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For $d=2$, triangular $k$-reptiles exist for all $k$ of the form $a^2, 3a^2$ or $a^2 + b^2$ and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only $k$-reptile simplices that are known for $d \\ge 3$, have $k = m^d$, where $m$ is a positive integer. We substantially simplify the proof by Matou\\v{s}ek and the second author that for $d=3$, $k$-reptile tetra"},"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":"1602.04668","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-02-15T13:24:37Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"090b616d9b8f62ea903dc2a2ace1f5c4804bda5836b62ed51d69c0ea956feb14","abstract_canon_sha256":"7a7fd866905c9f07df4f1f5169d425ce6789b994b4eee129ea938946e2795e31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:17.580679Z","signature_b64":"PJvsQj4I9OHH/T/EmjMEKvmyJD9KO5hwnGaL7FIviCyKVQ3pSrVUjMyx412GphVgrCklWEdfxQ3RJEqeRf4MDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c9f6fbbddf75ee1bf5c6b98563ce01ff4ed199d2d63f8c1532cc636fbfde0770","last_reissued_at":"2026-05-18T00:40:17.580081Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:17.580081Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the nonexistence of $k$-reptile simplices in $\\mathbb R^3$ and $\\mathbb R^4$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Jan Kyn\\v{c}l, Zuzana Pat\\'akov\\'a","submitted_at":"2016-02-15T13:24:37Z","abstract_excerpt":"A $d$-dimensional simplex $S$ is called a $k$-reptile (or a $k$-reptile simplex) if it can be tiled by $k$ simplices with disjoint interiors that are all mutually congruent and similar to $S$. 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