Inside Box XII: INQ's D60

Talking about dice is one of our favorite things: how often do you get to think about  geometry, probability, and gaming all at the same time? Almost everybody is familiar with a standard die: the cube marked with little dots called “pips” that is used in everything from Monopoly to Vegas-style Craps games. And of course, a lot of people also know the more unusual dice shapes that have been used in roleplaying games like Dungeons & Dragons for decades. 

The standard compliment of dice in many RPGs is a d4, d6, d8, d10, d12, and d20. Those particular dice (except the d10) are modeled on a set of polygons called the Platonic solids. The Platonic solids are basically the three-dimensional versions of regular polygons: the triangle, square, and the pentagon. One can prove mathematically that there are exactly five of them, the tetrahedron (d4), cube (d6), octahedron (d8), dodecahedron (d12), and icosahedron (d20). Platonic solids are the only three-dimensional shapes that are perfectly symmetrical, with every internal angle and side length identical. 

They’re called Platonic because the study of these shapes goes back to ancient Greece–in this case, they’re named for Plato, who in his famous book the Timaeus suggested that each of the classical elements were made of one of these solids. In case you’re wondering their elemental companions, the d4 is fire because of it’s sharp points, the d6 is earth because it’s “the most stable,” easily stacked and packed, the d8 is air based on Plato’s version of the Jan Brady Principle (it was in the middle between fire and water in terms of size and shape) and the d20 is water, because it’s so rounded that it just flows off your hand. The d12? Actually, Plato didn’t really have an element for that, he just sort of vaguely said that the gods used shaped the heavens in that shape. Poor d12, it always felt like the least-used die in D&D, too.

Your d60 is a special polygon called a deltoidal hexecontahedron–a hexecontahedron, of course, being a polygon with 60 faces. But wait, didn’t we say there were only 5 Platonic solids? Yup. The hexecontahedron is one of a class of polygons called the Catalan solids, named after Belgian mathematician Eugène Catalan, who first described them in the 1800s. What’s amazing about the Catalan solids is that they are basically created by taking one of the Platonic solids and pulling out the faces to make new “sides.” A hexecontahedron can be created from either a dodecahedron (12-sided polygon) or an icosahedron (a 20-sided polygon); specifically, from pulling out one side and placing four “kites”–4-sided polygons–along it in a 3-sided pyramid shape. For what it’s worth, there are actually four different types of hexecontahedron, so you could actually come up with four different types of d60s.

One interesting twist is that, there is a an actual geometric reason that we call this a “d60” instead of a “60-sided die.” That’s because it is possible to make dice from polygons that have a certain number of numbered faces, but a lot more sides. For example, it’s not hard to find dice that geometrically speaking are basically elongated prisms, with tapered edges that the die can’t land on. One of these dice would have say, 20 numbered sides, but 40 actual sides; leading to the strange case where you have a d20, but not a 20-sided die. 

Finally, what makes a d60 amazing for games is that 60 is what is known as a “highly divisive number.”  Essentially that means it can be divided by more whole numbers than any number smaller that it is; in this case, 60 can be divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and of course itself.  In dice terms, that means it can stand in for almost all the other dice in a polyhedral set! Need to roll a regular, boring d6? Bust out your amazing d60, roll it, and just reassign the numbers: 1-10 is the d6’s one, 11-20 is the d6’s two, and so on.  If you’re missing your d20, then 1-3 takes the places of 1, 4-6 is 3, etc. We can’t guarantee that the dealers in Vegas will accept this substitution, but your games of Monopoly may get a lot more interesting. And next time you’re prepping to play your favorite RPG, forget bulky bags of Platonic solids – show up to the game session with a Catalan solid d60 and watch people’s eyes pop with wonder at this geometric miracle!