Talk:Space-filling curve

A curve filling a countably dimensional hypercube
You should describe such a curve. Also, you should make a link to a description of the discontinuous mapping of the unit interval onto the unit square, discovered by Georg Cantor. -- Leocat 14:13, 25 October 2006 (UTC)

Space-filling curves are not bijections
So this space-filling curve is a continuous bijection, right? Why isn't it a homemomorphism (it certainly can't be). Because its inverse isn't continuous? That must be it. Lethe | Talk 07:42, Mar 3, 2005 (UTC)


 * Space filling curves aren't one to one, so aren't bijections.(Balthamos 23:35, 11 June 2006 (UTC))

Any space-filling curve must be densely self-intersecting
Wasn't one of the attributes of the Peano curve the property of non self-intersection? WLD 13:14, 23 Apr 2005 (UTC)


 * There cannot be any non-self-intersecting (i.e. injective) continuous curve filling up the unit square, because that will make the curve a homeomorphism from the unit interval onto the unit square (using the fact that any continuous bijection from a compact space onto a Hausdorff space is a homeomorphism), but the unit-square (which has no cut-point) is not homeomorphic to the unit interval (all points of which, except the endpoints, are cut-points).
 * A.D.


 * The problem may be unfamiliarity with the definition of "self-intersecting". Here it just means that curve is not simple (not injective). Two curves intersect if the intersection of their images is non-empty, and if a curve is not simple we can find two "subcurves" of the curve (obtained by taking the restrictions to two disjoint segments of its domain) that intersect. But, in analogy to the intersection of lines, one might be tempted to think (incorrectly) that the meaning of "curves intersecting" is that they cross each other, going to the other side. And for the Peano curve (and also the Hilbert curve), whereever two subcurves intersect, they don't cross. --Lambiam 02:52, 11 November 2007 (UTC)

Self-avoiding property of finite approximations
Under the Properties heading, the article says


 * Peano's original construction is self-contacting at all finite approximations, but Hilbert's construction is self-avoiding for all finite approximations.

But this is not true: First, Peano's original construction does not even use finite approximations. Instead, it is defined directly in terms of the trinary digits of the argument (!). Second, his construction could be described in terms of self-avoiding finite approximations, as the figure in the article clearly shows. And third, properties of finite approximations are not properties of the curve, so they don't belong under the Properties heading anyway. Accordingly, I am going to delete the statement. Hanche 16:37, 2 September 2007 (UTC)

Proof that a square and its side contain the same number of points
The formal proof kindly provided by JRSpriggs (around which I built this section) involves the definition of a right-inverse which seems not to be a ("proper") function, because it yields nonunique results... This is because the original space-filling curve is self-intersecting. Was I clear enough? Am I right? Is this right inverse compatible with Cantor–Bernstein–Schroeder theorem? Paolo.dL 15:19, 7 September 2007 (UTC)


 * Using the axiom of choice, we can get a right inverse to any surjective function. In fact, asserting the existence of an inverse to every surjective function is AC. I sense there might be away to get around AC by using directly some sort of dual to C-B-S, but I doubt there's actually a constructive proof. Depends what you call "proper" I guess. --192.75.48.150 17:40, 7 September 2007 (UTC)

OK, here is the figure with the explanation of the word "proper" or "true" (from the article Function). Paolo.dL 18:26, 7 September 2007 (UTC)


 * Right. In your picture, let's say f(a) = 1, f(d) = 2, f(b) = f(c) = 3. f is a surjection from Y to X. You can use either
 * g(1) = a, g(2) = d, g(3) = b or
 * h(1) = a, h(2) = d, h(3) = c
 * as a right inverse - they are both injections from X to Y. But of course you also see there's no injection from Y to X. For the infinite case, to actually define the right inverse and make the proof work, you need to assume the axiom of choice (see that article for more details). --192.75.48.150 19:03, 7 September 2007 (UTC)

Yes! In other words, a space-filling curve, being self-intersecting, is similar to the function $$f: Y \to X$$ obtained by just reversing the arrows in the figure. This suggests that, paradoxically, there could be even "more points" in the side of the square (Y) than in the square (X)! By the way, that's plausible somehow: the tangent function proves that a finite-length segment can contain the same number of points as an infinitely long line... [of course I know that the two infinite sets have the same cardinality $$\mathbf c$$, but I am trying to describe the concept in a more intuitive way].

