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I understand the two enantiomorphs which implement the opposite faces convention (aka the n+1 rule). I am interested in a much larger picture:

  • history
  • which market offers which enantiomorph?
    • casino dice
    • precision dice
    • backgammon dice
    • drugstore dice (US), shop dice (UK)
  • the 15 distinct configurations
  • the derivation from 6! permutations down to the two configurations
  • why those two?
  • is there a rival configuration?
4
  • What is the question? The subject line doesn't seem to match the body of your "question". Sep 1 '21 at 17:42
  • The first paragraph of my answer directly addresses the question in the subject line. Beyond that, I am hoping that those who read the remainder of my answer will be prompted to share from their knowledge additional facts related to the topics I covered.
    – nuatthis27
    Sep 2 '21 at 0:04
  • "Tell me all about dice" isn't really a question though.
    – Nij
    Sep 3 '21 at 22:47
  • In light of the comments, I am rethinking my goal and how to make it conform with the standards of boardgames.stackexchange.
    – nuatthis27
    Sep 8 '21 at 10:42
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There is no industry-wide standard for mapping numbers 1...6 across the faces of a regular hexahedral (cubic) d6 die. Over the centuries, makers of dice have used nearly all of the possible configurations. Their choices appear in the historical record, and in private collections here and here.

On the other hand, 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 cubic d6, the sum is 7. Bosch, Fathauer, and Segerman 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. Since so many modern dice makers have been using this convention, it might be considered to be the de facto standard; however, this convention says nothing about how the face pairs are distributed across the die. More precisely, the term standard d6 refers to the two-member set of cubic dice which implement the opposite-faces convention. The two configurations are mirror images. Here are their nets illustrated in the stereotypical cruciform:

                               Enantiomorphs
           -----------------------------------------------------
              left-handed                     right-handed
                 *****                           *****
                 * 3 *                           * 3 *
             *************                   *************
             * 2 * 1 * 5 *                   * 5 * 1 * 2 *
             *************                   *************
                 * 4 *                           * 4 *
                 *****                           *****
                 * 6 *                           * 6 *
                 *****                           *****

The cubic d6 has only 15 pairs of enantiomorphs. What are they? Who used them? Is there a pair with statistics which might rival the opposite-faces convention? To answer these questions, start counting configurations.

The Combinatorics

There are 6!=720 ways to map numbers 1...6 onto the faces of the cubic d6 die, but many of these configurations are rotationally equivalent. The size of the rotational symmetry group |G|=24. So, there are

                         6!/|G| = 720/24
                                = 30

rotationally distinct configurations of the cubic d6 die. This count includes mirror images. In a perfect world, dice makers may choose any one of these 30 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 6 ways to place "6" on an arbitrary face; this placement determines the placement of "1" on the opposite face.
  • There are 4 ways to place "5" on one of the remaining faces; this placement determines the placement of "2" on the opposite face.
  • Finally, there are 2 ways to place "4" on one of the remaining faces; this placement determines the placement of "3" on the opposite face.

So, there are 6!!/|G| = (642)/24 = 48/24 = 2

rotationally distinct configurations implementing the opposite-faces convention . This count includes the two mirror images illustrated above.

The 30 configurations appear here and half of them appear here (note the correction in the Description and the silent correction in the video shortly after 10:00). Parker listed the 15 configurations in the same order as they appear in Table 1 in de Voogt et al. and with face "1" consistently appearing at the center of the cruciform. The following table is a similar listing but shows the configurations as paired mirror images.

                                    mirror image
                                    -------------

     1.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 3 * 1 * 4 *          * 4 * 1 * 3 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****


     2.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 3 * 1 * 5 *          * 5 * 1 * 3 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****


     3.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 3 * 1 * 6 *          * 6 * 1 * 3 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 2 *                  * 2 *
                 *****                  *****


     4.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****


     5.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****


     6.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 3 *                  * 3 *
                 *****                  *****


     7.          *****                  *****
                 * 5 *                  * 5 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****


     8.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****


     9.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 4 *                  * 4 *
                 *****                  *****


    10.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****


    11.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 6 *                  * 6 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****


    12.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 6 *          * 6 * 1 * 2 *
             *************          *************
                 * 4 *                  * 4 *
                 *****                  *****
                 * 5 *                  * 5 *
                 *****                  *****



    13.          *****                  *****
                 * 4 *                  * 4 *
             *************          *************
             * 2 * 1 * 3 *          * 3 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****



    14.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 4 *          * 4 * 1 * 2 *
             *************          *************
                 * 5 *                  * 5 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****


    15.          *****                  *****
                 * 3 *                  * 3 *
             *************          *************
             * 2 * 1 * 5 *          * 5 * 1 * 2 *
             *************          *************
                 * 4 *                  * 4 *
                 *****                  *****
                 * 6 *                  * 6 *
                 *****                  *****

(Readers are invited to compare their d6s against the 15 pairs above.)

Parker asked (Question #6, 11:34), "How good are our original, normal, garden-variety, six-sided dice when it comes to the values around any one verticee(sic)?" He analyzed the 15 configurations with a proof by exhaustion and provided his results as a table of statistics in the Description. He concluded that while Configuration #15 implements the opposite-faces convention (all three pairs of opposite faces sum to 7), it is the worst in terms of numerically balanced vertices. Configuration #1 is the best in terms of numerically balanced vertices but the worst in terms of the convention; Configuration #1 has only a single pair of opposite faces summing to 7. (The configuration circled in gold in the video is the left-handed version of Configuration #15. The configuration circled in purple is Configuration #1.)

In the end there are two competing configurations. Configuration #1 mitigates air bubbles; configuration #15 mitigates oblateness. The mass market chose #15. Some manufacturers chose the right-handed version of Configuration #15; others chose the left-handed version.

But wait! There's more! The cubic d6 is not the only marketed d6:

  • triangular bipyramid
    • Pyramid d6, The Dice Lab
  • symmetric trigonal trapezohedron
    • Slant Dice, The Dice Lab
    • Hedron D6, Tiny Tokens
    • Isodie 10, Geometrix
  • asymmetric trigonal trapezohedron
    • Skew Dice, The Dice Lab
  • modified triangular prism (dreidel)
    • Top d6, Wombat
  • modified triangular antiprism
    • "Crystal" d6, The Dice Lab
    • Barrel Dice RPG Set, Diceforge
  • (any others?)

The reader is invited to discover the configurations applied to these alternate d6s.

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  • For completeness, there's also the modified hexagonal prism D6 Oct 29 '21 at 0:24

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