Dense packings of the Platonic and Archimedean solids

Dense packings of polyhedra: Platonic and Archimedean solids.

Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical, and biological sciences. The preponderance of previous work has focused on spherical particles and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of nonoverlapping, nontiling polyhedra within an adaptive fundamental cell...

Maximum and minimum stable random packings of Platonic solids.

Motivated by the relation between particle shape and packing, we measure the volume fraction ϕ occupied by the Platonic solids which are a class of polyhedrons with congruent sides, vertices, and dihedral angles. Tetrahedron-, cube-, octahedron-, dodecahedron-, and icosahedron-shaped plastic dice were fluidized or mechanically vibrated to find stable random loose packing ϕ(rlp)=0.51,0.54,0.52,0...

The Platonic Solids

The tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. From a first glance, one immediately notices that the Platonic Solids exhibit remarkable symmetry. They are the only convex polyhedra for which the same same regular polygon is used for each face, and the same number of faces meet at each vertex. Their symmetries are aesthetically pleasing, like those of stones cu...

Maximally random jammed packings of Platonic solids: hyperuniform long-range correlations and isostaticity.

We generate maximally random jammed (MRJ) packings of the four nontiling Platonic solids (tetrahedra, octahedra, dodecahedra, and icosahedra) using the adaptive-shrinking-cell method [S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009)]. Such packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid. The MRJ packing fr...

Symmetry Groups of Platonic Solids