Peano curve
E403670
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Peano curve canonical | 2 |
| Lebesgue space-filling curve | 1 |
| Sierpiński curve | 1 |
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
ⓘ
Peano curve self-linksurface differs ⓘ
surface form:
Lebesgue space-filling 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 ⓘ |
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Sierpiński curve
this entity surface form:
Lebesgue space-filling curve