And you are right: contrary to what I wrote, the two right inverses of f (g and h) are by all means true (proper) functions. I was confusing the right inverse of f with the multivalued function shown in the figure on the right, which would be a "quasi-inverse" of f, rather than one of its right inverses.

Thanks a lot. It was very kind of you to explain. What's more important is that the proof in the article is correct. This is nice because the proof can be intuitively interpreted as I wrote above and in the article (even by those who do not totally understand it). It provides good food for thought. :-) Paolo.dL 20:55, 7 September 2007 (UTC)

Thanks also to JRSpriggs for his latest enlightening contribution to this section. In this context, it is amazing that Hilbert was able to associate up to four points in the side to each point in the square! (see also the discussion Intuitively acceptable examples about "larger than infinite", from which the idea of creating this section originated) Paolo.dL 09:33, 9 September 2007 (UTC)

Is it so interesting to use the space filling curve to prove that [0,1] and [0,1]2 have the same cardinality? It can be done easily without space-filling curves or the axiom of choice: given x in [0,1], write it in binary (so that it has an infinite number of 1s, in the case that x has 2 binary representations, just to remove ambiguity) and let y be the bits in even positions and z be the bits in odd positions. i.e. if x = (.y1 z1 y2 z2 ...) in base 2, then define f(x) = ( (.y1 y2 ...), (.z1 z2 ...)). then f is obviously surjective and the preimage of any point in the codomain has size ≤ 4. (since y, z each have at most 2 representations in binary) I claim this is no less intuitive than using the space-filling curve to prove this result (in fact it is much more intuitive) since the proof that the Cantor set C is homeomorphic to C2, a fact used to prove the that the space-filling curve actually fills up the space, I think uses the exact same idea anyway. Unique-k-sat (talk) 01:10, 17 December 2009 (UTC)

Bad illustration
The first illustration on this page has the caption: "3 iterations of the Peano curve, a space-filling curve" But the illustration is actually just another picture of the Hilbert curve. The Hilbert curve is a Peano curve, but as far as I can gather it is not the curve the Peano came up with. So either the illustration should be replaced with an illustration like this: http://mathworld.wolfram.com/PeanoCurve.html Which I believe is 'The Peano Curve' Or we should change the illustration caption to say 'a Peano curve' instead of 'The Peano curve', but if we're going to just change the caption the illustration should probably just be removed, 'cause it is redundant with a hilbert curve illustration directly below it. 128.97.68.15 21:35, 10 October 2007 (UTC)


 * Let us use "Peano curve" for the actual curve invented by Peano. Then the Hilbert curve and the Peano curve are different things. The Peano curve consists of 3×3 reduced copies of itself glued together, while the Hilbert curve is the union of 2×2 reduced copies of itself. (Here, reduced = reduced in scale). The first image does indeed show the construction of the actual first Peano curve, and shows a different curve than the next image, the Hilbert curve. --Lambiam 22:42, 10 October 2007 (UTC)


 * I must agree with Lambiam on this. If you check out Peano's paper from 1890 (referenced in the article), you will not see a picture, but a description of the parametrization of the curve based on the base 3 representation of the parameter. It takes a bit of work to puzzle it out, but the picture in the article as it is now is correct. Hanche 17:10, 12 October 2007 (UTC)

Relationship of Space-filling to Tiling and tessellation
Comparing the Sierpinski Triangle and Carpet to tiling arrangements, it would appear that the technique is applicable to repetitive tiling arrangement; triangle, square, hexagon being the simplest. It would seem impossible to apply it to other than rep-tile arrangements. —Preceding unsigned comment added by 86.160.138.236 (talk) 12:05, 2 June 2008 (UTC)


 * I think you are right (except that "impossible" is, in general, too strong). However, the Wikipedia verifiability policy requires that we only put content in our articles that has been published before in reliable sources. Even if true, content based on original research must not be added. Therefore I have removed this addition from the article. --Lambiam 18:00, 2 June 2008 (UTC)

Peano Curve And Uncountability Of The Real Number System
1 About Peano's deduction

(1). About Peano curve: Peano constructed a infinite curve sequence: P = {p1, p2, p3, ...}, and all the points in the unit square will be the limit points of the curve sequence P. Therefore Peano has a conclusion that the "limit curve" of the sequence P is filling the unit square.

