Collection Spotlight: Polyhedral Dice

My dice collection takes dice-collecting to a whole new level, with extra added math!

D20 Die

I’ve always enjoyed collecting dice as a kid, but after I figured out how many different sets existed and the sheer number that I’d have to buy, I pretty much gave up on that hope. However, when I started adding mathematical objects to my collection, I figured a dice set would be the perfect addition.

Dice are really old. In face, the word “dice” refers to the plural form of the word “die”. We do not know the origin of these things. However some people estimate them to have existed as early as 2400 BC. Although at the time, the opposite faces of a six-sided die did not add up to 7, but had consecutive numbers on them. The general design of a die intentionally sets the highest possible number opposite the lowest possible number, the second-highest possible number opposite the second-lowest possible number and so-forth. This is to reduce the effects of any defects in the making of the die. So, if a die has a slightly higher probability of landing on one face, the probability of landing on the exact opposite face also goes up. This means that the overall average roll of the die is does not change.

D4 Dice

We usually refer to dice by the number of faces they have. For example, we would describe a six-sided die as a D6. The most popular type of die is the D6, however, you will see D4’s, D8’s, D10’s and other dice used in Role Playing Games (RPG’s) such as Dungeons & Dragons.

However, what I wanted to talk about particularly was some of the mathematics behind the dice. To be more specific, I want to talk about what some of these dice represent. I first want to talk about polyhedrons, which you can think of as 3D shapes. Mathematicians call a small set of these polyhedrons Platonic Solids. They are Regular, Convex polyhedrons. The definitions are below:

  • Regular: All faces are congruent (identical) and arranged the same way around each vertex
  • Convex: A line drawn between any two points of this polyhedron either lies on or inside the polyhedron itself

In 3D, we have a total of five possible polyhedrons (also referred to as “polytopes) that meet this criteria, shown below:

Platonic Solids

From top to bottom, left to right, with the number of sides, they are:

  • Tetrahedron (4)
  • Octahedron (8)
  • Icosahedron (20)
  • Hexahedron (6)
  • Dodecahedron (12).

As you go to a higher number of dimensions, the number of possible polytopes change. For example, in 4D, there are a total of six possible polytopes. However, in higher dimensions (5 and above), this number goes down to three and stays there. The three that remain are the higher-dimensional equivalents of the Tetrahedron, Hexahedron and Octahedron.

If you’re interested in learning more about these polytopes, you can check out this Numberphile video, which illustrates the concepts really well.

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