# Dice of Sundry Sides

Here’s a brain teaser for you: What would two-sided dice look like? How about one-sided dice? Think about those for a moment. The answers are below, but first let’s look at some more common dice with different numbers of sides.

### Platonic Solids

The six-sided die is the most common, probably simply because a cube has six sides, and cubes with rounded corners are perfect for rolling dice. A die is essentially a random number generator. But what if, instead of numbers from one to six, you want to generate a random number between one and four? Then we need a solid shape with four equal sides.

And we have one: the triangular pyramid, or tetrahedron. While this doesn’t roll as easily as a six-side die, it has four identical sides and works well as a random number generator. Since no side faces up when the die is at rest, the numbers have to be placed differently.

We can also create an eight-sided die by attaching the square sides of two square pyramids, so that we have eight triangular faces: an octahedron. The tetrahedron (four sides), cube (six sides) and octahedron (eight sides) are three of the five Platonic solids. The other two are the dodecahedron (12 sides) and the icosahedron (20 sides).

The five Platonic solids are the only shapes where each side is the same regular polygon, and the same number of sides meet at each vertex. Dice have been made in these five shapes since the dawn of civilization, and a typical set of dice for modern role-playing games will include these shapes. When these dice are made properly, the opposite sides will add up to the total number of sides plus one.

### Prism Dice and Other Shapes

It’s still possible to make dice with other numbers of sides. They will be random number generators, even if they are not Platonic solids. One design method is based on the dreidel, a four-sided top played with during Hanukkah.

A die with three or more sides can be made by the same principle, creating a prism shape. Viewed from the top, the die will look like a triangle (three sides), square (four sides) pentagon (five sides), hexagon (six sides), heptagon (eight sides), octagon (eight sides), nonagon (nine sides), decagon (ten sides), hendecagon (eleven sides), or dodecagon (twelve sides). The ends can be rounded to avoid having the die land on its end.

Of course, dice with four, six, eight or twelve sides are already taken care of by the Platonic solids. And for the other numbers of sides, a top, prism or barrel is not always the desired style. Instead, we can follow the example of the octahedron and simply attach the bases of two pyramids, with the base being a polygon of half the number of sides desired for the die. This would be a bipyramid. However, with the ten-sided die pictured to the left, the faces are staggered, so that a side always faces up, and each side is a kite shape rather than a triangle, making this a trapezohedron.

### How Many Sides Can Dice Have?

In order for dice to be fair, there must be an equal chance of each side landing up when they are thrown. If a shape is a convex isohedron, it has the symmetry required to create a fair die. The five Platonic solids and the 13 Catalan solids are isohedra, as are bipyramids and trapezohedra. An isohedron will always have an even number of faces. One example is the 60-sided die pictured to the right. It is a deltoidal hexecontahedron, one of the Catalan solids, with each side the shape of  kite. The Catalan solid with the largest number of faces is the disdyakis triacontahedron, with 120 sides. However, there is no limit to the number of sides a bipyramid or trapezohedron may have. Below is a video by The Dice Lab about their 120-sided die.

### Two-Sided and One-Sided Dice

Have you figured out what a two-sided die would look like? Here’s a hint: Think about what you would use it for. Basically, you would want to choose between two options, by chance. That happens all the time, and there is an object we use to do it.

That’s right: a coin is effectively a two-sided die. If you want a two-sided die that looks more like a die, then there is one pictured to the right for you. But now consider the harder question: What would a one-sided die look like? First of all, we should point out that a one-sided die would be pointless, as the outcome would never be in doubt, and when there is only one option available, we do not need help making a choice. But as a thought experiment, what would a one-sided die look like? Can you think of a physical object with only one side?

That’s right: a Möbius strip! The one-sided die pictured at left is a Möbius strip, meaning that it has only one side. If you were to start where the number 1 is engraved and follow the surface along the strip, you would return to where you started without ever having crossed an edge. So if you ever need to determine the outcome when there is only one option, simply roll this die. You could also say that a sphere is a one-sided die.

If you want to try another brain teaser involving dice, check out the Two Dice Wager.

### Three-Sided Coins

By searching for a two-sided die, we arrived at a coin. However, a coin technically has three sides: heads, tails and the edge. In fact, a flipped coin will sometimes — very infrequently — land on its edge. This is more likely for coins with thicker edges. What if we wanted to expand the thickness of a coin until there was a probability of exactly one-third that the coin would land on heads, tails, or its side? Merrill Flood of the RAND Corporation was intrigued by this problem, and had mathematicians calculate the appropriate dimensions and machine shops mill some of the coins. Supposedly, the legendary John von Neumann was able to work out the design formula for a fair three-sided coin in his head. However, there is some debate about the difference between the theoretical physics of a fair three-sided coin and how one functions in practice. For more about von Neumann and game theory, I highly recommend The Prisoner’s Dilemma, by William Poundstone.