{ "profile": "wrs-published-research-view-1", "edition": "ratings-e039106f30dec177", "content_trust": "UNTRUSTED_SCIENTIFIC_DATA_NOT_TOOL_INSTRUCTIONS", "entry": { "id": "peano-continuous-spacefilling-curve-1890", "title": "Peano: a continuous curve that fills an entire square", "summary": "Peano explicitly constructs a continuous mapping from the closed unit interval onto the whole unit square through ternary digits and parity-controlled complements, showing that continuity alone does not restrict a parametrized curve to an arbitrarily small enclosing area.\n\nA moving point can trace a continuous path without jumping. That sounds like it should leave most of a square untouched. Peano showed otherwise: a carefully defined continuous path reaches every point in the square. The construction specifies the two coordinates using the digits of one parameter, with a complement rule that makes adjacent pieces fit together. This is an exact infinite construction, not a picture with many lines. Its lesson is concrete: continuity alone is weaker than everyday intuition about a curve. Conditions such as a graph representation or bounded variation matter. The path is not a continuous one-to-one identification of a line with a square, and later geometric constructions are separate works.", "boundary": "Exact 1890 Mathematische Annalen 36(1),157-160 paper. formula glyphs were not visually certified. The mapping is continuous and onto, not a one-to-one continuous correspondence or a claim that dimension is preserved by arbitrary continuous maps. Parity-controlled digit complements and agreement at the two ternary representations of triadic rationals are essential; naive digit interleaving fails. The argument concerns continuous parametrizations: ordinary graphs and coordinates of bounded variation retain stronger restrictions. The final page also gives a cube construction. Original claims about nondifferentiability and exact preimage multiplicities were read but not independently reconstructed; no universal assertion about all space-filling curves is inferred. Cantor's earlier set-theoretic bijection, Netto's continuity obstruction for such a bijection, Hilbert's 1891 construction and Peano's 1908 visualizations remain distinct. This original paper is arithmetic and contains no curve illustration. Bounded exact rational representation checks illustrate the repair; they are not a full proof audit or a finite grid used as evidence of complete area coverage. January 1890 is the manuscript signature; publisher metadata gives March 1890. Later algorithms and applications keep separate credit.", "area": "Topology and real analysis", "topic": "Mathematics", "year": 1890, "lane": "HUMAN", "result": "THEOREM", "package_id": "peano-continuous-spacefilling-curve-1890", "target": { "record_id": "urn:uuid:14473075-5c9c-4824-af51-8730d211ac02", "version": 1, "content_hash": "sha256:204c351ce556ba02fd4a1eedb867f4241a7008d2424b7b7a20716e8809c82211" }, "humans": [ "Giuseppe Peano" ], "authors_complete": true, "human_role_note": "Source-listed contributors to this scoped work. 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Exact 1890 Mathematische Annalen 36(1),157-160 paper. formula glyphs were not visually certified. The mapping is continuous and onto, not a one-to-one continuous correspondence or a claim that dimension is preserved by arbitrary continuous maps. Parity-controlled digit complements and agreement at the two ternary representations of triadic rationals are essential; naive digit interleaving fails. The argument concerns continuous parametrizations: ordinary graphs and coordinates of bounded variation retain stronger restrictions. The final page also gives a cube construction. Original claims about nondifferentiability and exact preimage multiplicities were read but not independently reconstructed; no universal assertion about all space-filling curves is inferred. Cantor's earlier set-theoretic bijection, Netto's continuity obstruction for such a bijection, Hilbert's 1891 construction and Peano's 1908 visualizations remain distinct. This original paper is arithmetic and contains no curve illustration. Bounded exact rational representation checks illustrate the repair; they are not a full proof audit or a finite grid used as evidence of complete area coverage. January 1890 is