Is there any standard way of numbering the faces of the 20-sided die?
There is no industry-wide standard for mapping numbers 1...20 across the faces of a regular icosahedral d20 die. Manufacturers have used several configurations. Their choices appear in dice collections here and
If there is no standard, is there any set of standard ways?
The term "standard die" is often used as a generic term to refer to a die which implements the opposite-faces convention: values on opposite faces sum to one more than the number of faces. In the case of the d20, the sum is 21. Since so many dice makers use this convention, it might be considerd to be the de facto standard; however, this convention says nothing about how the face pairs are distributed across the die. Actually, the term "standard die" refers to a member of the set of dice which implement the opposite-faces convention. Rainbolt refers to one such member with a net from the Dice Collector; Bosch, Fathauer, and Segerman offer another net from the same set. In these two cases and others, different numbers surround face "20":
|Bosch et al.
Bosch et al. offered a thought experiment to explain how the opposite-faces convention preserves averages for imperfectly shaped dice. In a word, they explained how the convention mitigates oblateness.
Some reasoning would be the (sic) take further step the effort to make it fair; however, I don't have any rule in mind that will achieve this end.
Bosch et al. suggested just such a rule. They argued that there are additional measures which manufacturers could take to make d20s which are even more numerically balanced. They offered additional thought experiments to support numerically balanced vertex sums and numerically balanced face sums to mitigate the presence of an air bubble. They stated a mathematical optimization problem and coded an integer program (IP) to solve it. They found an optimal solution with numerically balanced vertices and faces.
To see how large this optimization problem is, start counting configurations. There are 20!=2432902008176640000 ways to map numbers 1...20 onto the faces of the regular icosahedral d20 die, but many of these configurations are rotationally equivalent. The size of the rotational symmetry group |G|=60. So, there are
20!/|G| = 2432902008176640000/60
rotationally distinct configurations of the d20 die. This count includes mirror images. In a perfect world, dice makers may choose any one of the many rotationally distinct configurations; however, the world is not perfect. The opposite-faces convention helps to mitigate some of that imperfection. How many configurations implement the opposite-faces convention?
- There are 20 ways to place "1" on an arbitrary face; this placement determines the placement of "20" on the opposite face.
- There are 18 ways to place "2" on one of the remaining faces; this placement determines the placement of "19" on the opposite face.
- Finally, there are 2 ways to place "10" on one of the remaining faces; this
placement determines the placement of "11" on the opposite face.
So, there are
20!!/|G| = (20*18*16*14*12*10*8*6*4*2)/60
rotationally distinct configurations implementing the opposite-faces convention
. This count includes the mirror images. Among these are (an unknown number of?) configurations with balanced vertices and faces. With a computer search, Bosch et al. found one of them. This configuration appears on the mass market as the Magic-numbered d20 and on a 3D-printed d20 as the ENUMERATED BALANCED ICOSAHEDRON.
I end with a simple question. Among the 61,931,520 "standard d20s", how many implement balanced vertices and faces? Is the configuration which Bosch et al. found the only one?