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infinite sequence and the process of taking the limit of the fundamental sequence. Itâs known that there are countabe infinite rational numbers in , these rational numbers can be produced by an infinite recursion process. And all the irrational numbers in are the limits of the rational number set in . 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, itâs 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.
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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
1918:, 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.
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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.
736:...)). 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 C, a fact used to prove the that the space-filling curve actually fills up the space, I think uses the exact same idea anyway.
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863:(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.
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1603:. 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. --
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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 . Due to every point of intersection has a number, it means a one-to-one mapping between the real numbers in and natural number set. Its contrary to uncountability of the real number system.
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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 , the graphics is a curve but not a square; after taking the limit, the intersections will cover all the number in , and at that time the graphics is a square but not a curve.
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2073:.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).
2137:, 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
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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 to the
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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
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in the construction, every point in every segment has at least one coordinate equal to 2. 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
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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
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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
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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 . As the discussion in section 1, Hilbert Curve also can not covers the
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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.
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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
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Is it so interesting to use the space filling curve to prove that and have the same cardinality? It can be done easily without space-filling curves or the axiom of choice: given x in , write it in binary (so that it has an infinite number of 1s, in the case that x has 2 binary representations, just
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GREG BREINHOLT and CHRISTOPH SCHIERZ gave a computer recursion algorithm to generate Peano curve . 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
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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
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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 3Ă3 reduced copies of itself glued together, while the Hilbert curve is the union of 2Ă2 reduced copies of itself. (Here, reduced = reduced in
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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
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By using the reduction to absurdity, it can be proved that there is not a oneto- one mapping between a line segment and a square * 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).
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There is a problem. Draw a coordinatized line across the median line of the initial square. Assume that the interval in the square is . 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
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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
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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
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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 , hence the image is a compact; being dense in the square it is the square itself: stop. Possibly
Qiuzhihong ignores that the
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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
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If a sequence {an} satisfied this condition, then Cantor said that âthe sequence has a definite limit b.â 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
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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
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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
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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
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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
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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 squaresâ codes are generated by adding two bits after the former squareâs code. For example, in the second split, the four squares that
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Though the process of constructing the Peano curve is infinite steps, itâs not certain means the process include taking the limit of Peano curve. And though the limit of Peano curve is a square, itâs not enough to prove there is a one-toone mapping between the curve and the square. Base on the
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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, itâs meaningful to talk about the different between the process of constructing the
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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.
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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
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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
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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
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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 , that the sequence S (and itâs sub sequence) will include all the numbers in . Its contrary to uncountability of the real number system.
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This coding is just similar to the real numberâs 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 {HÂ ? 1(1/2),HÂ ? 2(1/2), . . . ,HÂ ? N(1/2), . . .}, and the center of
860:(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.
1291:(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, itâs 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.
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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.
1216:âThe first section of Cantorâs 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.
1294:(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.
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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 x to by sending (.x1x2x3... , .y1y2y3)-:
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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.
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It is the curve named âthe Peano curveâ 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.
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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
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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.
623:)! 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
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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 .
2285:"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.
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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 :
1571:(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.
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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?
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Though Cantorâs definition of irrational number didnât contain such paradox, but at times it is misunderstanded as the form that contain some paradox. Itâs meaningful to talk about how to avoid such misunderstanding.
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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.
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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
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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).
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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
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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 .
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The proof has used the induction unconspicuous. As the discussion in section 2, what the induction proved is the membersâ attribute in the fundamental sequence, not include the limit of the sequence.
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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 itâs a one
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equal to a set, the set has an element that a itself. The set will lead to the Russellâs paradox. This paper also discusses that Peano curve and some other plane filling curves contain such paradox.
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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).
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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).
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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.
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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 âsymbolâ (Zeichen) to describe the status of b.
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All the real numbers in are the limit points of S, but it's not means that there is a "limit number sequence" containts all the real numbers in . Otherwise the real number system is countable.
