What is recorded here?
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. A 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.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
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.
Read the original work
- Peano, Sur une courbe, qui remplit toute une aire plane; exact1890 original hosted by University of Edinburgh
- Crossref identity for the exact1890 Peano space-filling paper
- University of Florence mathematical history entry: Giuseppe Peano and the1890 space-filling note
This English explanation is an editorial reading view. It does not replace the complete original work or certify its science.
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People and systems
Giuseppe Peano
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English catalogue narrative
Peano: a continuous curve that fills an entire square
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.
A 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.
Scientific scope
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.
People and systems
Giuseppe Peano