{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2009:ANUKNTRKHMA3YZBVXDQVCQTBB3","short_pith_number":"pith:ANUKNTRK","schema_version":"1.0","canonical_sha256":"0368a6ce2a3b01bc6435b8e15142610edee44bbe02a73fded7b20624a82958f3","source":{"kind":"arxiv","id":"0909.0940","version":3},"attestation_state":"computed","paper":{"title":"Dense Packings of Polyhedra: Platonic and Archimedean Solids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"S. Torquato, Y. Jiao","submitted_at":"2009-09-04T19:43:51Z","abstract_excerpt":"We formulate the problem of generating dense packings of nonoverlapping, non-tiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space R3 , except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodeca"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0909.0940","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2009-09-04T19:43:51Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"5028a262217c2f02822a36681e2eb25e0152fdd97da3fa0ecea022fd770837d0","abstract_canon_sha256":"8a73c4091e82b0d0fa711890fa62d06c3941186c4a8f1533f5bd7ca7fc497ea6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:11:59.617511Z","signature_b64":"cSopaux0pcuXvA47CS2tKoXU4MveFlY71GcR4//5ocsVZ44GJms2tnC6I0XTnEbCZTQL8gkuxAiI9JRHPKOmAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0368a6ce2a3b01bc6435b8e15142610edee44bbe02a73fded7b20624a82958f3","last_reissued_at":"2026-05-18T02:11:59.616632Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:11:59.616632Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dense Packings of Polyhedra: Platonic and Archimedean Solids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"S. Torquato, Y. Jiao","submitted_at":"2009-09-04T19:43:51Z","abstract_excerpt":"We formulate the problem of generating dense packings of nonoverlapping, non-tiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space R3 , except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodeca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0909.0940","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"0909.0940","created_at":"2026-05-18T02:11:59.616784+00:00"},{"alias_kind":"arxiv_version","alias_value":"0909.0940v3","created_at":"2026-05-18T02:11:59.616784+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0909.0940","created_at":"2026-05-18T02:11:59.616784+00:00"},{"alias_kind":"pith_short_12","alias_value":"ANUKNTRKHMA3","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_16","alias_value":"ANUKNTRKHMA3YZBV","created_at":"2026-05-18T12:25:58.837520+00:00"},{"alias_kind":"pith_short_8","alias_value":"ANUKNTRK","created_at":"2026-05-18T12:25:58.837520+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3","json":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3.json","graph_json":"https://pith.science/api/pith-number/ANUKNTRKHMA3YZBVXDQVCQTBB3/graph.json","events_json":"https://pith.science/api/pith-number/ANUKNTRKHMA3YZBVXDQVCQTBB3/events.json","paper":"https://pith.science/paper/ANUKNTRK"},"agent_actions":{"view_html":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3","download_json":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3.json","view_paper":"https://pith.science/paper/ANUKNTRK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0909.0940&json=true","fetch_graph":"https://pith.science/api/pith-number/ANUKNTRKHMA3YZBVXDQVCQTBB3/graph.json","fetch_events":"https://pith.science/api/pith-number/ANUKNTRKHMA3YZBVXDQVCQTBB3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3/action/storage_attestation","attest_author":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3/action/author_attestation","sign_citation":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3/action/citation_signature","submit_replication":"https://pith.science/pith/ANUKNTRKHMA3YZBVXDQVCQTBB3/action/replication_record"}},"created_at":"2026-05-18T02:11:59.616784+00:00","updated_at":"2026-05-18T02:11:59.616784+00:00"}