404:") 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
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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. --
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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 . . .
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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. --
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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.
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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.
2036:.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 -: -->
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I agree that information *should* be in Knowledge, and yet is not in this "space-filling curve" article. But now that information (including an algorithm in C) is in the
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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}:
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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
1892:) in itself. So the fixed point is in that set too. The HĂślder regularity of these curves seems to me relevant enough to be included, should one find a reference.--
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G. Breinholt, C Schierz, Algorithm 781: generating Hilbertâs space-filling curve by recursion, ACM Transactions on Mathematical Software (TOMS), 24 (1998) 184-189.
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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. Â --
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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).
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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:
1414:-to-one mapping, it has x = 1Â ? x = 1/2. That is, all the points in the median line are map to 1/2, itâs inconsistent with the one-to-one mapping postulate.
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correctedness of Peano's 1890 note in Mathematische Annalen has been checked by virtually any mathematician for the last 120 years (including Hilbert). --
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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. â
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Peano's original construction is self-contacting at all finite approximations, but Hilbert's construction is self-avoiding for all finite approximations.
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the square will be coded as 1/2. But according the definition, every point in sequence {HÂ ? 1(1/2),HÂ ? 2(1/2), . . . ,HÂ ? N(1/2), . . .} is a constant
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It can be proved by using the induction that the limit not in the fundamental sequence has some mathematical reason, rather than subjective definition.
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Russellâs 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 :
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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 ("
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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?)
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Thanks also to JRSpriggs for his latest enlightening contribution to this section. In this context, it is amazing that Hilbert was able to associate
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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. Â --
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whole square. This section should talk about another thing, the one-to-one mapping between a line segment and a square * base on Hilbert Curve.
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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.)
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1288:(1)P1 = 3.1, is a rational number; and 101 is an integer. m/101 is a ratio of two integers, according the definition, itâs a rational number.
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1950:"Roughly speaking, differentiability puts a bound on how fast the curve can move." - Turn would be more appropriate than move here, IMO.
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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.
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Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton, USA, 1990.
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This coding is hoped to give every point in the square a single code in , to finish the mapping the square ďż˝ to the line segment .
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D. Hilbert, Ueber die stetige Abbildung einer Line auf ein Fl?chenstck, Mathematische Annalen, 38 (1891) 459-460.
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to remove ambiguity) and let y be the bits in even positions and z be the bits in odd positions. i.e. if x = (.y
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and (2) that contraction takes the set of all curves satysfying a certain HĂślder condition (with a fixed
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G. Peano, Sur une courbe, qui remplit toute une aire plane, Mathematische Annalen, 36 (1890) 157-160.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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Yes! In other words, a space-filling curve, being self-intersecting, is similar to the function
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Do you have a reference for this? I also wonder if it's of sufficient interest to be included.
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pointing here, but all redirects which contained "Peano" and "curve" have been redirected to
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OK, here is the figure with the explanation of the word "proper" or "true" (from the article
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This relation is not a proper function (one input has two outputs). It is sometimes called a
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I think you are right (except that "impossible" is, in general, too strong). However, the
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generated from the square coded 0.00, will be coded coded as (0.00000.00010.00100.0011).
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That is, the limit of the sequence {H-1(x), H-2(x), . . . , H-N(x), . . . } is point p.
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Wasn't one of the attributes of the Peano curve the property of non self-intersection?
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requires that we only put content in our articles that has been published before in
436:
2150:
2142:
2091:
2074:
2039:
1494:). If you have a question about an issue that may need clarification, you can use
254:
2008:
1992:
1373:
2. The process of taking the limit may not keep the one-to-one mapping relation.
660:) are by all means true (proper) functions. I was confusing the right inverse of
2241:
2205:
2201:
2197:
2177:
2173:
2158:
2138:
2016:
1893:
1884:
A general way to see it, is that (1) these curve are fixed points of a suitable
1867:
1673:
1646:
1604:
1509:
1370:
1. The infinite process is not certain include the process of taking the limit;
791:
387:
286:
102:
2336:
of a countable union of measure-0 line segments is 0. Am I missing something?
