§1. In the play of games that involve chance, dice (singular die) have been used for millennia to produce unpredictable outcomes. Although the cube is the most popular shape for a die, other shapes are certainly possible, as exemplified by dice with various numbers of faces. This report examines sixteen particular eight-faced dice shaped as regular octahedra.

There are always limitations when rendering three-dimensional objects in a two-dimensional medium such as a computer screen, but the choice made here is to display the dice as polyhedral nets:

• Advantages of nets are that all faces are shown in full, and adjacencies of one face to another either are shown, or can be inferred. Also, the net serves as a cutting pattern for anyone who wants to prepare from cardboard a model of the die.
• A disadvantage is that the adjacencies to be inferred may not be immediately evident. However, a mitigation is provided in §4 below.

There are eleven choices for the net, as shown at MathWorld; while any of them would be correct for this purpose, the one chosen was the most convenient. Chart one shows the sixteen dice, each a different permutation of the numbers one through eight:

chart one

Each die has a label in the format "Li" or "Rj", where "L" and "R" stand for left and right. Two dice with the same number are mirror images of each other, but beyond that, the labeling scheme is arbitrary. The label is displayed as a caption in the chart, and is printed on the die itself. Because all the dice are different, but are similar-looking, displaying the label directly on the die helps game players to ensure that they have a correct set of dice — or more likely, a correct subset, as most games will call for fewer than sixeen dice. Quantities such as eight, four, or two might well be preferred. For a quantity of three or six, the alternative dice of §8 below might be preferred.

§2. Some patterns happen to emerge with the particular numbering plan chosen for this report, as shown in chart two:

chart two
	swap these faces …
	1 ↔ 8	2 ↔ 7	3 ↔ 6	4 ↔ 5
… to swap these dice:	L1 ↔ R8 L2 ↔ R7 L3 ↔ R6 L4 ↔ R5 L5 ↔ R4 L6 ↔ R3 L7 ↔ R2 L8 ↔ R1	L1 ↔ L5 L2 ↔ L6 L3 ↔ L7 L4 ↔ L8 R1 ↔ R5 R2 ↔ R6 R3 ↔ R7 R4 ↔ R8	L1 ↔ L3 L2 ↔ L4 L5 ↔ L7 L6 ↔ L8 R1 ↔ R3 R2 ↔ R4 R5 ↔ R7 R6 ↔ R8	L1 ↔ L2 L3 ↔ L4 L5 ↔ L6 L7 ↔ L8 R1 ↔ R2 R3 ↔ R4 R5 ↔ R6 R7 ↔ R8

Many other plans would be equally valid.

§3. In chart three below, die X1 is like die L1 except that some of the numerals have been printed in a different rotation. For the purposes here, however, X1 and L1 are considered completely equivalent.

chart three

To distinguish two dice on merely the basis of the rotations in which their numerals are printed would quickly lead to an unwieldy profusion in the quantity of dice.

§4. Chart four below is similar to chart one above. One change is that tabs have been appended to some faces to aid a person who cuts the pattern from cardboard and glues the octahedron together. The tab includes the number of the face under which it attaches, to help prevent connecting the tab to the wrong face. Another change is that the die faces are now colored in two examples of many possible schemes. In some games a red three and a green three, for instance, might have different significance.

chart four
four colors			two colors

§5. It is helpful to note that in a regular octahedron, each face is:

• edgewise adjacent to three faces;
• cornerwise adjacent to three faces;
• completely non-adjacent to one face, called the opposite face.

chart five

Chart five contains an example. Face 2 (yellow) of die L6 is:

• edgewise adjacent to faces 5, 6, and 8 (green);
• cornerwise adjacent to faces 1, 3, and 4 (yellow);
• opposite face 7 (green).

In this design of octahedral dice, the numbers on opposite faces always sum to nine (the "opposite-sum-nine" rule). Similarly, the numbers on opposite faces of an ordinary cubic die sum to seven.

§6. Some games require a die numbered one through four. However, the obvious choice, a regular tetrahedron, is inconvenient because when it comes to rest after a roll, no one side is on top. An alternative is to use an octahedral die with each number from one through four appearing twice, on opposite faces. The dice of this report can easily be thus transformed, yielding two dice, mirror images of each other. In this report the results are labeled L9 and R9 for lack of a better designation. Example:

chart six-a
ordinary L1	ordinary R1
after reduction: 5 → 4; 6 → 3; 7 → 2; 8 → 1

Here is a two-value version, where edgewise-adjacent faces have different numbers. "T" stands for "two":

chart six-b

§7. Diebuilders who experiment with different nets should beware of the following. There is one particular net than can be folded into two different shapes:

• an ordinary regular octahedron, like die L1;
• an octahedron that is not only irregular but indeed concave. The concavity appears between faces 1 and 6 when constructed as in chart seven below.

This second figure is called a tritetrahedron because it is the union of three equal regular tetrahedra. This illustrates why printing numbers on the tabs is helpful, and sometimes essential.

chart seven
folded into regular octahedron	folded into tritetrahedron

chart eight

Observe that faces 1 and 8 are always adjacent, as are 2 and 7; 3 and 6; 4 and 5. The coloring emphasizes this. This characteristic was not intentional, but rather a consequence of the vertex-sum-eighteen rule.

Contrast this with the opposite-sum-nine dice of §1-7, among which face 1 never adjoins face 8; 2 never 7; 3 never 6; 4 never 5. Hence the "M" and "S" dice are all distinct from the "L" and "R" dice.