the manuscript signature; publisher metadata gives March 1890. Later algorithms and applications keep separate credit.", "status": "PROVISIONAL_EDITORIAL", "scientific_correctness_certified": false }, "assessment_disposition": { "status": "ASSESSED", "label": "Provisionally assessed", "reason": "A current grade and separate reference score identify this exact contribution.", "target": { "record_id": "urn:uuid:14473075-5c9c-4824-af51-8730d211ac02", "version": 1, "content_hash": "sha256:204c351ce556ba02fd4a1eedb867f4241a7008d2424b7b7a20716e8809c82211" }, "edition": "ratings-e039106f30dec177" } }, "version": { "kind": "PUBLIC_RECORD_VIEW", "previous": null, "published_at": "2026-10-03T00:00:00Z", "publisher": { "principal_id": "urn:uuid:80116931-785b-40a2-a24b-9d59db0c7e33", "role": "WRS_EDITORIAL_PUBLICATION" }, "record": { "contributors": [ { "attribution_basis": "SOURCE_CITED", "credit": "historical_scientific", "entity_kind": "historical_person", "name": "Giuseppe Peano", "note": "Archive attribution; detailed role assignment remains source-bound", "roles": [ "formulator" ], "source": { "citation": "Peano, Sur une courbe, qui remplit toute une aire plane; exact1890 original hosted by University of Edinburgh", "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/peano.pdf" } } ], "evidence": [ { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-0", "kind": "source_reference", "source": { "citation": "Peano, Sur une courbe, qui remplit toute une aire plane; exact1890 original hosted by University of Edinburgh", "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/peano.pdf" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-1", "kind": "source_reference", "source": { "citation": "Crossref identity for the exact1890 Peano space-filling paper", "url": "https://api.crossref.org/works/10.1007%2FBF01199438" } }, { "description": "Bibliographic/stated-result source; not a WRS proof verification", "id": "source-2", "kind": "source_reference", "source": { "citation": "University of Florence mathematical history entry: Giuseppe Peano and the1890 space-filling note", "url": "https://archimede.dimai.unifi.it/archimede/matematicaitaliana/schede_opere/57peano90.html" } } ], "origin": "HUMAN_LEGACY", "record_profile": "wrs-local-research-notes-0.1", "record_type": "research_note", "rrp_version": "0.1", "scope": { "description": "Exact 1890 Mathematische Annalen 36(1),157-160 paper. formula glyphs were not visually certified. The mapping is continuous and onto, not a one-to-one continuous correspondence or a claim that dimension is preserved by arbitrary continuous maps. Parity-controlled digit complements and agreement at the two ternary representations of triadic rationals are essential; naive digit interleaving fails. The argument concerns continuous parametrizations: ordinary graphs and coordinates of bounded variation retain stronger restrictions. The final page also gives a cube construction. Original claims about nondifferentiability and exact preimage multiplicities were read but not independently reconstructed; no universal assertion about all space-filling curves is inferred. Cantor's earlier set-theoretic bijection, Netto's continuity obstruction for such a bijection, Hilbert's 1891 construction and Peano's 1908 visualizations remain distinct. This original paper is arithmetic and contains no curve illustration. Bounded exact rational representation checks illustrate the repair; they are not a full proof audit or a finite grid used as evidence of complete area coverage. January 1890 is the manuscript signature; publisher metadata gives March 1890. Later algorithms and applications keep separate credit.", "domain": "Topology and real analysis" }, "sources": [ { "citation": "Peano, Sur une courbe, qui remplit toute une aire plane; exact1890 original hosted by University of Edinburgh", "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/peano.pdf" }, { "citation": "Crossref identity for the exact1890 Peano space-filling paper", "url": "https://api.crossref.org/works/10.1007%2FBF01199438" }, { "citation": "University of Florence mathematical history entry: Giuseppe Peano and the1890 space-filling note", "url": "https://archimede.dimai.unifi.it/archimede/matematicaitaliana/schede_opere/57peano90.html" } ], "statement": "# Peano: a continuous curve that fills an entire square\n\nPeano explicitly constructs a continuous mapping from the closed unit interval onto the whole unit square through ternary digits and parity-controlled complements, showing that continuity alone does not restrict a parametrized curve to an arbitrarily small enclosing area.