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is replaced by a contiguous subsequence of the centers of these nine smaller squares. This subsequence is formed by grouping the nine smaller squares into three columns, ordering the centers contiguously within each column, and then ordering the columns from one side of the square to the other, in
244:
In the definition of the Peano curve, it is possible to perform some or all of the steps by making the centers of each row of three squares be contiguous, rather than the centers of each column of squares. These choices lead to many different variants of the Peano curve.
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such a way that the distance between each consecutive pair of points in the subsequence equals the side length of the small squares. There are four such orderings possible:
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A "multiple radix" variant of this curve with different numbers of subdivisions in different directions can be used to fill rectangles of arbitrary shapes.
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is a simpler variant of the same idea, based on subdividing squares into four equal smaller squares instead of into nine equal smaller squares.
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112:. Because of this example, some authors use the phrase "Peano curve" to refer more generally to any space-filling curve.
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was the first point in its ordering, then the first of these four orderings is chosen for the nine centers that replace
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This article is about a particular curve defined by
Giuseppe Peano. For other curves with similar properties, see
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Right three centers top to bottom, middle three centers bottom to top, and left three centers top to bottom
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Left three centers top to bottom, middle three centers bottom to top, and right three centers top to bottom
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Right three centers bottom to top, middle three centers top to bottom, and left three centers bottom to top
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Left three centers bottom to top, middle three centers top to bottom, and right three centers bottom to top
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of the centers of the squares, from the set and sequence constructed in the previous step. As a base case,
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is chosen in such a way that the distance between the first point of the ordering and its predecessor in
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340:, Texts in Computational Science and Engineering, vol. 9, Springer, pp. 25–27,
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Three iterations of a Peano curve construction, whose limit is a space-filling curve.
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Cole, A. J. (September 1991), "Halftoning without dither or edge enhancement",
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is partitioned into nine smaller equal squares, and its center point
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Peano's curve may be constructed by a sequence of steps, where the
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275:(1890), "Sur une courbe, qui remplit toute une aire plane",
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is the one-element sequence consisting of its center point.
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of the curves through the sequences of square centers, as
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also equals the side length of the small squares. If
237:Peano curve with the middle line erased creates a
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104:. Peano was motivated by an earlier result of
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334:Bader, Michael (2013), "2.4 Peano curve",
310:, Courier Dover Publications, p. 3,
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195:Among these four orderings, the one for
145:consists of the single unit square, and
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968:List of fractals by Hausdorff dimension
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327:
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304:Gugenheimer, Heinrich Walter (1963),
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108:that these two sets have the same
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950:How Long Is the Coast of Britain?
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38:
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59:Two iterations of a Peano curve
1023:Theory of continuous functions
974:The Fractal Geometry of Nature
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297:
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217:The Peano curve itself is the
1:
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81:in 1890. Peano's curve is a
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990:Chaos: Making a New Science
346:10.1007/978-3-642-31046-1_2
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131:of squares, and a sequence
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73:is the first example of a
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124:th step constructs a set
982:The Beauty of Fractals
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307:Differential Geometry
278:Mathematische Annalen
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77:to be discovered, by
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928:Lewis Fry Richardson
923:Hamid Naderi Yeganeh
713:Burning Ship fractal
645:Weierstrass function
337:Space-Filling Curves
171: − 1
100:, however it is not
686:Space-filling curve
663:Multifractal system
546:Space-filling curve
531:Sierpinski triangle
371:The Visual Computer
87:continuous function
75:space-filling curve
21:space-filling curve
16:Space-filling curve
913:Aleksandr Lyapunov
893:Desmond Paul Henry
857:Self-avoiding walk
852:Percolation theory
496:Iterated function
437:Fractal dimensions
383:10.1007/BF01905689
291:10.1007/BF01199438
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225:goes to infinity.
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956:Coastline paradox
933:Wacław Sierpiński
918:Benoit Mandelbrot
842:Fractal landscape
750:Misiurewicz point
655:Strange attractor
536:Apollonian gasket
526:Sierpinski carpet
239:Sierpinski carpet
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873:Michael Barnsley
740:Lyapunov fractal
598:Sierpiński curve
551:Blancmange curve
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377:(5): 235–238,
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79:Giuseppe Peano
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603:Z-order curve
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573:Hilbert curve
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91:unit interval
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1002:Chaos theory
997:Kaleidoscope
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898:Gaston Julia
878:Georg Cantor
703:Escape-time
635:Gosper curve
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583:Lévy C curve
568:Dragon curve
447:Box-counting
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116:Construction
106:Georg Cantor
70:
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993:(1987 book)
985:(1986 book)
977:(1982 book)
963:Fractal art
883:Bill Gosper
847:Lévy flight
593:Peano curve
588:Moore curve
474:Topological
459:Correlation
110:cardinality
98:unit square
71:Peano curve
1017:Categories
801:Orbit trap
796:Buddhabrot
789:techniques
777:Mandelbulb
578:Koch curve
511:Cantor set
259:References
83:surjective
908:Paul Lévy
787:Rendering
772:Mandelbox
718:Julia set
630:Hexaflake
561:Minkowski
481:Recursion
464:Hausdorff
273:Peano, G.
102:injective
89:from the
818:fractals
705:fractals
673:L-system
615:T-square
423:Fractals
229:Variants
155:In step
67:geometry
767:Tricorn
620:n-flake
469:Packing
452:Higuchi
442:Assouad
866:People
816:Random
723:Filled
691:H tree
610:String
498:system
352:
314:
69:, the
942:Other
219:limit
350:ISBN
312:ISBN
251:The
96:the
94:onto
379:doi
342:doi
287:doi
163:of
65:In
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