(2). A fault of Peano's deduction: Assume that all the points on (P1, P2, P3 ...) construct a point set. The limit point of a point set is not certain in the point set, for example, all the irrational numbers are the limits of rational number set, but irrational numbers are not in the rational number set. But the "limit curve" is a member of sequence P. It has no reason to say that the "limit curve" is equal to the limit point set.

(3). A counterexample of Peano's deduction:

Assume that a sequence of number sequence:



S = {s1, s2, s3, ...}.

s1, s2, ... are number sequences:

s1:

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

s2:

0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09

0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19

......

0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99

s3:

0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009

0.011, 0.012, 0.013, 0.014, 0.015, 0.016, 0.017, 0.018, 0.019

......



All the real numbers in [0, 1] are the limit points of S, but it's not means that there is a "limit number sequence" containts all the real numbers in [0, 1]. Otherwise the real number system is countable.

2 Cantor’s definition of real number

Russell’s paradox points out that a set which is a member of itself will lead to some paradox. Analyse the definition of real number given by Cantor [1]:

”The first section of Cantor’s paper of 1872 was devoted to the real numbers, in particular to the irrationals. In a paper published somewhat later, he stressed explicitly the objections he had to previous attempts to define irrational numbers in terms of infinite series.

Thus Cantor set himself the task of developing a satisfactory theory of the irrationals which in no way presupposed their existence. He followed Weierstrass in beginning with the rational numbers. The set of all rationals (A), including zero, thus provided a foundation for all further concepts of number.

Cantor then was explicit that whenever he spoke of a numerical quantity in any further sense he did so above all in terms of infinite sequences of rational numbers {an}:

Definition: The infinite sequence a1, a2, ..., an, ... is said to be a fundamental sequence if there exists an integer N such that for any positive, rational valuve of e, |an+m ? an| < e, for whatever m and for all n > N.

If a sequence {an} satisfied this condition, then Cantor said that ”the sequence has a definite limit b.” This had a very special meaning for Cantor and must be taken as a convention to express, not that the sequence {an} actually had the limit b, or that b was presupposed as the limit, but merely that with each such sequence {an} a definite symbol a was associated with it. At this point in his exposition Cantor was explicit in using the word ”symbol” (Zeichen) to describe the status of b.

. ..

A rational number a (where a could be defined by the constant sequence {a}, which was clearly a fundamental sequence and for which the a was actually the value reached by the sequence in the limit).

. ..

The definitions to which Cantor referred were those for which arithmetic operations might be extended from the domain of rational numbers to the new numbers b (Cantor now called them numbers instead of symbols).

. ..

Consequently an element b 2 B should be considered as a number only for the sake of convenience. Ultimately it only represented a fundamental sequence. ”

Obviously, Cantor used a rational number set to define a real number. According the definition, a real number is equal to a rational number sequence, that is:

(1). In the real number field, a rational number a is equal to a sequence {a};

(2). In the real number field, a irrational number b is equal to a sequence {b1, b2,. . ., bn,. . .}.

According the set theory, a number sequence is correspond to a number set rather than a number member. Therefore, the definition of rational number in the real number field, a rational number a is equal to a sequence {a}. It will construct a set that containts itself: number a is equal to a set, the set has an element that a itself. It will lead to the Russell’s paradox.

Though Cantor’s definition of irrational number didn’t contain such paradox, but at times it is misunderstanded as the form that contain some paradox. It’s meaningful to talk about how to avoid such misunderstanding.

