dice – Is GameScience’s non-cylindrical seven-sided die a balanced/fair die?

No, they are not fair unless you ignore the sides

There are a few requirements for a uniform solid die to be fair.

The active sides must be face-transitive.

Dice are only fair if all of the sides that are being used are equally likely to be landed on. In order for this to be true, it needs to be face-transitive, meaning all the sides are the same shape. More specifically…

Isohedral Figure

In geometry, a polytope of dimension 3 (a polyhedron) or higher is isohedral or face-transitive when all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any faces A and B, there must be a symmetry of the entire solid by rotations and reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.

Isotoxal Figure

Regular Polyhedra are isohedral (face-transitive), isogonal (vertex-transitive) and isotoxal (edge transitive).

This 7 sided die is neither of those things. But it is if we ignore every result on the pentagonal sides.

Put another way, given a face on the die, there must be a rotation (at least one) that results in every other face, edge, and vertex being mapped onto the same place as a different face, edge, and vertex, respectively. Let’s try it in 2-d.

Equilateral Triangle

This makes a good 2 dimensional die. Rotating the triangle 120 degrees around the center maps every vertex and edge of the triangle to another one. Let’s take that to 3 dimensions, say a cube. A d6. We’re all familiar. A d6 is a fair die because there exists at least one rotation that results in each face, edge, and vertex being mapped onto the location of a different one. One of those rotations would obviously be a rotation that can be represented by “90 degrees on one axis, and 90 degrees on another”. Or, in Euler Angles, 90, 90, 0. Or, if it helps, 90 degrees pitch, and 90 degrees yaw. Or any combination of pitch, yaw, and roll.

All other fair dice have this property. A rotation exists that maps every face, edge, and vertex of a d4 onto a different face, edge, and vertex. There exists one for a d20. There are in fact, many rotations that do this for these fair dice. But there is no rotation that does this for a d7. You could spin it 180 degrees around the “up” axis (sitting on neither 6 or 7), but then the top edge would not have been translated into the position of another edge. You could lay it flat on 6 and spin it 72 degrees, but then the pentagonal faces would not have been translated into another face.

The center of each face must be equidistant to the center of mass.

When it comes to (fair) dice, the centre of mass is in the exact centre of the object. This means all the faces are equidistant from it. The result of this is, after a roll, every face has equal opportunity to come up. However, if the centre of mass is moved from the geographical centre of the die, then the axis of rotation is changed, and the die is no longer fair. source

Changing the center of mass is known as weighting the die. As the centre of mass is moved further from the middle of the die, the effectively lighter face will roll upwards more often than not.

Making fair dice by ignoring faces

Dice with an odd number of flat faces can be made as “long dice”.(26) They are based on an infinite set of prisms. All the (rectangular) faces they may actually land on are congruent, so they are equally fair. (The other 2 sides of the prism are rounded or capped with a pyramid, designed so that the die never actually rests on those faces) Source

That last sentence is the most important part. This 7 sided die is fair for ranges 1-5, provided you ignore the 6 and 7th face. As we read above, any prism can be fair provided the ends are “capped” or ignored (see Long Dice). So, a real d7 would be made of a heptagonal prism. So, ignoring the ends, there exists a rotation that maps every face, vertex, and edge onto the location of a different face, edge, and vertex. Lets go back to that example above. We lay it flat on the 6th edge and spin it 72 degrees. Voila! Each of the faces is now in the location of where a face used to be, each edge is in the place of where a different edge was, and each vertex is in the place of a different vertex. Except for the caps, which we’ve ignored.

More recently, you may have noticed barrel dice. They use the same basic principle, except their sides are triangles rather than rectangles.

Barrel Dice

Why don’t non-symmetric unorthodox shapes work?

The result of the die being face-transitive and having a center of mass equidistant from the centers of the faces is that it requires the same amount of force in one direction to turn it over no matter what face it has landed on. When we look back at the d7, we can easily guess that applying force to go from face 1 to face 2 is the same amount of force that will change it from face 2 to face 3 as it rests on the table. This is due to the fact that the angles between the faces are the same, and because the faces are the same on those sides. There is as much surface area touching the table when “1” is up as there is when “2” is up. Let us consider faces 6 and 7.

When face 6 is up, face 7 is down. There is now a greater surface area on the table. Moreover, the angle between face 6 and any other face touching it is greater (90 degrees versus 72 degrees). Both of these mean that it requires more force to push it onto one of the other faces. So when the die is tumbling and face 6 or 7 hits the table near the end of the tumble and loses some of its velocity and rotational velocity, it is more likely that X amount of force will not result in the die tumbling over that face to land on 1-5.