@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}
}

{"_id":"mABycxPL9qkbda9Dq","bibbaseid":"torquato-jiao-densepackingsoftheplatonicandarchimedeansolids-2009","downloads":0,"creationDate":"2015-12-14T17:20:22.824Z","title":"Dense packings of the Platonic and Archimedean solids","author_short":["Torquato, S.","Jiao, Y."],"year":2009,"bibtype":"article","biburl":null,"bibdata":{"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\nof liquid, glassy and crystalline states of matter(1-4), granular\nmedia(3,5), heterogeneous materials(3) and biological systems(6-8).\nProbing the symmetries and other mathematical properties of the densest\npackings is a problem of interest in discrete geometry and number\ntheory(9-11). Previous work has focused mainly on spherical\nparticles-very little is known about dense polyhedral packings. Here we\nformulate the generation of dense packings of polyhedra as an\noptimization problem, using an adaptive fundamental cell subject to\nperiodic boundary conditions (we term this the `adaptive shrinking cell'\nscheme). Using a variety of multi-particle initial configurations, we\nfind the densest known packings of the four non-tiling Platonic solids\n(the tetrahedron, octahedron, dodecahedron and icosahedron) in\nthree-dimensional Euclidean space. The densities are 0.782..., 0.947...,\n0.904... and 0.836..., respectively. Unlike the densest tetrahedral\npacking, which must not be a Bravais lattice packing, the densest\npackings of the other non-tiling Platonic solids that we obtain are\ntheir previously known optimal (Bravais) lattice packings. Combining our\nsimulation results with derived rigorous upper bounds and theoretical\narguments leads us to the conjecture that the densest packings of the\nPlatonic and Archimedean solids with central symmetry are given by their\ncorresponding densest lattice packings. This is the analogue of Kepler's\nsphere conjecture for these solids.","bibtype":"article","author":"Torquato, S and Jiao, Y","journal":"NATURE","number":"7257","bibtex":"@article{\n title = {Dense packings of the Platonic and Archimedean solids},\n type = {article},\n year = {2009},\n identifiers = {[object Object]},\n pages = {876-U109},\n volume = {460},\n month = {8},\n publisher = {NATURE PUBLISHING GROUP},\n city = {MACMILLAN BUILDING, 4 CRINAN ST, LONDON N1 9XW, ENGLAND},\n id = {a7801ecd-f6cf-3f7f-85c0-d2cb7c7ed2f0},\n created = {2015-12-14T19:51:29.000Z},\n file_attached = {false},\n profile_id = {3187ec9d-0fcc-3ba2-91e0-3075df9b18c3},\n group_id = {d75e47fd-ff52-3a4b-bf1e-6ebc7e454352},\n last_modified = {2017-03-14T12:30:08.401Z},\n read = {false},\n starred = {false},\n authored = {false},\n confirmed = {true},\n hidden = {false},\n citation_key = {ISI:000268938300034},\n source_type = {article},\n user_context = {Article},\n private_publication = {false},\n abstract = {Dense particle packings have served as useful models of the structures\nof liquid, glassy and crystalline states of matter(1-4), granular\nmedia(3,5), heterogeneous materials(3) and biological systems(6-8).\nProbing the symmetries and other mathematical properties of the densest\npackings is a problem of interest in discrete geometry and number\ntheory(9-11). Previous work has focused mainly on spherical\nparticles-very little is known about dense polyhedral packings. Here we\nformulate the generation of dense packings of polyhedra as an\noptimization problem, using an adaptive fundamental cell subject to\nperiodic boundary conditions (we term this the `adaptive shrinking cell'\nscheme). Using a variety of multi-particle initial configurations, we\nfind the densest known packings of the four non-tiling Platonic solids\n(the tetrahedron, octahedron, dodecahedron and icosahedron) in\nthree-dimensional Euclidean space. The densities are 0.782..., 0.947...,\n0.904... and 0.836..., respectively. Unlike the densest tetrahedral\npacking, which must not be a Bravais lattice packing, the densest\npackings of the other non-tiling Platonic solids that we obtain are\ntheir previously known optimal (Bravais) lattice packings. Combining our\nsimulation results with derived rigorous upper bounds and theoretical\narguments leads us to the conjecture that the densest packings of the\nPlatonic and Archimedean solids with central symmetry are given by their\ncorresponding densest lattice packings. This is the analogue of Kepler's\nsphere conjecture for these solids.},\n bibtype = {article},\n author = {Torquato, S and Jiao, Y},\n journal = {NATURE},\n number = {7257}\n}","author_short":["Torquato, S.","Jiao, Y."],"bibbaseid":"torquato-jiao-densepackingsoftheplatonicandarchimedeansolids-2009","role":"author","urls":{},"downloads":0},"search_terms":["dense","packings","platonic","archimedean","solids","torquato","jiao"],"keywords":[],"authorIDs":[]}