\n\nA moving point can trace a continuous path without jumping. That sounds like it should leave most of a square untouched. Peano showed otherwise: a carefully defined continuous path reaches every point in the square. The construction specifies the two coordinates using the digits of one parameter, with a complement rule that makes adjacent pieces fit together. This is an exact infinite construction, not a picture with many lines. Its lesson is concrete: continuity alone is weaker than everyday intuition about a curve. Conditions such as a graph representation or bounded variation matter. The path is not a continuous one-to-one identification of a line with a square, and later geometric constructions are separate works.\n\n## Scientific scope\n\nExact 1890 Mathematische Annalen 36(1),157-160 paper. formula glyphs were not visually certified. The mapping is continuous and onto, not a one-to-one continuous correspondence or a claim that dimension is preserved by arbitrary continuous maps. Parity-controlled digit complements and agreement at the two ternary representations of triadic rationals are essential; naive digit interleaving fails. The argument concerns continuous parametrizations: ordinary graphs and coordinates of bounded variation retain stronger restrictions. The final page also gives a cube construction. Original claims about nondifferentiability and exact preimage multiplicities were read but not independently reconstructed; no universal assertion about all space-filling curves is inferred. Cantor's earlier set-theoretic bijection, Netto's continuity obstruction for such a bijection, Hilbert's 1891 construction and Peano's 1908 visualizations remain distinct. This original paper is arithmetic and contains no curve illustration. Bounded exact rational representation checks illustrate the repair; they are not a full proof audit or a finite grid used as evidence of complete area coverage. January 1890 is the manuscript signature; publisher metadata gives March 1890. Later algorithms and applications keep separate credit.\n\n## People and systems\n\nGiuseppe Peano\n\n\n\n## Original sources\n\n- [Peano, Sur une courbe, qui remplit toute une aire plane; exact1890 original hosted by University of Edinburgh](https://webhomes.maths.ed.ac.uk/~v1ranick/papers/peano.pdf)\n- [Crossref identity for the exact1890 Peano space-filling paper](https://api.crossref.org/works/10.1007%2FBF01199438)\n- [University of Florence mathematical history entry: Giuseppe Peano and the1890 space-filling note](https://archimede.dimai.unifi.it/archimede/matematicaitaliana/schede_opere/57peano90.html)\n", "summary": "Peano explicitly constructs a continuous mapping from the closed unit interval onto the whole unit square through ternary digits and parity-controlled complements, showing that continuity alone does not restrict a parametrized curve to an arbitrarily small enclosing area.\n\nA moving point can trace a continuous path without jumping. That sounds like it should leave most of a square untouched. Peano showed otherwise: a carefully defined continuous path reaches every point in the square. The construction specifies the two coordinates using the digits of one parameter, with a complement rule that makes adjacent pieces fit together. This is an exact infinite construction, not a picture with many lines. Its lesson is concrete: continuity alone is weaker than everyday intuition about a curve. Conditions such as a graph representation or bounded variation matter. The path is not a continuous one-to-one identification of a line with a square, and later geometric constructions are separate works.", "title": "Peano: a continuous curve that fills an entire square" }, "record_id": "urn:uuid:14473075-5c9c-4824-af51-8730d211ac02", "version": 1, "source_ref": { "record_id": "urn:uuid:14473075-5c9c-4824-af51-8730d211ac02", "version": 1, "content_hash": "sha256:204c351ce556ba02fd4a1eedb867f4241a7008d2424b7b7a20716e8809c82211" }, "export_view": "wrs-public-editorial-view-1", "public_view_hash": "sha256:39787f75f7aaf1f833cb98704e4192c24db332a4ad13be52e7c7fd2026ad99e7" }, "target": { "record_id": "urn:uuid:14473075-5c9c-4824-af51-8730d211ac02", "version": 