1257:(1). In the real number field, a rational number a is equal to a sequence {a};
842:
must not be added. Therefore I have removed this addition from the article. Â --
757:
2374:
2359:
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2315:
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2154:
2114:
2082:
2066:
2024:
1965:
1938:
1901:
1875:
1860:
1765:
Could somebody add to article that most of these curves are (1/2)-HĂślder i.e.
1754:
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79:
1599:@"Qiuzhihong", "113.90.251.32", etc: the right place to put your question is
756:
either the illustration should be replaced with an illustration like this:
2350:
Already there are uncountably many linear functions on a unit interval. â
2157:, I would rename this article Peano curve, at least for homogeneity...
1264:
set has an element that a itself. It will lead to the Russellâs paradox.
1282:{Pn} = {3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592, 3.1415926, . . .}
648:
And you are right: contrary to what I wrote, the two right inverses of
668:
shown in the figure on the right, which would be a "quasi-inverse" of
2090:
I'm leaning towards removing this section entirely. Any objections?
207:
645:, but I am trying to describe the concept in a more intuitive way].
1490:
Knowledge is not the place for contributing original research (see
2129:
689:
in the side to each point in the square! (see also the discussion
396:
Proof that a square and its side contain the same number of points
2125:
This is in a sense a minor point; thus, I feel, not irrelevant.
1353:
0.011, 0.012, 0.013, 0.014, 0.015, 0.016, 0.017, 0.018, 0.019
1350:
0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009
1053:
0.011, 0.012, 0.013, 0.014, 0.015, 0.016, 0.017, 0.018, 0.019
1050:
0.001, 0.002, 0.003, 0.004, 0.005, 0.006, 0.007, 0.008, 0.009
311:
Space filling curves aren't one to one, so aren't bijections.(
15:
1851:. I think this is true for Hilbert's curve and Peano's curve.
691:
Intuitively acceptable examples about "larger than infinite"
253:
2256:
No it doesn't. The citation style it uses is standard. See
1297:
3 Peano Curve and uncountability of the real number system
693:, from which the idea of creating this section originated)
2280:
Article fails to address the topic of curves filling space
2053:
Your "standard proof" is a space-filling curve proof: see
1914:
For database retrieval, one doesn't use the Hilbert Curve
1279:
For example, the fundamental sequence associated with Pi:
1676:" article, which defines it and goes into more detail. --
323:
Any space-filling curve must be densely self-intersecting
853:
Peano Curve And Uncountability Of The Real Number System
803:
Relationship of Space-filling to Tiling and tessellation
1844:{\displaystyle \|f(x)-f(y)\|_{2}<c{\sqrt {|x-y|}}}
1771:
1344:
0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99
1338:
0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19
1335:
0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09
1044:
0.91, 0.92, 0.93, 0.94, 0.95, 0.96, 0.97, 0.98, 0.99
1038:
0.11, 0.12, 0.13, 0.14, 0.15, 0.16, 0.17, 0.18, 0.19
1035:
0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09
629:
586:
1367:
Therefore it has two simple but useful conclusions:
101:, a collaborative effort to improve the coverage of
613:
obtained by just reversing the arrows in the figure
546:as a right inverse - they are both injections from
1843:
637:
604:
281:A curve filling a countably dimensional hypercube
372:Under the Properties heading, the article says
368:Self-avoiding property of finite approximations
8:
1803:
1772:
1329:0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
1029:0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
866:(3). A counterexample of Peano's deduction:
758:http://mathworld.wolfram.com/PeanoCurve.html
206:, which collaborates on articles related to
869:Assume that a sequence of number sequence:
19:
1436:It finished a mapping from y to (x1, x2).
152:
47:
2232:This article need help to convert old to
2172:updated: in the meanwhile, a new article
1834:
1820:
1818:
1806:
1770:
672:, rather than one of its right inverses.
