Peano curve

E403670

continuous surjection curve in the plane fractal curve mathematical object space-filling curve

The Peano curve is a space-filling fractal curve that continuously maps a one-dimensional interval onto a two-dimensional area, demonstrating that a line can completely fill a square.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (3)

Label	Occurrences
Peano curve canonical	2
Lebesgue space-filling curve	1
Sierpiński curve	1

How this entity was disambiguated

This entity first appeared as the object of triple T3995509 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Peano curve
Context triple: [Giuseppe Peano, notableWork, Peano curve]

A. Archimedes' spiral
Archimedes' spiral is a classical mathematical curve that winds outward from a fixed point at a constant rate as it revolves around that point.
B. Ulam spiral
The Ulam spiral is a graphical arrangement of the positive integers in a spiral pattern that reveals striking diagonal alignments of prime numbers, suggesting unexpected structure in their distribution.
C. Jordan curve theorem
The Jordan curve theorem is a fundamental result in topology stating that any simple closed curve in the plane divides the plane into exactly two distinct regions, an "inside" and an "outside."
D. Menger sponge
The Menger sponge is a classic three-dimensional fractal object characterized by infinite surface area and zero volume, constructed by recursively removing cubes from a larger cube.
E. Weierstrass function
The Weierstrass function is a classic example in mathematical analysis of a continuous function that is nowhere differentiable, illustrating the counterintuitive behavior possible in real-valued functions.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Peano curve
Target entity description: The Peano curve is a space-filling fractal curve that continuously maps a one-dimensional interval onto a two-dimensional area, demonstrating that a line can completely fill a square.

A. Archimedes' spiral
Archimedes' spiral is a classical mathematical curve that winds outward from a fixed point at a constant rate as it revolves around that point.
B. Ulam spiral
The Ulam spiral is a graphical arrangement of the positive integers in a spiral pattern that reveals striking diagonal alignments of prime numbers, suggesting unexpected structure in their distribution.
C. Jordan curve theorem
The Jordan curve theorem is a fundamental result in topology stating that any simple closed curve in the plane divides the plane into exactly two distinct regions, an "inside" and an "outside."
D. Menger sponge
The Menger sponge is a classic three-dimensional fractal object characterized by infinite surface area and zero volume, constructed by recursively removing cubes from a larger cube.
E. Weierstrass function
The Weierstrass function is a classic example in mathematical analysis of a continuous function that is nowhere differentiable, illustrating the counterintuitive behavior possible in real-valued functions.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	continuous surjection ⓘ curve in the plane ⓘ fractal curve ⓘ mathematical object ⓘ space-filling curve ⓘ
cardinalityOfFibers	uncountable for many points in the square ⓘ
codomain	[0,1]×[0,1] with the Euclidean topology ⓘ
demonstrates	that a one-dimensional interval can be mapped continuously onto a two-dimensional area ⓘ that there exists a continuous surjection from [0,1] onto [0,1]×[0,1] ⓘ that topological dimension differs from intuitive geometric dimension ⓘ
dimension	1 ⓘ
domain	[0,1] with the usual topology ⓘ
field	fractal geometry ⓘ real analysis ⓘ set theory ⓘ topology ⓘ
hasConstruction	iterative subdivision of the unit square into 3×3 grids ⓘ limit of a sequence of polygonal paths ⓘ
hasProperty	fills a two-dimensional region ⓘ has self-intersections ⓘ image has nonempty interior ⓘ image has positive Lebesgue measure ⓘ image is compact ⓘ image is connected ⓘ image is locally connected ⓘ is a continuous surjection between compact metric spaces ⓘ is a counterintuitive example in topology ⓘ is not injective ⓘ preserves compactness ⓘ
introducedBy	Giuseppe Peano ⓘ
is	a continuous mapping from [0,1] onto the unit square ⓘ a surjective mapping from [0,1] onto the unit square ⓘ nowhere differentiable almost everywhere ⓘ
isExampleOf	continuous surjection that is not a homeomorphism ⓘ curve whose image has higher topological dimension than its domain ⓘ path whose image is a Peano continuum ⓘ
mapsFrom	closed interval [0,1] ⓘ
mapsTo	unit square [0,1]×[0,1] ⓘ
namedAfter	Giuseppe Peano ⓘ
relatedTo	Hilbert curve ⓘ Lebesgue space-filling curve ⓘ linked to: Peano curve Peano continuum ⓘ space-filling curves ⓘ
topologicalDimensionOfImage	2 ⓘ
usedAs	example in courses on fractal geometry ⓘ example in courses on real analysis ⓘ example in courses on topology ⓘ
yearProposed	1890 ⓘ

How these facts were elicited

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Giuseppe Peano → notableWork → Peano curve ⓘ

Giuseppe Peano → developed → Peano curve ⓘ

Wacław Sierpiński → notableIdea → Sierpiński curve ⓘ

linked to: Peano curve

Peano curve → relatedTo → Lebesgue space-filling curve ⓘ