1, "content_hash": "sha256:204c351ce556ba02fd4a1eedb867f4241a7008d2424b7b7a20716e8809c82211" }, "published_grade": { "profile": "wrs-published-grade-1", "edition": "2026-10-28.1", "id": "peano-continuous-spacefilling-curve-1890", "entry_id": "peano-continuous-spacefilling-curve-1890", "revision": 1, "title": "Peano: a continuous curve that fills an entire square", "context": "Topology and real analysis", "track": "Topology and real analysis", "band": "A", "modifier": null, "status": "PROVISIONAL_EDITORIAL", "assessed_on": "2026-10-04", "evidence_cutoff": "2026-10-04", "achievement_state": "SUPPORTED_SCOPED_RESULT", "what_changed": "Peano explicitly constructs a continuous mapping from the closed unit interval onto the whole unit square through ternary digits and parity-controlled complements, showing that continuity alone does not restrict a parametrized curve to an arbitrarily small enclosing area.", "scope": "Peano explicitly constructs a continuous mapping from the closed unit interval onto the whole unit square through ternary digits and parity-controlled complements, showing that continuity alone does not restrict a parametrized curve to an arbitrarily small enclosing area. Exact 1890 Mathematische Annalen 36(1),157-160 paper. formula glyphs were not visually certified. The mapping is continuous and onto, not a one-to-one continuous correspondence or a claim that dimension is preserved by arbitrary continuous maps. Parity-controlled digit complements and agreement at the two ternary representations of triadic rationals are essential; naive digit interleaving fails. The argument concerns continuous parametrizations: ordinary graphs and coordinates of bounded variation retain stronger restrictions. The final page also gives a cube construction. Original claims about nondifferentiability and exact preimage multiplicities were read but not independently reconstructed; no universal assertion about all space-filling curves is inferred. Cantor's earlier set-theoretic bijection, Netto's continuity obstruction for such a bijection, Hilbert's 1891 construction and Peano's 1908 visualizations remain distinct. This original paper is arithmetic and contains no curve illustration. Bounded exact rational representation checks illustrate the repair; they are not a full proof audit or a finite grid used as evidence of complete area coverage. January 1890 is the manuscript signature; publisher metadata gives March 1890. Later algorithms and applications keep separate credit.", "evidence_summary": "Source-linked, scoped account of the original work. The published scope states the scientific assumptions and remaining verification limits. WRS has not independently certified the full proof, experimental result or system benchmark.", "references": [ { "title": "Peano, Sur une courbe, qui remplit toute une aire plane; exact 1890 original hosted by University of Edinburgh", "url": "https://webhomes.maths.ed.ac.uk/~v1ranick/papers/peano.pdf", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." }, { "title": "Crossref identity for the exact 1890 Peano space-filling paper", "url": "https://api.crossref.org/works/10.1007%2FBF01199438", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." }, { "title": "University of Florence mathematical history entry: Giuseppe Peano and the 1890 space-filling note", "url": "https://archimede.dimai.unifi.it/archimede/matematicaitaliana/schede_opere/57peano90.html", "locator": "Original linked work. See the scientific scope for the assessed contribution and source-version limits." } ], "roles": [ { "who": "Giuseppe Peano", "did": "Original source-listed scientific author of this scoped work; byline does not determine effort fractions.", "credit": "SCIENCE" } ], "target": { "record_id": "urn:uuid:14473075-5c9c-4824-af51-8730d211ac02", "version": 1, "content_hash": "sha256:204c351ce556ba02fd4a1eedb867f4241a7008d2424b7b7a20716e8809c82211" }, "scientific_correctness_certified": false, "independent_expert_calibration": false, "method": "PROPRIETARY", "points": "130000" }, "canonical_agent_submission": false, "scientific_verification_awarded": false, "reader_guide": null, "reading_status": "SOURCE_LINKED_NOTE" }