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If I have a perfectly balanced and thus fair cubic die, then polish 3 adjacent faces (so that their coefficient of friction is effectively zero) and roughen the remaining faces (so that their coefficient of friction is, say, one) but otherwise leave the die still perfectly balanced, how will it affect the fairness of the die, assuming it is used normally, i.e. by throwing it on a horizontal surface with random velocity and spin?

More generally, other than balance (macro geometry and density), what factors would affect the fairness of regular polyhedron used as a die?

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Great question. It occurs to me that the edge shaping will impact the answer to the question. With sharp edges, it seems the polish would have less of an effect; with rounded edges, it will make a difference how far the polish extends around the curvature. –  AdamRedwine Mar 1 '12 at 19:49
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3 Answers

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It should have no effect, as long as there are a large number of rolls which obey the full-sticking condition during instances of face-surface contact. The coefficient of friction is not the cause of energy loss in the die as it rolls, rather this is a case where you have (approximately) nondifferentiable kinematic constraints, rolling a square, and under these circumstances, there is energy loss from turning a corner. The way to calculate the energy loss is to note that the linear momentum and angular momentum around the pivot line or pivot point, but the energy is lost when the die suddenly changes direction of roll. The mechanism of loss is largely independent of the surface properties, unless you make some of the surfaces sticky.

To make a biased die. you want to round out some of the edges and corners to allow rolling away from some faces easier than others.

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This is more intution than calculation:

If some sides are polished perfectly the velocity component parallel to the table does not create any friction and therefore no torque or elastic effects. This could shift the probabilities a bit towards resting on the polished sides.

A surface treatment could also influence the elasticity of the sides. In an extreme case you could create something like hooks used for velcro that stick to the table or some other surface that absorbs a lot of momentum due to deformation. Both cases would increase the probability for resting on that side quite a bit.

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For the second part, elasticity could be a factor. For example, if it were possible to make a surface completely inelastic e.g. with glue, it would cause the opposite face to come up more often as the dice stops moving as soon as the inelastic face came in contact with the surface.

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