630:
628:
585:
2306:the unit square. Do I have that right?
2234:more standard Knowledge reference style
720:...) in base 2, then define f(x) = ( (.y
554:. But of course you also see there's no
435:
2029:
570:(see that article for more details). --
295:Space-filling curves are not bijections
154:
49:
1481:) 05:27, June 21, 2008 (UTC) â Please
351:of "curves intersecting" is that they
1210:2 Cantorâs definition of real number
7:
1601:Knowledge:Reference_desk/Mathematics
1496:Knowledge:Reference desk/Mathematics
262:This article is within the field of
200:This article is within the scope of
95:This article is within the scope of
38:It is of interest to the following
1023:s1, s2, ... are number sequences:
462:Right. In your picture, let's say
406:CantorâBernsteinâSchroeder theorem
14:
2403:Mid-priority mathematics articles
838:. Even if true, content based on
115:Knowledge:WikiProject Mathematics
2418:Systems articles in chaos theory
2398:Start-Class mathematics articles
2227:Please use standard <ref: -->
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118:Template:WikiProject Mathematics
82:
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2413:Mid-importance Systems articles
1421:5 Cantorâs dimension reduction
240:This article has been rated as
135:This article has been rated as
2274:16:18, 14 September 2019 (UTC)
2251:10:46, 14 September 2019 (UTC)
2128:The example given in 1890 by
2030:What's the point of Section 5?
1835:
1821:
1799:
1793:
1784:
1778:
1492:Knowledge:No original research
832:Knowledge verifiability policy
596:
1:
2167:13:38, 30 November 2012 (UTC)
2072:separated by a distance : -->
1902:08:15, 24 February 2011 (UTC)
1726:16:53, 11 November 2008 (UTC)
1706:16:53, 11 November 2008 (UTC)
1518:15:45, 19 December 2009 (UTC)
746:01:10, 17 December 2009 (UTC)
698:09:33, 9 September 2007 (UTC)
681:20:55, 7 September 2007 (UTC)
575:19:03, 7 September 2007 (UTC)
457:18:26, 7 September 2007 (UTC)
431:17:40, 7 September 2007 (UTC)
413:15:19, 7 September 2007 (UTC)
391:16:37, 2 September 2007 (UTC)
360:02:52, 11 November 2007 (UTC)
220:Knowledge:WikiProject Systems
109:and see a list of open tasks.
2423:WikiProject Systems articles
2408:Start-Class Systems articles
2025:06:07, 2 February 2012 (UTC)
1939:02:59, 13 January 2009 (UTC)
1876:15:21, 9 February 2010 (UTC)
1861:20:29, 8 February 2010 (UTC)
1655:08:32, 5 November 2008 (UTC)
1639:18:36, 29 October 2008 (UTC)
795:17:10, 12 October 2007 (UTC)
777:22:42, 10 October 2007 (UTC)
766:21:35, 10 October 2007 (UTC)
638:{\displaystyle \mathbf {c} }
290:14:13, 25 October 2006 (UTC)
223:Template:WikiProject Systems
1433:x2 = 0.b1b2 . . . bn . . .
1430:x1 = 0.a1a2 . . . an . . .
