Experiments with cylindrical dice
Last update: Oct 10 2005

Daniel B. Murray, Associate Professor
Physics
University of British Columbia Okanagan
Kelowna, BC, Canada     Email: daniel dot murray at ubc dot ca

Normally dice are isohedral in shape. Part of the definition of an isohedron is that all faces are the same shape, but the definition also specifies that each face have the same geometrical relationship to the center of mass of the die. By their nature, vigorously rolled isohedral dice will land with equal probability on all of their faces. All common dice shapes, including the cube, tetrahedron, dodecahedron (12-sided) and the icosahedron (20-sided) are fair dice. An article by Ed Pegg Jr. has systematically listed all possible isohedral dice shapes. (His thesis "A Complete List of Fair Dice" Master's Thesis 1997 is posted here in Word format (812 KB) with his kind permission). The detailed dynamics of the dice roll, considering such details as friction, angular momentum and moment of inertia, are not important to equal probabilities of a roll coming up with one of the faces. Such a dice can be understood to be fair from geometrical considerations alone.
This article looks at an elementary type of dice that is not isohedral in shape and thus not automatically fair by symmetry. For simplicity, I choose the cylinder (specifically, a cylinder with circular cross section) for the dice shape. Such a dice has three faces. Two are flat circles. The other face is curved. A coin with non-vanishing thickness approximates this shape.
I arbitrarily selected commercially available nylon rod of diameter 19.1 mm and cut it into dice of varying heights. Thus all dice in this study have the same diameter. The heights varied from 3.5 mm to 47 mm, thus exploring the full range of possibilities.
A glass surface was chosen for the table on with the die was thrown. To avoid losing the dice, walls surrounded a 10 cm by 10 cm region of the glass surface. Two different tables were used. One table was made of glass 2.40 mm thick. The other table was made of glass 15.50 mm thick. The qualitative motion and sound of the dice rolling is very different for the two tables. The motion on the 15.50 mm glass table is much more lively and it takes more time for the die to stop moving. The motion on the 2.40 mm glass table slows down much more quickly and there is much more noise made.
The figure at the right shows the results of about 70,000 rolls. The vertical axis is the probability that the die lands with either of the flat circles pointing up. The horizontal axis shows the height of the die in terms of the inverse tangent of the ratio of height to diameter. Data points obtained on each of the two tables are plotted in yellow (15.50 mm) and light blue (2.40 mm). Error bars show one standard deviation of uncertainty based on the number of rolls made, which was normally from 1000 to 2000 rolls.
The most prominent feature of this graph is that the probability is quite different for the two different tables. No matter what their height, the cylindrical dice are more likely to land with a round end up if they are thrown on the 15.50 mm thick glass table. The 2.40 mm glass table favours the die landing on the curved surface.
Both being glass, both tables have the same amount of friction with the die. They differ in their coefficient of restitution. In the past, coefficient of restitution has been identified as a factor important in the theory of dice rolls [Eugene M. Levin, "Experiments with loaded dice", American Journal of Physics, volume 51, 1983, pages 149-152] [Edward T. Pegg Jr., "A Complete List of Fair Dice", 1997 Master's Thesis] and experimental work plus computer simulations of coin tosses [Daniel B. Murray and Scott W. Teare, "Probability of a tossed coin landing on edge", Physical Review E, volume 48, 1993, pages 2547-2552].

The following links lead to further details:

How the dice are made
Varying the dice material
Details on the table surface
Automated dice throwing mechanism
Recording the outcome of the rolls
Analysis of data when height/diameter is small
Analysis of data when diameter/height is small
Dimensions of a fair three sided dice
The Pegg Geometrical Model (solid angle theory)
General comments and conclusions
Raw data for rolls made on 15.50 mm glass table
Raw data for rolls made on 2.40 mm glass table