In the definition, a irrational number b is associated with a fundamental sequence {bn}, and all the members in the fundamental sequence {bn} is a rational number. Obviously, according the definition, the irrational number b as the limit of the fundamental sequence is not in the fundamental sequence.

It can be proved by using the induction that the limit not in the fundamental sequence has some mathematical reason, rather than subjective definition.

For example, the fundamental sequence associated with Pi:

{Pn} = {3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926,. . .}

Let m be an integer that 0 <= m <= 9, write the numbers in the sequence in the form of the sum of the previous number and a decimal fraction, for example, 3.14 = 3.1 + 0.04.

(1)P1 = 3.1, is a rational number; and 101 is an integer. m/101 is a ratio of two integers, according the definition, it’s a rational number.

(2)For any n belong to N, assume that 10n is an integer, then 10(n+1) is an integer, and m/10(n+1) is a ratio of two integers, it’s a rational number. Assume that Pn is a rational number, then P(n+1) = Pn+m/10(n+1), is the sum of two rational numbers, is a rational number.

(3)Then for any n belong to N, Pn is a rational number. What the induction proved is the attribute of a infinite system that has a one-to-one mapping with the natural number system. As a irrational number, the limit of the fundamental sequence is not in the system.

3 Peano Curve and uncountability of the real number system

In 1877, Cantor proved in theory that there is a one-to-one mapping between a line and a plane. 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].

Fig. 1. Peano Curve

GREG BREINHOLT and CHRISTOPH SCHIERZ gave a computer recursion algorithm to generate Peano curve [3]. Though the methods of curve generation given by Peano is split every square into smaller squares, it seems a parallel processing, due to the number of squares produced by every split is finite, the nth division produces 4n smaller squares, it can be prescribed a serial order to produce those squares one by one. It will change the parallel processing generation to a serial one. It means the cure that generated by the computer recursion algorithm given by GREG BREINHOLT and CHRISTOPH SCHIERZ is strictly equal to the curve given by Peano, every point that can be covered by the Peano curve also can be covered by the curve generated by the computer recursion algorithm.

There is a problem. Draw a coordinatized line across the median line of the initial square. Assume that the interval in the square is [0, 1]. Then generate the Peano curve according the recursion algorithm step by step. Therefore we can number all the points of intersection one by one according the order of generation. If the Peano curve covers all the points in the square, the points of intersection between the Peano curve and the coordinatized line will cover all the points in [0, 1]. Due to every point of intersection has a number, it means a one-to-one mapping between the real numbers in [0, 1] and natural number set. Its contrary to uncountability of the real number system.

Talk about this question in the decimal system, assume that a square split into tenths, that is every square has 9 new intersection with the coordinatized line. Assume the intersections sequence that produced by the first split is s1, the intersections sequence that produced by the second split is s2,. ..

It is:

s1:

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9

s2:

0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09

0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19

. ..

0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99

s3:

0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009

0.011, 0.012, 0.013, 0.014, 0.015, 0.016, 0.017, 0.018, 0.019

. ..

Let S be a sequence that S = {s1, s2, s3,. . .}. If the intersections between the Peano curve and the coordinatized line will cover all the points in [0, 1], that the sequence S (and it’s sub sequence) will include all the numbers in [0, 1]. Its contrary to uncountability of the real number system.

It seems that the focus is whether the irrational number points on the coordinatized line are covered by the Peano curve, or the Peano curve covers the rational number points only. Before come to this question, it’s meaningful to talk about the different between the process of constructing the infinite sequence and the process of taking the limit of the fundamental sequence. It’s known that there are countabe infinite rational numbers in [0, 1], these rational numbers can be produced by an infinite recursion process. And all the irrational numbers in [0, 1] are the limits of the rational number set in [0, 1]. It means that the infinite process of constructing the rational numbers is not include the process of taking the limit of the rational number set. Also, it’s known that though the real number set can be produced by taking the limit of the rational number set, there is not a one-to-one mapping between the rational number set and the real number set. It means the process of taking the limit may not keep the one-to-one mapping relation.

