{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:KAQIATYVSDNFTDUQYPMVGSEDGZ","short_pith_number":"pith:KAQIATYV","canonical_record":{"source":{"id":"math-ph/0606028","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2006-06-09T04:26:48Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"d8894efbd5c604714f93c468f51d775a95579118215477008addca97c8240114","abstract_canon_sha256":"1a0884d288c2748361947e2a1c71450baf436fbb929d43142abf39e7b6bab6f5"},"schema_version":"1.0"},"canonical_sha256":"5020804f1590da598e90c3d9534883364dcfe243cbb256303b8498b20c5a01aa","source":{"kind":"arxiv","id":"math-ph/0606028","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0606028","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0606028v1","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0606028","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"pith_short_12","alias_value":"KAQIATYVSDNF","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"KAQIATYVSDNFTDUQ","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"KAQIATYV","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:KAQIATYVSDNFTDUQYPMVGSEDGZ","target":"record","payload":{"canonical_record":{"source":{"id":"math-ph/0606028","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math-ph","submitted_at":"2006-06-09T04:26:48Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"d8894efbd5c604714f93c468f51d775a95579118215477008addca97c8240114","abstract_canon_sha256":"1a0884d288c2748361947e2a1c71450baf436fbb929d43142abf39e7b6bab6f5"},"schema_version":"1.0"},"canonical_sha256":"5020804f1590da598e90c3d9534883364dcfe243cbb256303b8498b20c5a01aa","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:07.386980Z","signature_b64":"cqRedJaXj7tMNtgBdVoFM/N0cc+vIgJFyFTNGmq5vmOVbFuXcRPaYVJJNRjQUde2TXfdr2l7tKNsKWqycNJ2CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5020804f1590da598e90c3d9534883364dcfe243cbb256303b8498b20c5a01aa","last_reissued_at":"2026-05-18T04:13:07.386386Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:07.386386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math-ph/0606028","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:13:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wi2ubDzKhnNWoeGZthXgRfvIFXHiJMpa5zazoku5HUAUJw8Ljh6Jgf9ygL/Vn6kIG8Z8A1Q6nog6R3eNrkG0Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T13:24:09.718441Z"},"content_sha256":"747e58aef86925231c11dba7ad426c91b821f434b0c16bdd62a38d44eaede49f","schema_version":"1.0","event_id":"sha256:747e58aef86925231c11dba7ad426c91b821f434b0c16bdd62a38d44eaede49f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:KAQIATYVSDNFTDUQYPMVGSEDGZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions","license":"","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Helen Au-Yang, Jacques H.H. Perk","submitted_at":"2006-06-09T04:26:48Z","abstract_excerpt":"We show that generalized Penrose tilings can be obtained by the projection of a cut plane of a 5-dimensional lattice into two dimensions, while 3-d quasiperiodic lattices with overlapping unit cells are its projections into 3d. The frequencies of all possible vertex types in the generalized Penrose tilings, and the frequencies of all possible types of overlapping 3-d unit cells are also given here. The generalized Penrose tilings are found to be nonconvertable to kite and dart patterns, nor can they be described by the overlapping decagons of Gummelt."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0606028","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:13:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7r42Gz7ClpOY+vVJaTD9zYe/ANA3di6qB/YDNjLCNcId1+wXPiz70InDj/yNutMLIvxc296NGf8FEwsnlUVfAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T13:24:09.718792Z"},"content_sha256":"625cb2dc42cfecfcca5b3a50bd06a8752e6f053f573387619ef16b3de0082f63","schema_version":"1.0","event_id":"sha256:625cb2dc42cfecfcca5b3a50bd06a8752e6f053f573387619ef16b3de0082f63"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KAQIATYVSDNFTDUQYPMVGSEDGZ/bundle.json","state_url":"https://pith.science/pith/KAQIATYVSDNFTDUQYPMVGSEDGZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KAQIATYVSDNFTDUQYPMVGSEDGZ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T13:24:09Z","links":{"resolver":"https://pith.science/pith/KAQIATYVSDNFTDUQYPMVGSEDGZ","bundle":"https://pith.science/pith/KAQIATYVSDNFTDUQYPMVGSEDGZ/bundle.json","state":"https://pith.science/pith/KAQIATYVSDNFTDUQYPMVGSEDGZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KAQIATYVSDNFTDUQYPMVGSEDGZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:KAQIATYVSDNFTDUQYPMVGSEDGZ","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":"1a0884d288c2748361947e2a1c71450baf436fbb929d43142abf39e7b6bab6f5","cross_cats_sorted":["math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2006-06-09T04:26:48Z","title_canon_sha256":"d8894efbd5c604714f93c468f51d775a95579118215477008addca97c8240114"},"schema_version":"1.0","source":{"id":"math-ph/0606028","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math-ph/0606028","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"arxiv_version","alias_value":"math-ph/0606028v1","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math-ph/0606028","created_at":"2026-05-18T04:13:07Z"},{"alias_kind":"pith_short_12","alias_value":"KAQIATYVSDNF","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"KAQIATYVSDNFTDUQ","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"KAQIATYV","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:625cb2dc42cfecfcca5b3a50bd06a8752e6f053f573387619ef16b3de0082f63","target":"graph","created_at":"2026-05-18T04:13:07Z","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":"We show that generalized Penrose tilings can be obtained by the projection of a cut plane of a 5-dimensional lattice into two dimensions, while 3-d quasiperiodic lattices with overlapping unit cells are its projections into 3d. The frequencies of all possible vertex types in the generalized Penrose tilings, and the frequencies of all possible types of overlapping 3-d unit cells are also given here. The generalized Penrose tilings are found to be nonconvertable to kite and dart patterns, nor can they be described by the overlapping decagons of Gummelt.","authors_text":"Helen Au-Yang, Jacques H.H. Perk","cross_cats":["math.MP"],"headline":"","license":"","primary_cat":"math-ph","submitted_at":"2006-06-09T04:26:48Z","title":"Quasicrystals: Projections of 5-d Lattice into 2 and 3 Dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math-ph/0606028","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":"sha256:747e58aef86925231c11dba7ad426c91b821f434b0c16bdd62a38d44eaede49f","target":"record","created_at":"2026-05-18T04:13:07Z","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":"1a0884d288c2748361947e2a1c71450baf436fbb929d43142abf39e7b6bab6f5","cross_cats_sorted":["math.MP"],"license":"","primary_cat":"math-ph","submitted_at":"2006-06-09T04:26:48Z","title_canon_sha256":"d8894efbd5c604714f93c468f51d775a95579118215477008addca97c8240114"},"schema_version":"1.0","source":{"id":"math-ph/0606028","kind":"arxiv","version":1}},"canonical_sha256":"5020804f1590da598e90c3d9534883364dcfe243cbb256303b8498b20c5a01aa","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5020804f1590da598e90c3d9534883364dcfe243cbb256303b8498b20c5a01aa","first_computed_at":"2026-05-18T04:13:07.386386Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:13:07.386386Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cqRedJaXj7tMNtgBdVoFM/N0cc+vIgJFyFTNGmq5vmOVbFuXcRPaYVJJNRjQUde2TXfdr2l7tKNsKWqycNJ2CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:13:07.386980Z","signed_message":"canonical_sha256_bytes"},"source_id":"math-ph/0606028","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:747e58aef86925231c11dba7ad426c91b821f434b0c16bdd62a38d44eaede49f","sha256:625cb2dc42cfecfcca5b3a50bd06a8752e6f053f573387619ef16b3de0082f63"],"state_sha256":"eb8518f023261922a8f0e7eea518f10eac5fb9c190eef90dcc1f0a0b37a2615d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ci8mTjGT9W7YtJHMXzhuC6WvWn+IH82Bed+5yfvjsoO7SKPX31iYdxe27ZWNK3sbllSRcU9On50b89CqzUy+CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T13:24:09.720936Z","bundle_sha256":"cc2e9809102aca081d6d81d4835f77033b937891e6118764ab72a1f6ce9c2ca4"}}