abstract bibtex

Dense particle packings have served as useful models of the structures
of liquid, glassy and crystalline states of matter(1-4), granular
media(3,5), heterogeneous materials(3) and biological systems(6-8).
Probing the symmetries and other mathematical properties of the densest
packings is a problem of interest in discrete geometry and number
theory(9-11). Previous work has focused mainly on spherical
particles-very little is known about dense polyhedral packings. Here we
formulate the generation of dense packings of polyhedra as an
optimization problem, using an adaptive fundamental cell subject to
periodic boundary conditions (we term this the `adaptive shrinking cell'
scheme). Using a variety of multi-particle initial configurations, we
find the densest known packings of the four non-tiling Platonic solids
(the tetrahedron, octahedron, dodecahedron and icosahedron) in
three-dimensional Euclidean space. The densities are 0.782..., 0.947...,
0.904... and 0.836..., respectively. Unlike the densest tetrahedral
packing, which must not be a Bravais lattice packing, the densest
packings of the other non-tiling Platonic solids that we obtain are
their previously known optimal (Bravais) lattice packings. Combining our
simulation results with derived rigorous upper bounds and theoretical
arguments leads us to the conjecture that the densest packings of the
Platonic and Archimedean solids with central symmetry are given by their
corresponding densest lattice packings. This is the analogue of Kepler's
sphere conjecture for these solids.

@article{ title = {Dense packings of the Platonic and Archimedean solids}, type = {article}, year = {2009}, identifiers = {[object Object]}, pages = {876-U109}, volume = {460}, month = {8}, publisher = {NATURE PUBLISHING GROUP}, city = {MACMILLAN BUILDING, 4 CRINAN ST, LONDON N1 9XW, ENGLAND}, id = {a7801ecd-f6cf-3f7f-85c0-d2cb7c7ed2f0}, created = {2015-12-14T19:51:29.000Z}, file_attached = {false}, profile_id = {3187ec9d-0fcc-3ba2-91e0-3075df9b18c3}, group_id = {d75e47fd-ff52-3a4b-bf1e-6ebc7e454352}, last_modified = {2017-03-14T12:30:08.401Z}, read = {false}, starred = {false}, authored = {false}, confirmed = {true}, hidden = {false}, citation_key = {ISI:000268938300034}, source_type = {article}, user_context = {Article}, private_publication = {false}, abstract = {Dense particle packings have served as useful models of the structures of liquid, glassy and crystalline states of matter(1-4), granular media(3,5), heterogeneous materials(3) and biological systems(6-8). Probing the symmetries and other mathematical properties of the densest packings is a problem of interest in discrete geometry and number theory(9-11). Previous work has focused mainly on spherical particles-very little is known about dense polyhedral packings. Here we formulate the generation of dense packings of polyhedra as an optimization problem, using an adaptive fundamental cell subject to periodic boundary conditions (we term this the `adaptive shrinking cell' scheme). Using a variety of multi-particle initial configurations, we find the densest known packings of the four non-tiling Platonic solids (the tetrahedron, octahedron, dodecahedron and icosahedron) in three-dimensional Euclidean space. The densities are 0.782..., 0.947..., 0.904... and 0.836..., respectively. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Combining our simulation results with derived rigorous upper bounds and theoretical arguments leads us to the conjecture that the densest packings of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This is the analogue of Kepler's sphere conjecture for these solids.}, bibtype = {article}, author = {Torquato, S and Jiao, Y}, journal = {NATURE}, number = {7257} }

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