Therefore it has two simple but useful conclusions:

1. The infinite process is not certain include the process of taking the limit;

2. The process of taking the limit may not keep the one-to-one mapping relation.

Though the process of constructing the Peano curve is infinite steps, it’s not certain means the process include taking the limit of Peano curve. And though the limit of Peano curve is a square, it’s not enough to prove there is a one-toone mapping between the curve and the square. Base on the sequence S given above, it can has a conclusion that before taking the limit, the intersections between the Peano curve and the coordinatized line cover the rational number set in [0, 1], the graphics is a curve but not a square; after taking the limit, the intersections will cover all the number in [0, 1], and at that time the graphics is a square but not a curve.

Fig. 2. Hilbert curve

4 Hilbert curve

The generation of Hilbert Curve is similar with the generation of Peano curve, it is continually divide a square into fourths and connected the four smaller squares centers as Figure.2, the result of this curve is a Hilbert curve [4]. As the discussion in section 1, Hilbert Curve also can not covers the whole square. This section should talk about another thing, the one-to-one mapping between a line segment [0,1] and a square [0,1]*[0,1] base on Hilbert Curve.

Coding the squares in binary system. The four squares that generated by first split are coded clockwise as (0.00, 0.01, 0.10, 0.11). After a split, the smaller squares’ codes are generated by adding two bits after the former square’s code. For example, in the second split, the four squares that generated from the square coded 0.00, will be coded coded as (0.00000.00010.00100.0011).

This coding is hoped to give every point in the square a single code in [0, 1], to finish the mapping the square [0, 1] � [0, 1] to the line segment [0,1].

This coding is just similar to the real number’s definition given by Cantor, which using the limit of a sequence to define a point. For example, the point that the center of the square is defined by the sequence {H ? 1(1/2),H ? 2(1/2), . . . ,H ? N(1/2),. . .}, and the center of

the square will be coded as 1/2. But according the definition, every point in sequence {H ? 1(1/2),H ? 2(1/2), . . . ,H ? N(1/2),. . .} is a constant

point, every point is the limit of itself, that is every point in the sequence will be coded as 1/2. Therefore the number 1/2 will map to a infinite square, but not only the center of the square.

By using the reduction to absurdity, it can be proved that there is not a oneto- one mapping between a line segment [0, 1] and a square [0,1]*[0,1] base on Hilbert Curve. Assume that there is a one-to-one mapping, and a point p in the median line that parallel to the y-axis, point p is map to x(0 <= x <= 1). That is, the limit of the sequence {H-1(x), H-2(x),. . ., H-N(x),. . . } is point p.

Due to the Hilbert Curve is symmetry about the median line that parallel to the y-axis, the limit of the sequence {H-1(1-x), H-2(1-x),. . ., H-N(1-x),. . . } is point p, too. Because of it’s a one

-to-one mapping, it has x = 1 ? x = 1/2. That is, all the points in the median line are map to 1/2, it’s inconsistent with the one-to-one mapping postulate.

This reduction to absurdity not only works in the initial square, but also works in any smaller square that generated by Hilbert Curve. Also, the absurdity is not caused by self-intersecting, or it can not be eliminated by permit to self-intersecting.

5 Cantor’s dimension reduction

In 1877, Cantor proved in theory that there is a one-to-one mapping between a line and a plane. His proof is like this: Assume y is a real number, and y = 0.a1b1a2b2. . . anbn. ..

Let

x1 = 0.a1a2. . . an. ..

x2 = 0.b1b2. . . bn. ..

It finished a mapping from y to (x1, x2).

The proof has used the induction unconspicuous. As the discussion in section 2, what the induction proved is the members’ attribute in the fundamental sequence, not include the limit of the sequence.

6 Conclusion

In Cantors definition of real number, using a fundamental sequence to definite a number in the real number field. This paper points out that according the definition, in the real number field, a rational number a is equal to a sequence {a}. It will construct a set that containts itself: number a is equal to a set, the set has an element that a itself. The set will lead to the Russell’s paradox. This paper also discusses that Peano curve and some other plane filling curves contain such paradox.

References

[1] Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton, USA, 1990.

[2] G. Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, 36 (1890) 157-160.

