Designing Fair 3D-Printed Dice with Computational Geometry
What if your next game of Dungeons & Dragons or Settlers of Catan could feature dice shaped like dragons, armadillos, or even kittens—all while delivering true randomness? A team led by Hossein Baktash (Carnegie Mellon University) and Keenan Crane has developed an algorithmic framework to predict and engineer the rest probabilities of arbitrarily shaped rigid bodies, paving the way for exotic, yet fair, 3D-printed dice.
Introduction to Irregular Dice
Traditional dice rely on regular polyhedra, such as D6 cubes or D20 icosahedra, whose symmetry guarantees uniform face probabilities. But when you stray into whimsical shapes—kittens, knucklebones, dragons—geometry rules everything. The researchers published their method in ACM Transactions on Graphics, demonstrating that by mapping an object’s convex hull onto a unit sphere and computing spherical Voronoi regions, one can analytically derive the probability a tossed object lands on each face.
Core Algorithmic Foundations
- Convex Hull Extraction: Generate a triangular mesh and compute its convex hull using CGAL libraries.
- Spherical Mapping: Project face normals onto the unit sphere to form a spherical polygon layout.
- Spherical Voronoi Computation: Partition the sphere into regions whose areas correspond to rest-state probabilities.
- Numerical Integration: Calculate each region’s solid angle with an O(n log n) GPU-accelerated routine.
The team implemented a CUDA version that can compute probabilities for complex models in under 3 ms on modern GPUs, and they’re currently collaborating with Adobe Research to integrate this into upcoming versions of Blender and Adobe Substance.
3D Printing Advances: Materials, Tolerances, and Calibration
Producing these dice with precision matters. The researchers trialed two manufacturing methods:
- SLA Resin Printing at 25–50 µm resolution for smooth surfaces and tight geometric fidelity.
- FDM with ABS/PLA at 100–200 µm layer height for robustness and affordability.
They also experimented with embedded weighting using tungsten inserts to adjust center-of-mass tolerances to ±0.2 mm. This ensures that material density variations do not introduce unwanted bias.
Experimental Validation
Seven shapes—including a D6+D6 double die, binomial coins (two- and three-flip simulators), and three three-outcome dice (kitten, armadillo, dragon)—were 3D printed and tossed 1,000 times each. Observed rest frequencies deviated by less than 4% from predictions, well within statistical expectations for low-momentum scenarios.
“Our low-momentum assumption holds in many practical contexts—from dice on a gaming table to parts settling in additive-manufacturing powder beds,” explains Keenan Crane. “By baking orientation bias directly into geometry, we open up new design possibilities.”
Manufacturing Considerations and Quality Control
Scaling this from lab to market involves:
- Quality assurance via 3D scanning to verify mesh fidelity.
- Automatic tolerance compensation in slicing software to maintain center-of-gravity alignment.
- Standardized test protocols (1000+ tosses) for batch certification.
Applications Beyond Tabletop Gaming
While gamers gain novel accessories, industries stand to benefit:
- Underwater Robotics: Low-momentum calibration helps subsea modules orient upon deployment.
- Aerospace: Satellite tumbling prediction using convex geometry rather than costly dynamical simulations.
- Geotechnical Engineering: Modeling how individual soil grains or ballast stones settle and compact.
Expert Opinions and Future Directions
Oklahoma State University mathematician Henry Segerman notes that the method omits friction and bounce, yet agrees it provides “a highly efficient baseline model”. Meanwhile, NVIDIA Research is exploring Tensor Core acceleration for real-time in-game simulations of falling objects.
Upcoming releases aim to add:
- Multi-body interactions for dice colliding side by side.
- Stochastic dynamics to incorporate bounce and slip factors.
- Integration with popular game engines (Unity, Unreal) for live physics previews.
Conclusion
By merging computational geometry, high-precision 3D printing, and GPU acceleration, this research redefines randomness. Whether you’re rolling a dragon or designing the next robotics component, understanding how shape dictates behavior opens a new frontier in both play and practical engineering.