In what way is the Peano curve not one-to-one with $[0,1]^2$?
The construction of the Peano curve is a recursive refinement of a particular path that discretely looks one-to-one, in that it touches every coordinate point at a given scale in a bijection. In the limit there's no bijection, but at every step there is a bijection between the curve so far and the coordinates of points within $[0,1]^2 ...
real analysis - How to prove Peano curve is surjective?
Jul 14, 2018 · Instead, the Peano curve is defined by. P(x) = limn→∞Pn(x) P (x) = lim n → ∞ P n (x) for all x ∈ I x ∈ I. While it is true that every point along the curves Pn P n has at least one of its two coordindates rational, this does not apply their limit, as every irrational number is the limit of a sequence of rational numbers. The key ...
What is the topological dimension of the Peano curve?
So one can conclude that the topological dimension of the Peano curve must be one. But how could it be? I thought that in this context when we say curve, we mean the image of the curve. But the image of the Peano curve is $[0,1]^2$ and the topological dimension of $[0,1]^2$ is two.
Drawing the Peano curve - Code Golf Stack Exchange
Nov 18, 2018 · In geometry, the Peano curve is the first example of a space-filling curve to be discovered, by Giuseppe Peano in 1890. 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 cardinality.
real analysis - Area of Peano Curve - Mathematics Stack Exchange
If f f is a space-filling curve like the Peano curve, then, the area is 1 1. There certainly are other definitions of “curve”. For instance, you might say that a curve in S S is a subset that is locally the union of finitely many analytic curves. Such a set would be nowhere dense, so its area would be zero. If you accept this definition ...
fractals - How would I calculate points on the Peano curve ...
Jul 20, 2016 · $\begingroup$ The Peano curve (more precisely, the sequence of curves converging to it) is typically described in terms of its image, rather than the actual function that yields the image. In other words, even the approximating (non-space-filling) curves don't usually come with a specific formula that tells you what the image of particular ...
analysis - Is it true that a space-filling curve cannot be injective ...
In the case of the ordinary Peano-Hilbert curve, it does not help much: the construction has infinitely (only countably, though) many inherent points of overlap (discussed here: In what way is the Peano curve not one-to-one with $[0,1]^2$?), but I can't really see any reason why there could not be a space-filling curve without such points.
Does the Peano curve have length? - Mathematics Stack Exchange
May 6, 2018 · For the highly non-differentiable Peano curve, we'd have to go back to (1) (1) of course. It is easy to see that the supremum is infinite. Indeed, partition the square into N2 N 2 squares of side length 1 N 1 N. For each square, pick a ti ∈ [0, 1] t i ∈ [0, 1] such that γ(ti) γ (t i) hits its centre. After arranging these ti t i into ...
Pushing Lebesgue measure with Peano curve - Mathematics Stack …
Jun 5, 2018 · You can also get singular measures by composing the Cantor-Lebesgue function with Hilbert/Peano curve. A good reference for these facts is "Space filling curves" by Hans Sagan. $\endgroup$ – hot_queen
path connected - Why can't a knot be a space filling curve ...
May 29, 2021 · The key is in this part of the definition of a knot: (and the function is injective otherwise) This cannot be satisfied by a space-filling curve.