Path planning for the Platonic solids on prescribed grids by edge-rolling

Area of Science:

• Computational Geometry
• Robotics
• Discrete Mathematics

Background:

• The five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) have diverse applications in science, art, and mathematics.
• Path planning for these solids, particularly their rolling motion, has been proposed but largely unvalidated, with exceptions for cube-based puzzles.

Purpose of the Study:

• To develop and validate a path-planning algorithm for all five Platonic solids.
• To determine the shortest path for achieving a desired pose (position and orientation) through edge-rolling on specific grids.

Main Methods:

• A path-planning algorithm was developed utilizing the breadth-first search (BFS) algorithm.
• The algorithm operates on prescribed grids, employing edge-rolling mechanics for movement.
• Penrose tiling was selected for regular-pentagon grids due to its inherent five-fold symmetry.

Main Results:

• The BFS-based algorithm successfully generated shortest paths for Platonic solids to reach target poses.
• A key finding is that a tetrahedron can only achieve a single orientation for any given position on the Penrose tiling.
• The study validates path planning for Platonic solids beyond simple cubic dice puzzles.

Conclusions:

• The developed path-planning algorithm provides a validated method for determining movement strategies for Platonic solids.
• The unique orientation constraint of the tetrahedron on Penrose tilings highlights specific geometric limitations.
• This research contributes to computational geometry and robotics by addressing complex object path planning.