2439:
2331:05:11, 31 March 2024 (UTC)
2316:03:55, 31 March 2024 (UTC)
1711:Finding the Hilbert factor
1624:
1613:01:22, 14 March 2010 (UTC)
1581:12:59, 12 March 2010 (UTC)
1554:12:33, 12 March 2010 (UTC)
857:1 About Peano's deduction
246:project's importance scale
2375:06:35, 5 April 2024 (UTC)
2360:19:19, 3 April 2024 (UTC)
2346:18:30, 3 April 2024 (UTC)
2222:19:42, 12 July 2013 (UTC)
2186:18:10, 4 March 2013 (UTC)
2115:03:31, 6 April 2012 (UTC)
2100:15:41, 5 April 2012 (UTC)
2083:03:05, 4 April 2012 (UTC)
2067:01:28, 4 April 2012 (UTC)
2048:01:03, 4 April 2012 (UTC)
1966:15:28, 18 July 2010 (UTC)
1692:Densely Self-intersecting
1503:06:24, 27 June 2008 (UTC)
331:13:14, 23 Apr 2005 (UTC)
316:23:35, 11 June 2006 (UTC)
307:07:42, Mar 3, 2005 (UTC)
261:
239:
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134:
67:
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2295:12:25, 18 May 2022 (UTC)
2196:I left two redirects to
847:18:00, 2 June 2008 (UTC)
824:12:05, 2 June 2008 (UTC)
605:{\displaystyle f:Y\to X}
141:project's priority scale
1755:20:29, 5 May 2023 (UTC)
1686:19:13, 5 May 2023 (UTC)
1020:S = {s1, s2, s3, ...}.
98:WikiProject Mathematics
2121:Renaming this article?
1845:
1380:Fig. 2. Hilbert curve
639:
619:) than in the square (
606:
445:
258:
195:Systems science portal
28:This article is rated
1846:
1469:comment was added by
640:
607:
490:. You can use either
482:is a surjection from
439:
257:
1769:
1308:Fig. 1. Peano Curve
666:multivalued function
627:
584:
442:multivalued function
121:mathematics articles
203:WikiProject Systems
1997:pms:Curva ĂŤd Peano
1981:fr:Courbe de Peano
1910:Database retrieval
1841:
1625:Peano's 1890 curve
785:I must agree with
635:
602:
446:
259:
90:Mathematics portal
34:content assessment
2001:pt:Curva de Peano
1985:it:Curva di Peano
1977:es:Curva de Peano
1956:comment added by
1942:
1925:comment added by
1839:
1540:comment added by
1486:
840:original research
826:
814:comment added by
687:up to four points
278:
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151:
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2250:_citations": -->
2218:
2213:
2183:
2164:
2147:SierpiĹski curve
1968:
1941:
1919:
1899:
1850:
1848:
1847:
1842:
1840:
1838:
1824:
1819:
1811:
1810:
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1736:
1735:
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1666:
1665:
1610:
1556:
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1383:4 Hilbert curve
836:reliable sources
809:
751:Bad illustration
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642:
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611:
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2367:174.160.238.145
2365:square. Hmm.
2338:174.160.238.145
2308:174.160.238.145
2287:GlutenFreePizza
2282:
2230:
2216:
2211:
2194:
2181:
2162:
2123:
2032:
2009:sv:Peanos kurva
2005:ru:ĐŃĐ¸Đ˛Đ°Ń ĐоанО
1993:pl:Krzywa Peano
1974:
1951:
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1912:
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1761:HĂślder property
1733:
1731:
1713:
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1663:
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1608:
1535:
1483:sign your posts
1465:âThe preceding
855:
805:
753:
735:
731:
727:
723:
719:
715:
711:
707:
625:
624:
582:
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568:axiom of choice
420:axiom of choice
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32:on Knowledge's
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2380:
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2352:David Eppstein
2323:David Eppstein
2281:
2278:
2277:
2276:
2266:David Eppstein
2229:
2225:
2212:Randall Bart
2193:
2190:
2189:
2188:
2122:
2119:
2118:
2117:
2107:David Eppstein
2105:Not from me. â
2088:
2087:
2086:
2085:
2059:David Eppstein
2031:
2028:
2013:vi:ÄĆ°áťng Peano
1973:
1970:
1958:67.183.113.131
1947:
1944:
1927:FillerBrushMan
1911:
1908:
1907:
1906:
1905:
1904:
1879:
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1837:
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849:
816:86.160.138.236
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749:
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721:
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242:Mid-importance
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167:Midâimportance
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3:
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2170:
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2160:
2156:
2152:
2148:
2144:
2140:
2136:
2135:Hilbert curve
2131:
2126:
2120:
2116:
2112:
2108:
2104:
2103:
2102:
2101:
2097:
2093:
2084:
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2076:
2070:
2069:
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2064:
