The Peano curve is a plane-filling function introduced by Peano (1890) that maps a unit interval
continuously onto a unit square .
The fractal curve illustrated
above can be written as a Lindenmayer system .
Wunderlich (1973) systematized the Peano-type constructions obtained by recursively subdividing the unit square into a array of subsquares. The
resulting patterns fall into switch-back and meandering types, with the original
Peano curve belonging to the switch-back type (Sagan 1994).
The iteration
of the Peano curve illustrated above is implemented in the Wolfram
Language as PeanoCurve n ].
See also Dragon Curve ,
Hilbert Curve ,
Lindenmayer System ,
Moore
Curve ,
Sierpiński Curve
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References Dickau, R. M. "Two-Dimensional L-Systems." https://www.robertdickau.com/lsys2d.html . Hilbert,
D. "Über die stetige Abbildung einer Linie auf ein Flachenstück."
Math. Ann. 38 , 459-460, 1891. Mandelbrot, B. B. The
Fractal Geometry of Nature. Peano, G. "Sur une courbe, qui remplit une aire plane."
Math. Ann. 36 , 157-160, 1890. https://www.youtube.com/watch?v=3qbZadltTCI .
Companion notebook: https://community.wolfram.com/t/23888 .Sagan,
H. Space-Filling
Curves. Wagon,
S. Mathematica
in Action. Wunderlich,
W. "Über Peano-Kurven." Elem. Math. 28 , 1-10, 1973.
https://doi.org/10.5169/seals-29447 . Referenced
on Wolfram|Alpha Peano Curve
Cite this as:
Weisstein, Eric W. "Peano Curve." From
MathWorld https://mathworld.wolfram.com/PeanoCurve.html
Subject classifications
array of subsquares. The
resulting patterns fall into switch-back and meandering types, with the original
Peano curve belonging to the switch-back type (Sagan 1994).
th iteration
of the Peano curve illustrated above is implemented in the Wolfram
Language as PeanoCurve[n].
Pegg, E. Jr. Mathematical
Games. Episode 16: "Space-Filling Curves." Apr. 18, 2024.