[3] G. Breinholt, C Schierz, Algorithm 781: generating Hilbert’s space-filling curve by recursion, ACM Transactions on Mathematical Software (TOMS), 24 (1998) 184-189.

[4] D. Hilbert, Ueber die stetige Abbildung einer Line auf ein Fl?chenstck, Mathematische Annalen, 38 (1891) 459-460.

—The preceding unsigned comment was added by Qiuzhihong (talk • contribs) 05:27, June 21, 2008 (UTC) – Please sign your posts!


 * Wikipedia is not the place for contributing original research (see No original research). If you have a question about an issue that may need clarification, you can use Reference desk/Mathematics.  --Lambiam 06:24, 27 June 2008 (UTC)
 * Besides, Qiuzhihong's criticism (points 2 and 3) to Peano's construction shows that (s)he actually has not a great understanding of Peano curves. The curve is continuous on [0,1], hence the image is a compact; being dense in the square it is the square itself: stop. Possibly Qiuzhihong ignores that the correctedness of Peano's 1890 note in Mathematische Annalen has been checked by virtually any mathematician for the last 120 years (including Hilbert). --pma (talk) 15:45, 19 December 2009 (UTC)


 * The point is whether the limit of a curves sequence also a curve. The proof above did not mention this point and treat it as an axiom.
 * Think about this curves sequence: f(n): x*x + y*y = 1/n. (A sequence of concentric circles). The limit of f(n) is just a single point (0, 0). Does it mean a single point and a curve contain the same number of points? — Preceding unsigned comment added by 113.90.251.32 (talk • contribs) 12:33, 12 March 2010 (UTC)


 * (1) It is not treated as an axiom: it is treated as background knowledge. (2) The number of points is irrelevant. (3) Much of the above has little to do with editing the article. This is not a general discussion forum. JamesBWatson (talk) 12:59, 12 March 2010 (UTC)


 * @"Qiuzhihong", "113.90.251.32", etc: the right place to put your question is Reference_desk/Mathematics. Many experts there will help you clarifying your doubts in mathematics. BUT, first you have to learn that you have to sign your posts. And, you have to understand that it make no sense writing things that you have not understood inside articles. This way you are making a damage. Why did you put here the long "article" above? It is no-value garbage and only makes more difficult to follow the discussions in the talk page. --pm a  01:22, 14 March 2010 (UTC)

Peano's 1890 curve
What curve did Peano discover (define?) in 1890 that was space filling? The article doesn't seem to define it or even name it (does it have a name?) RJFJR (talk) 18:36, 29 October 2008 (UTC)


 * It is the curve named “the Peano curve” in the picture at the top of the article. (Actually, it might be the mirror image.) Peano's definition was arithmetical in nature, defining the parametrization in terms of the ternary expansion of the curve parameter. Hanche (talk) 08:32, 5 November 2008 (UTC)


 * ✅ Yes, this article now specifically captions that picture "the Peano curve construction", and later specifically names "In 1890, Peano discovered a continuous curve, now called the Peano curve", with a link to the "Peano curve" article, which defines it and goes into more detail. --DavidCary (talk) 19:13, 5 May 2023 (UTC)

Densely Self-intersecting
There is a comment above that clarifies what is meant by "densely self-intersecting". We need to get that into the article! Or add a link. (Thank you for the above comment because I was lost). RJFJR (talk) 16:53, 11 November 2008 (UTC)

Finding the Hilbert factor
There is a short comment in the article about using the Hilbert curve to cluster data. Apparently for two points you find how far along the curve you have to go to get to that point and the closer the length along the curve the closer the points are considered. But it doesn't tell me how to find the length to the point. Can someone help me out or point me to material on this? (I'd add it to the article but I don't understand it yet.) RJFJR (talk) 16:53, 11 November 2008 (UTC)


 * ✅ I agree that information *should* be in Wikipedia, and yet is not in this "space-filling curve" article. But now that information (including an algorithm in C) is in the Hilbert curve article. --DavidCary (talk) 20:29, 5 May 2023 (UTC)

