Want To Use Dragons As Dice? Now You Can, Thanks To Math

Why are dice cubes? Okay, nerds, yes, some are tetrahedra, or dodecahedra, or icosahedra, or whatever else your TTRPG calls for – but the point is, they’re all regular, symmetrical, perfectly balanced… in a word: boring. Why is that?

The obvious answer is “to make them fair”. Ever since we figured out that dice obey the laws of probability rather than the whims of the divine, it’s made sense for them to be all as standardized – and therefore as fair and balanced – as possible. But here’s the thing: you don’t need to be a Platonic or Aristotelian solid for that to be the case. Wouldn’t it be more fun if you could play a game of Pig by tossing a miniature pig? Or Dungeons and Dragons with actual dragons?

Well, thanks to a group of researchers hailing from Carnegie-Mellon University (CMU), Nvidia Research, and Adobe Research, you can. 

A diversity of dice

“We started from the idea of: ‘If you look at an object, can you tell its resting probabilities?’” CMU computer science professor Keenan Crane told New Scientist last week. Translated from math-speak, that’s a seemingly simple question: if you shake and roll a particular shape, what are the probabilities of the various positions it can land in?

Now, there’s an obvious way to go about calculating this: simply pay a bunch of grad students to toss various shapes around and record how they land. Unfortunately, this method is overly time-intensive, and unlikely to secure funding – so Crane and his colleagues went a different route. At first.

“We adopt the philosophy that, from a statistical perspective, the rest behavior of a rolling object is largely a function of its geometry and little else,” write the team. “In particular we assume that momentum is negligible, and that dynamics hence follow the gradient of a gravitational potential.” 

“This assumption enables us to analyze rolling via techniques from computational geometry, rather than those from physical simulation,” they explain. “Unlike a dynamic simulation, our method does not predict the final resting pose of an object from a given initial height, orientation, and momentum. Instead, it estimates the probability distribution over stable resting configurations, based purely on geometry.”

In short, they created a computer program that could run simulated rolls of these “dice” through geometry alone – not by recreating the shape of the die itself, but by mapping its corners and edges onto a sphere. That way, “we can just bypass the simulation entirely and understand these probabilities from a much simpler geometric picture,” Crane explained.

The theory dealt with, the team then “tossed” the dice using a rigid body simulator – dice, and to be honest most other objects, are “rigid bodies” in physics as they do not deform their shape. From the results, they were able to tweak the designs slightly – make a kitty nose slightly pointier; shift a bunny’s center of mass a little further south, that kind of thing – so as to get the desired range of probabilities.

Then, after modeling seven of these peculiarly shaped dice, the team went traditional: they 3D-printed the designs and tested them manually. That’s right: they rolled them. Up to 1,000 times for each die.

And guess what? It worked! “Even though the experiments have a lot of momentum and the die go through lot of bounces in each rolling instance, we observe that our purely geometric algorithm is a good prediction of the resting probabilities,” the team reported, with real-world experiments producing results that were within 3 or 4 percent of the probabilities they had predicted.

“If we were able to time travel back and gamble in ancient Greece, we might be able to make a lot of money (in BC-adjusted dollar),” Crane told Ars Technica. Indeed, one of the dice they made was a talus-shape, similar to the sheep- or goat-knucklebone astragali used for gambling and divination in the ancient world – back then, “people [would] bet on different outcomes, using past experience to build up their intuition about which sides are most likely,” he explained, but the new technique “can do this much more directly: By just looking at the shape, it can directly give you some pretty accurate probabilities.”

Are they really dice, though?

Now, we know what you’re thinking: what about that 3 or 4 percent difference, though? If the probabilities aren’t even, then these aren’t fair dice, right?

Well, you’re not wrong – but, Crane believes, you’re also being a little bit picky for someone who’s just been told they can use armadillos as dice if they want. 

“On the one hand, [critics] are absolutely correct that our model does not make perfect predictions from the perspective of idealized geometry and physics,” he told Ars Technica. “On the other hand, if your goal is to literally make dice for tabletop games, it's perfectly reasonable to make these kinds of approximations.”

Think about any given game, he pointed out: chances are, you’re not rolling the dice all that many times. Even for a game like Settlers of Catan, well-known for taking hours to play, “I might roll the dice only about 100 times in the whole game,” Crane said. “Even if the dice are actually fair, the distribution of rolls I see throughout this short game might vary quite a bit from the idealized distribution – I might just happen to roll a lot more nines that game than usual.” 

“So, in a real game, there may not be a huge practical difference between using fair dice and dice that have a small bias,” he said. “The signal-to-noise ratio is so low that the practical effect over a short game is pretty small.”

Are the dice perfectly fair? No – but neither are normal dice, technically. If that’s what you’re after, go to Vegas, or enlist the services of a particularly sophisticated random number generator.

If, however, you want to use a bunny as a die in your next game of Monopoly, then this is the paper that can help.

“Not every initial choice will lead to a solution, or a solution might simply not exist,” the team caution in the paper. “It might be hard to find a good enough solution. That being said, our method is also far superior to the alternative of trying to design dice with target probabilities by hand.”

“To our knowledge, no previous solutions are known for many of the examples shown above, nor did we find algorithms for designing such dice,” they write. “The evaluation and differentiation steps of our method are both extremely fast and make an easy tool to investigate many design choices.”

The study will be published in the journal ACM Transactions on Graphics, and can be viewed here.