2060:
2056:
2055:Z-order curve
2052:
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2049:
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2041:
2027:
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2018:
2014:
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2006:
2002:
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1994:
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1986:
1982:
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1969:
1967:
1963:
1959:
1955:
1945:
1943:
1940:
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1928:
1924:
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1900:
1895:
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1887:
1883:
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1881:
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1873:
1869:
1865:
1864:
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1862:
1858:
1854:
1831:
1828:
1825:
1815:
1812:
1807:
1796:
1790:
1787:
1781:
1775:
1760:
1756:
1752:
1748:
1744:
1743:Hilbert curve
1739:
1730:
1729:
1728:
1727:
1723:
1719:
1710:
1708:
1707:
1703:
1699:
1691:
1687:
1683:
1679:
1675:
1669:
1660:
1659:
1656:
1652:
1648:
1643:
1642:
1641:
1640:
1636:
1632:
1614:
1611:
1606:
1602:
1598:
1597:
1596:
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1593:
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1566:
1565:
1564:
1563:
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1551:
1547:
1543:
1542:113.90.251.32
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1528:
1527:
1526:
1525:
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1519:
1515:
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1506:
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1480:
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1468:
1462:
1459:
1456:
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1450:
1447:
1443:
1442:6 Conclusion
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1419:
1415:
1412:
1409:
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1402:
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676:thought. :-)
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2143:Gosper curve
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1975:
1949:
1915:
1913:
1889:
1764:
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1667:
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1573:JamesBWatson
1536:â Preceding
1463:
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738:Unique-k-sat
701:
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348:intersection
326:
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264:Chaos theory
241:
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137:Mid-priority
136:
96:
62:Midâpriority
40:WikiProjects
2206:Peano curve
2202:Peano Space
2198:Peano space
2174:Peano curve
2139:Moore curve
1952:âPreceding
1921:âPreceding
1886:contraction
1745:article. --
1674:Peano curve
1449:References
810:âPreceding
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2392:Categories
2258:WP:CITEVAR
2155:Koch curve
1471:Qiuzhihong
1320:s2, . . .
761:below it.
418:Using the
2228:citations
2192:Redirects
1972:Interwiki
1853:David Pal
1747:DavidCary
1678:DavidCary
790:correct.
728:...), (.z
664:with the
556:injection
478:(c) = 3.
470:(d) = 2,
466:(a) = 1,
313:Balthamos
2236:, using
1954:unsigned
1946:Quibbles
1935:contribs
1923:unsigned
1550:contribs
1538:unsigned
1479:contribs
1467:unsigned
1220:number.
812:unsigned
695:Paolo.dL
678:Paolo.dL
454:Paolo.dL
450:Function
410:Paolo.dL
2262:WP:HARV
2238:ref-tag
2092:Wpegden
2075:Wpegden
2040:Wpegden
1500:Lambiam
1323:It is:
1056:......
1041:......
844:Lambiam
787:Lambiam
774:Lambiam
357:Lambiam
244:on the
217:Systems
208:systems
164:Systems
139:on the
2242:Krauss
2217:Talk
2017:Newone
1916:per se
1868:Hanche
1647:Hanche
1356:. . .
1341:. . .
1248:. . .
1241:. . .
1235:. . .
792:Hanche
538:(3) =
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512:(3) =
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402:proper
388:Hanche
287:Leocat
36:scale.
2130:Peano
1718:RJFJR
1698:RJFJR
1631:RJFJR
1498:. Â --
558:from
383:could
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301:Lethe
2371:talk
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2260:and
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2240:. --
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1751:talk
1738:Done
1722:talk
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1668:Done
1651:talk
1635:talk
1577:talk
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1475:talk
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305:Talk
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214:.
143:.
42::
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