, the 15.50 mm glass table allows the
die to bounce more before settling down. George Hart [private communication,
2002] has wondered what would happen in the situation where the die is
thrown and lands in water, where the effect of friction with the water would
be important. Letting a suspect die tumble through water is a recognized
method for detecting loading [John Scarne, "Scarne on Dice"]. Throwing the
die in a vaccuum would show whether or not air friction is important.
Perhaps this could be taken to the opposite limiting
extreme by tossing the die in a highly viscous medium such as oil or syrup. That limit is of interest since the speed would then be so low that mass
as well as the coefficient of restitution of the surface would no longer play
any role at all. A natural conjecture is that the probability of each face would be proportional to the solid angle that it subtends.
This page is still under construction.