{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:V27GRQEJEMRCJLJRPNCVCASWDD","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"033dcfde196ceca485a2cfa5ac87f8cdf38ae9155656ea6e51dfbec41e401278","cross_cats_sorted":["math.RA","physics.chem-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.mtrl-sci","submitted_at":"2013-06-07T08:03:27Z","title_canon_sha256":"589dc1f55218e6eec5427960f3a7efdd238a095c85c493c821b1103592b4e072"},"schema_version":"1.0","source":{"id":"1306.1646","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1306.1646","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"arxiv_version","alias_value":"1306.1646v1","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.1646","created_at":"2026-05-18T03:21:32Z"},{"alias_kind":"pith_short_12","alias_value":"V27GRQEJEMRC","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"V27GRQEJEMRCJLJR","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"V27GRQEJ","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:f5fe4149f8f6153d8d066f4202033384ffa18e50aec953335327097001806c95","target":"graph","created_at":"2026-05-18T03:21:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper discusses the geometry of $k$D crystal cells given by $(k+1)$ points in a projective space $\\R^{n+1}$. We show how the concepts of barycentric and fractional (crystallographic) coordinates, reciprocal vectors and dual representation are related (and geometrically interpreted) in the projective geometric algebra $Cl(\\R^{n+1})$ (see Grassmann H., edited by Engel F., Die Ausdehnungslehre von 1844 und die Geom. Anal., vol. 1, part 1, Teubner: Leipzig, 1894.) and in the conformal algebra $Cl(\\R^{n+1,1})$. The crystallographic notions of $d$-spacing, phase angle, structure factors, condit","authors_text":"Eckhard Hitzer","cross_cats":["math.RA","physics.chem-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.mtrl-sci","submitted_at":"2013-06-07T08:03:27Z","title":"Crystal planes and reciprocal space in Clifford geometric algebra"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.1646","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2ec1fbd278f4e1283d1eb8c3786f4598339d04ccdaba99826a30af87072d0eb2","target":"record","created_at":"2026-05-18T03:21:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"033dcfde196ceca485a2cfa5ac87f8cdf38ae9155656ea6e51dfbec41e401278","cross_cats_sorted":["math.RA","physics.chem-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cond-mat.mtrl-sci","submitted_at":"2013-06-07T08:03:27Z","title_canon_sha256":"589dc1f55218e6eec5427960f3a7efdd238a095c85c493c821b1103592b4e072"},"schema_version":"1.0","source":{"id":"1306.1646","kind":"arxiv","version":1}},"canonical_sha256":"aebe68c089232224ad317b4551025618d041ae4fca3b2c32d1f149ef99077368","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aebe68c089232224ad317b4551025618d041ae4fca3b2c32d1f149ef99077368","first_computed_at":"2026-05-18T03:21:32.843331Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:32.843331Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"9j5udisBhQSt/BGHUtAirlEzpSZY+u+oTd7n8/3ZQZlRByYXt7VRxo6ohpnuKmW5keh821YMkOk9ZfTknroOBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:32.843849Z","signed_message":"canonical_sha256_bytes"},"source_id":"1306.1646","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ec1fbd278f4e1283d1eb8c3786f4598339d04ccdaba99826a30af87072d0eb2","sha256:f5fe4149f8f6153d8d066f4202033384ffa18e50aec953335327097001806c95"],"state_sha256":"9975e7821e007ed2e5e557cdc5d5381cd2cbd935b8b8fca1e29c90dc8f590548"}