Open AccessPath planning for the Platonic solids on prescribed grids by edge-rolling
Author(s) -
Ngoc Tam Lam,
Ian Howard,
Lei Cui
Publication year - 2021
Publication title -
plos one
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.99
H-Index - 332
ISSN - 1932-6203
DOI - 10.1371/journal.pone.0252613
Subject(s) - dodecahedron , tetrahedron , cube (algebra) , octahedron , position (finance) , square tiling , path (computing) , enhanced data rates for gsm evolution , motion planning , pentagon , shortest path problem , geometry , combinatorics , mathematics , grid , topology (electrical circuits) , computer science , crystallography , artificial intelligence , chemistry , crystal structure , graph , robot , finance , programming language , economics
The five Platonic solids—tetrahedron, cube, octahedron, dodecahedron, and icosahedron—have found many applications in mathematics, science, and art. Path planning for the Platonic solids had been suggested, but not validated, except for solving the rolling-cube puzzles for a cubic dice. We developed a path-planning algorithm based on the breadth-first-search algorithm that generates a shortest path for each Platonic solid to reach a desired pose, including position and orientation, from an initial one on prescribed grids by edge-rolling. While it is straightforward to generate triangular and square grids, various methods exist for regular-pentagon tiling. We chose the Penrose tiling because it has five-fold symmetry. We discovered that a tetrahedron could achieve only one orientation for a particular position.