Hölder property
Could somebody add to article that most of these curves are (1/2)-Hölder i.e. $$\|f(x)-f(y)\|_2 < c\sqrt{|x-y|}$$. I think this is true for Hilbert's curve and Peano's curve.David Pal (talk) 20:29, 8 February 2010 (UTC)


 * Do you have a reference for this? I also wonder if it's of sufficient interest to be included. Hanche (talk) 15:21, 9 February 2010 (UTC)


 * A general way to see it, is that (1) these curve are fixed points of a suitable contraction and (2) that contraction takes the set of all curves satysfying a certain Hölder condition (with a fixed c) in itself. So the fixed point is in that set too. The Hölder regularity of these curves seems to me relevant enough to be included, should one find a reference.--pm a  08:15, 24 February 2011 (UTC)

Database retrieval
For database retrieval, one doesn't use the Hilbert Curve per se, one uses some approximation to it. The Hilbert Curve passes limit points more than once (it's a surjective mapping, after all). Consider, for instance, the Hilbert Curve on the unit square. The central point (1/2, 1/2) is converged upon by the curve from different directions. So there's more than one "length" from the origin to that point. Every point (x,y) whose coordinates are rational and have a denominator that is a power of 2 (there are countably infinite of them) will also suffer from the same multiple length condition. —Preceding unsigned comment added by FillerBrushMan (talk • contribs) 02:59, 13 January 2009 (UTC)

Quibbles
"Roughly speaking, differentiability puts a bound on how fast the curve can move." - Turn would be more appropriate than move here, IMO. —Preceding unsigned comment added by 67.183.113.131 (talk) 15:28, 18 July 2010 (UTC)

Interwiki
es:Curva de Peano, fr:Courbe de Peano, it:Curva di Peano, he:עקום פאנו, pl:Krzywa Peano, pms:Curva ëd Peano, pt:Curva de Peano, ru:Кривая Пеано, sv:Peanos kurva, vi:Đường Peano? Newone (talk) 06:07, 2 February 2012 (UTC)

What's the point of Section 5?
I don't understand the point of the section on cardinality. I feel it is not well-explained what is superior about this proof compared with the standard proofs. For example, a standard proof is to give an injection from [0,1]x[0,1] to [0,1] by sending (.x1x2x3..., .y1y2y3)-> .x1y1x2y2... (where, for example, you choose the two binary representations to be eventually 0 in cases where there are two representatives) and the trivial injection [0,1]->[0,1]x{0}, say, and then applying the Cantor-Bernstein theorem. None of this uses the axiom of choice and it all seems very intuitive (more so, I would argue, than anything involving space-filling curves). For example, I'm concerned that this section may confuse novices with regard to the superiority of using a continuous function to prove that the cardinalities are equal. Can someone explain to me what the advantage of this section is? Right now it seems to be detracting from the overall quality of the article. Wpegden (talk) 01:03, 4 April 2012 (UTC)
 * Your "standard proof" is a space-filling curve proof: see Z-order curve. —David Eppstein (talk) 01:28, 4 April 2012 (UTC)
 * The Z-curve is not a space-filling curve in the sense of this article (it is not a "curve" at all), since it is not continuous: the points .011111111...1 (finite number of 1's) and .1 are aribtrarily close together but get mapped to the points (.0111...1,.111...1) and (.1,.0), respectively, which are separated by a distance >.1111...1 (in the y-coordinate alone). (This should be clarified in that article.)  Part of what I don't like about this Section 5 is that it gives the impression that (continuous) space-filling curves are especially useful for proving the cardinality result, whereas for example, the "Z-curve" is just as good for this purpose (and simpler). Wpegden (talk) 03:05, 4 April 2012 (UTC)

I'm leaning towards removing this section entirely. Any objections? Wpegden (talk) 15:41, 5 April 2012 (UTC)
 * Not from me. —David Eppstein (talk) 03:31, 6 April 2012 (UTC)

Renaming this article?
This is in a sense a minor point; thus, I feel, not irrelevant.

