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A space-filling curve can be (everywhere) self-crossing if its approximation curves are self-crossing. A space-filling curve's approximations can be self-avoiding, as the figures above illustrate. In 3 dimensions, self-avoiding approximation curves can even contain knots. Approximation curves remain within a bounded portion of n-dimensional ...
In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. [1] Peano's curve is a surjective, continuous function from the unit interval onto the unit square, however it is not injective. Peano was motivated by an earlier result of Georg Cantor that these two sets have the same ...
Because it is space-filling, its Hausdorff dimension is 2 (precisely, its image is the unit square, whose dimension is 2 in any definition of dimension; its graph is a compact set homeomorphic to the closed unit interval, with Hausdorff dimension 1). The Hilbert curve is constructed as a limit of piecewise linear curves.
A fourth-stage Gosper curve The line from the red to the green point shows a single step of the Gosper curve construction. The Gosper curve, named after Bill Gosper, also known as the Peano-Gosper Curve [1] and the flowsnake (a spoonerism of snowflake), is a space-filling curve whose limit set is rep-7.
Peano curve: And a family of curves built in a similar way, such as the Wunderlich curves. 2: Moore curve: Can be extended in 3 dimensions. 2: Lebesgue curve or z-order curve: Unlike the previous ones this space-filling curve is differentiable almost everywhere. Another type can be defined in 2D. Like the Hilbert Curve it can be extended in 3D.
In 1890, Peano gave a space-filling curve, named Peano curve; and in 1891, Hilbert given an example of such curve, named Hilbert curve [1]. This section should talk about Peano curve first. The Peano curve is generated by continually split a square into fourths, as Figure.1, after infinite steps, the result of this curve is a Peano curve [2].
Lévy C curve; Sierpiński curve; Space-filling curve (Peano curve) See also List of fractals by Hausdorff dimension. Space curves/Skew curves. Conchospiral;
Moreover, in this case the map j is surjective, so that it provides a continuous onto function from the circle onto the 2-sphere, that is, a space-filling curve. Cannon and Thurston also explicitly described the map j : S 1 → S 2 {\displaystyle j:\mathbb {S} ^{1}\to \mathbb {S} ^{2}} , via collapsing stable and unstable laminations of the ...