In a single, solid rigid body, the Centroid defines the geometric center of an object. In the case of a perfect cube, the Centroid exists at the middle point inside the cube equidistant from each of the six faces of the cube. For a rigid body that has a consistent density throughout its structure, the Centroid also defines the Center of Mass. For a rigid body within a uniform gravitational field, the Center of Mass is equivalent to the Center of Gravity of that body.
For the special case of dice, rotational motion becomes important in the dynamics of a rotating cube. Rotational dynamics depend not on the Center of Mass but the rotational inertia or Mass Moment of Inertia. The Moment of Inertia measure the resistance of an object to rotational acceleration about an axis. Should the density or mass vary in distance from the rotational axis, the Moment of Inertia defines the resistance of that object to rotational acceleration.
When I decided to design Metal Dice, I wanted to minimize the rotational Moment of Inertia as a means of developing a die that would roll to stop at statistically even probabilities for each of the six possible outcomes. Using parasolid modeling, I was able to both readily calculate the Moment of Inertia and quickly make structural modifications that would reduce the differential between the Centroid, the Center of Mass and the Mass Moment of Inertia of a perfect cube. The Moment of Inertia is also driven by the form of the solid object, in this case a cube. I cannot change the basic cube form, so my approach is to minimize the offset of the center of mass from the Centroid and subsequently minimize the Mass Moment of Inertia within the limits of an ideal cube.
I designed the die to display the face features (‘pips’ or ‘dots’ numbering one through six) by drilling one or more spotfaces in the classic die pattern. By changing the drill depth of each pip to compensate for the mass of the material removed from the opposite face, I was able to match the geometric Centroid to the center of mass. This was readily accomplished in the ideal world of parasolid modeling.
In the real world, there are several issues to consider:
- The material used in the fabrication of the die is not exactly homogenome in density. The Center of Mass and Moment of Inertia could not be guaranteed even on a featureless cube.
- The depth of the spotfaces can only be controlled to within the fabrication tolerances of the CNC machines used. In most cases, this is within +/- 0.002 inches.
- The actual location and classic pattern of the pips on any given face is also limited by the tolerance of the CNC machine. This variation also causes the Moment of Inertia to drift from the Centroid.
All of the above issues led me to design a die that closely approximates a perfectly weighted ¾ inch cube. The variations from the ideal Centroid are: rx=0.0003, ry=0.0012, rz=0.0007 Variations in the material density homogeneity could half or double these figures.
To apply a common comparison, the average diameter of human hair is 0.003 inches.