The example given in 1890 by Peano had a very remarkable importance in the understanding of the new ideas in topology and analysis. It was the first example of a square-filling curve; not invented just as a mathematical curiosity: at the time many mathematicians did not even realize  that such phenomena where even possible, while extending the definition of curve from smooth to continuous, and this did lead to a number of mistakes and misunderstandings. It is one of the first fractal object ever studied in mathematics. Therefore, I would say, it deserves to be named after the name of its inventor (which is actually the way it is currently known in mathematics). Naming the object and the article by a generic "square-filling curve" is in a sense misleading: exactly in the sense that it tends to hide the historical importance of the author's contribution. It seems strange that the mother of all fractal curves is in an article with a generic name. Note that we have an article on the Hilbert curve, which is just a variation of Peano curve (these variations were even quickly mentioned in Peano's article; and a great mathematician like Hilbert wrote his article in order to call the attention of the mathematical world on Peano's observation, certainly not to stake a claim of originality); we even have one on the Moore curve, just a trivial variation of Hilbert's variation! Finally, since we also have articles on Gosper curve, Sierpiński curve, Osgood curve, Koch curve,  I would rename this article Peano curve, at least for homogeneity... pm a 13:38, 30 November 2012 (UTC)
 * updated: in the meanwhile, a new article Peano curve, on the specific example by Peano, has been created, which indeed seems the simplest solution of the issue. --pm a 18:10, 4 March 2013 (UTC)

Redirects
I left two redirects to Peano space and Peano Space pointing here, but all redirects which contained "Peano" and "curve" have been redirected to Peano curve.  Randall Bart    Talk   19:42, 12 July 2013 (UTC)

Please use standard citations
This article need help to convert old to more standard Wikipedia reference style, using ref-tag. --Krauss (talk) 10:46, 14 September 2019 (UTC)
 * No it doesn't. The citation style it uses is standard. See WP:CITEVAR and WP:HARV. —David Eppstein (talk) 16:18, 14 September 2019 (UTC)

Article fails to address the topic of curves filling space
How can a curve made up of 0-dimensional points or 1-dimensional line segments fill a 2-dimensional space? This is what I was hoping to learn from this article, but it was not addressed. There were only passing references to "counterintuitive" results, and praise for Peano's solution being "ground-breaking". All explanations are in jargon. No attempt is made to explain the assumptions required for the assertion that it is possible to fill space with a curve, or the limitations imposed by those assumptions. GlutenFreePizza (talk) 12:25, 18 May 2022 (UTC)


 * Yeah this is a confusing point, not made clear enough in the article, and the issue is raised in several of the other talk page sections. The Peano and Hilbert curves are made of many (infinitely many in the limit) tiny line segments. but at each stage k in the construction, every point in every segment has at least one coordinate equal to 2-k.  That is, each point in the limiting curve has at least one rational coordinate, so the curve itself has measure 0 in the plane.  It's space-filling in the sense that it comes arbitrarily close to every point in the unit square.  Do I have that right? 174.160.238.145 (talk) 03:55, 31 March 2024 (UTC)
 * You might start with a simpler question: how can 0-dimensional points fill up a 2-dimensional space? And yet all of the plane is filled by points. Once that seems natural, the ability to fill space by higher-dimensional line segments may not seem so strange. —David Eppstein (talk) 05:11, 31 March 2024 (UTC)
 * Continuous, non-self-intersecting curve may be harder, and anyway the examples given (Hilbert and Peano curves) definitely don't fill the space. How many continuous functions are there on the unit interval?  Countably many, since the curve is determined by its values at rational points.  The measure of a countable union of measure-0 line segments is 0.  Am I missing something? 174.160.238.145 (talk) 18:30, 3 April 2024 (UTC)
 * Already there are uncountably many linear functions on a unit interval. —David Eppstein (talk) 19:19, 3 April 2024 (UTC)
 * Oops, yes ;). But still, at each finite stage of the construction, the curve has measure 0.  So the union of all the finite stages has measure 0.  Going past that seems beyond the reach of classical analysis and I don't know what would even mean, since the curve is a continuous function from [0,1] to the square.  Hmm.  174.160.238.145 (talk) 06:35, 5 April 2024 (UTC)