Fig 1.
Five models of the Platonic solids and their unfolding geometry.
(a) Cube. (b) Tetrahedron. (c) Octahedron. (d) Icosahedron. (e) Dodecahedron.
Table 1.
Properties of polyhedron.
Fig 2.
Three different patterns of grid of the plane for the Platonic solids. (a) A square grid for the cube with 4 edge-rolling directions. (b) A triangular grid for the tetrahedron, octahedron, and icosahedron with 3 edge-rolling directions. (c) A pentagon grid using Penrose tiling for the dodecahedron with 5 edge-rolling directions.
Fig 3.
Patterns of the regular pentagon tiling.
(a), (b) Two patterns of pentagon tiling from Durer including a five-fold nucleus that is expanded by multiple twins of five-fold symmetry (reconstructed from [6]). (c)-(e) Three patterns of pentagon tiling in art proposed by Caris (reconstructed from [7]). (f) Penrose tiling with five-fold symmetry generated by attaching multiple groups of a pentagon to the initial one (reconstructed from [8]).
Fig 4.
Substitution rules for Penrose tiling (reconstructed from [9]).
(a) A pentagon is partially filled by 6 pentagons. (b) A pentacle is partially filled by 5 pentagons. (c) A half-pentacle is partially filled by 3 pentagons. (d) A rhombus is partially filled by 1 pentagon.
Table 2.
Geometrical parameters of the Platonic solids with inradius (ri), midradius (rm), circumradius (R), dihedral angles (α) and rolling angles (β).
In this table, all the edges of the Platonic solids have the same unit length (l = 1).
Fig 5.
Based on the BFS method, this algorithm starts from an initial pose represented by the Node I. The branches represent the rolling directions for each iteration. If some nodes are of the same pose, they are merged to reduce the search space (the green node). The algorithm stops when the desired pose, represented by the Node D, is reached and a shortest path is generated (colored in red).
Fig 6.
Platonic solids rolling through their edge MN with different rotation angles shown in Table 2.
A body frame (O − e1 e2 e3) is fixed at the center of each solid (left). After edge-rolling, each Platonic solid reaches a new pose, where the red curve represents the center’s trajectory of rolling. Path-planning results for the Platonic solids on a plane shown in the right side. (a) Tetrahedron. (b) Octahedron. (c) Icosahedron. (d) Cube. (e) Dodecahedron.
Fig 7.
Symmetric properties of a tetrahedron.
(a) A 3D view of edge-rolling 6 times around the vertex O where the red curve indicates the closed-path of rolling motion. (b) A top view of (a). The surface Sct of the tetrahedron is in contact with the plane in different cell after sequential rolling through the edges of NO, PO, and MO to reach the same pose. (c) The tetrahedron reaches only one orientation for each cell through edge-rolling.