Mathematically Fair N-Sided Dice via Equal Solid Angle Design
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Solution Overview
Problem
Existing polyhedral dice do not land with equal probability on any side, as the probability of landing varies significantly from one side to another due to their irregular shapes and spatial relationships.
Innovation Solution
Design and manufacturing methods for mathematically fair N-sided dice involve forming specific geometric shapes such as square pyramids, triangular prisms, pentagonal prisms, elongated double pyramids, and other polyhedrons, where each face subtends an equal solid angle from the center of mass, ensuring uniform probability of landing on any side.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If irregular polyhedral shapes are used for dice, then the number of sides can be varied arbitrarily, but the probability of landing on each side becomes uneven
Solution Approach 1:
The patent applies local quality by carefully controlling the spatial relationship between the center of mass and each face of the polyhedron. Specifically, it positions the center of mass at a distance from each face such that the solid angle subtended by each face is equal, ensuring uniform landing probability while allowing arbitrary numbers of sides
Solution Approach 2:
The patent changes geometric parameters (center of mass position, face distances, solid angles) to achieve mathematical fairness. By adjusting these parameters, the die can have any number of sides while maintaining equal landing probability on each face
2Reliability
If regular polyhedral shapes are used for dice, then equal landing probability is achieved, but the number of sides is limited to specific values (4, 6, 8, 12, 20)
Solution Approach 1:
The patent uses asymmetric irregular polyhedral shapes that are not regular polyhedra. By carefully designing asymmetric geometries where each face subtends an equal solid angle from the center of mass, it achieves both mathematical fairness and arbitrary numbers of sides
Solution Approach 2:
The patent moves beyond the limited set of regular polyhedra by introducing a new dimensional parameter - the position of the center of mass relative to each face. This allows creating irregular shapes with equal solid angles, enabling any number of sides while maintaining fairness
3Ease of manufacture
If the center of mass is positioned centrally in a polyhedron, then manufacturing is simplified, but faces at different distances from the center have different landing probabilities
Solution Approach 1:
The patent changes the position parameter of the center of mass from the conventional central position to a specific offset position. By calculating the precise distance from the center of mass to each face, it ensures equal solid angles are subtended by all faces, achieving uniform landing probability
Data Source
AI summary
Methods and systems for designing and manufacturing a mathematically fair N-sided die are disclosed. An example method can comprise selecting an irregular polyhedron to serve as a die, and determining a center of mass of the selected irregular polyhedron die. In an aspect, a size of each face of the irregular polyhedron can be selected such that a solid angle subtended by each face from the center of mass of the selected irregular polyhedron can be equal. In an aspect a system comprising an injection molding apparatus can be used to form the die.