Skew Dice™
 

Disclaimer: The commercial version of the Skew Dice™ was designed by Robert Fathauer and Henry Segerman. Indeed, Skew Dice™ is a trademark of their company: The Dice Lab. Please, go buy some of their amazing products!

These dice are asymmetric trigonal trapezohedrons, the least symmetric 6-sided face-transitive polyhedrons (they only have rotational 120º-symmetry). The face-transitivity property ensures that they are “fair by symmetry” in the sense of:

P. Diaconis & J. B. Keller, 1989 - “Fair Dice” The American Mathematical Monthly, Vol. 96, No. 4.

In the provided OpenScad module, size is a positive parameter that controls the size of the dice. More precisely, the radius of the dice (before stretching, see details below) is:

 size * sqrt(3)/2 (in mm) 

which is the same radius that a standard cubic dice of  size edge-length will have.

• size = 16 is the default, recommended setting.

The parameter phase, in the range [0,120], controls the skewness of the dice: the rotation (in degrees) of the three south edges with respect to the three north edges.

• phase =  0 & phase = 120 are extreme degenerate cases.
• phase = 30 is the default, recommended setting.
• phase = 60 gives you a normal cube.

The parameter stretch controls the vertical elongation of the dice. By default, all the vertices of these dice lie on a sphere. However, you can scale the dice vertically without losing their face-transitivity (and, thus, their fairness). 

For positive values of 'stretch', this value will be used as the vertical scale factor. 

For the special stretch = 0 value, the vertical scale factor will be automatically computed to make the north and south poles of the dice right-angled. This automatic stretch depends on the phase parameter and ranges from 1/sqrt(2) (for phase = 0 and phase = 120) to 1 (for phase=60).

• stretch > 1 provides vertically elongated dice.
• stretch = 1 is the default, recommended setting.
• stretch < 1 provides vertically flattened dice.
• stretch = 0 enforces right-angled N & S poles.

The parameter rounding determines the radius of the rounded corners of the polyhedron. In particular, it will be the radius (in mm) of the rounded corners of a die whose vertices lie on a sphere of 1 mm of radius.

• rounding = 0.00 provides a sharp polyhedron.
• rounding = 0.05 is the default, slightly rounded, value.
• rounding = 0.20 provides a heavily rounded polyhedron.

The array faces stores the numbers that will be engraved on the dice faces. You can place from 0 to 9 pips on each face.

In the default setting of faces = [1,2,3,4,5,6], the numbers are placed so opposite sides add to 7 and so the north and south caps are as balanced as possible (1+4+5 vs 2+3+6).

• faces = [0,0,0,0,0,0] for blank faces without pips.
• faces = [1,2,3,4,5,6] is the default, for normal dice.

The easiest way to print these dice is with the lowest pip-count face (the 1, on normal dice) lying directly on the print bed. But you can obtain more aesthetically pleasing results if you print them in the provided orientation (with the aid of supports).