The Denary Dice
Inside the Denary Dice set, our legendary first-of-it’s kind set of math rocks
One class of objects are the personifications of probability. The rulers of randomness. The avatars of odds. We’re talking of course about dice! And we love ‘em! The Denary Dice Set is one of the most popular items we have ever created; loved by gamers, mathematicians, and probability stans everywhere. This is the inside story of how we created and tested one of the world’s most curious dice sets.
Dice help illustrate scientific thought and question philosophy. They’re excellent tools for demonstrating important mathematical concepts, particularly in geometry and probability. But they are also about the human condition: do we believe in chance and free will, or fate and predestination? Does God play dice with the universe? Is there no fate but what we make for ourselves?
Dice have a long and glorious history. Plato said they were invented by the Egyptian god Thoth. The Roman Emperor Claudius loved them so much, he wrote a book called De arte aleae (“the art of dice”). Mozart created a game that used dice to compose music, and famous mathematicians like Fermat and Pascal weighed in on the rolling of dice. In 1620, Galileo wrote a scholarly paper about them. His thesis? Why rolling three 6-sided dice (3d6) gives a total of 10 more often than a total of 9, since people expect the average roll between 1–6 should be 3.
Of course, the purpose of dice is to roll random numbers, and we did a lot of work to ensure these were as fair as possible. For a die, being “fair” means it has precisely the same chance of landing on any given face (side), ensuring randomness. Now, truly fair dice are practically impossible to make; even casino dice, renowned as the fairest in the world, aren’t perfectly randomized. And the complex polyhedral dice found in many roleplaying games are, again, extremely fair, but not perfectly so. There are a range of factors that fight against perfect fairness: tiny imperfections in the shape and density of every die, the physical limitations that prevent each throw from being exactly the same (it’s hard to move your hand exactly the same way each time you toss the die), as well as air viscosity, friction, and imperfections in the surface where the die lands, just to name a few.
So while perfectly fair dice aren’t possible, reasonably fair ones are, based on the geometry of the solids. Our goal was to create these dice to be as fair as we could possibly make them for you. The shapes for the d4, d6, d8, and d10 are pretty standard, and recognized for basic fairness because of their shapes; Persi Diaconis, one of the foremost experts on randomness, calls them “fair by symmetry.”
The d1, d2 , and d3 are unusual, but not geometrically difficult to create. It was finding the right shapes for the d5 , d7 , and d9 that proved the real challenge. The easiest way to create a fair die that rolls odd numbers is to make it a barrel shape , with the desired number of sides on the long dimension and no numbers on the ends; in fact, these exist, and are called barrel dice.
But that seemed like the easy way out—we wanted to give you a set of polyhedrons that could land on the ends, too!
We knew that if you took a prism shape with the proper number of faces, you could experiment to find the right diameter for each end polygon, along with the length of the sides, to make them work. This idea creates dice that are called “fair by continuity,” because somewhere along a continuous sliding scale length of the long sides, you’ll find the fair dimension. To do so, we 3D printed multiple tests of each die, varying the relative dimensions, and went through a lot of trial and error—mostly error—to figure out the proper geometry for the faces.
This included rolling each prototype hundreds of times, and comparing the results for each side. Eventually, we found the right dimensions for each of the odd-sided dice to give a pretty random set of results upon rolling.
This included rolling each prototype hundreds of times, and comparing the results for each side. Eventually, we found the right dimensions for each of the odd-sided dice to give a pretty random set of results upon rolling. Note that dice that are fair by continuity can exhibit variations in fairness depending on the surface and way in which they’re rolled. In the end, it took over two years to develop the Denary Dice Set.
Because of their reputation for fairness, physicists have investigated the impact of how dice are rolled. A lot of people have wondered if it was possible to use a knowledge of the laws of physics to predict the outcome of a dice roll. And the amazing answer is: yes! One team theorized that by analyzing motion, probability, and a myriad of other factors, you would be able to use the classical laws of physical motion to precisely predict the movement of a thrown die for a set period of time. And when they calculated that with all factors accounted for, it turns out you can! And that the prediction is completely accurate for precisely 0.1 seconds after the throw.
So unless you can throw that die and have it land in less than 0.2 seconds, you’ve got a pretty random roll. But randomness isn’t just about the shape: there’s also the human element. Related studies have shown that people can very slightly influence the result, based on the way a die is rolled—adjusting the velocity, angle, and spin of a die as they roll it. This is known in casino circles as “dice control.” But we don’t recommend relying on gimmicks or tricks; as Diaconis once noted, “small changes in the initial conditions make for big changes on which side comes up.” Generally, randomness still wins out in the end: turns out it’s largely guaranteed because it is impossible to throw the same die in exactly the same way more than once.
We love click clack math rocks—or dice, as it turns out most people call them. There’s a lot of enjoyment in the games you can play, their relation to mathematical and physical laws of probability and motion, and the satisfying tactile sensation of holding and throwing a die.
