Talk:Space-filling curve

message: 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)
thread_title: A curve filling a countably dimensional hypercube

message: 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)
thread_title:  Space-filling curves are not bijections 

message: 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)
thread_title:  Self-avoiding property of finite approximations 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Bad illustration 

message: 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)
replies: text: 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)
thread_title: Relationship of Space-filling to Tiling and tessellation

message: 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)
replies: text: 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)

text: 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)
thread_title:  Peano's 1890 curve 

message: 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)
thread_title: Densely Self-intersecting

message: 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)
replies: text: 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)
thread_title: Finding the Hilbert factor

message: 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)
replies: text: 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)
replies: text: 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.--pma 08:15, 24 February 2011 (UTC)
thread_title: Hölder property

message: 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)
thread_title: Database retrieval

message: "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)
thread_title: Quibbles

message: :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)
thread_title:  Interwiki 

message: 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)
replies: text: Your "standard proof" is a space-filling curve proof: see Z-order curve. —David Eppstein (talk) 01:28, 4 April 2012 (UTC)
replies: text: 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)

text: Not from me. —David Eppstein (talk) 03:31, 6 April 2012 (UTC)
thread_title:  What's the point of Section 5? 

message: 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... pma 13:38, 30 November 2012 (UTC)
replies: text: 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. --pma 18:10, 4 March 2013 (UTC)
thread_title: Renaming this article?

message: 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)
thread_title:  Redirects 

message: This article need help to convert old to more standard  Wikipedia reference style, using ref-tag. --Krauss (talk) 10:46, 14 September 2019 (UTC)
replies: text: 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)
thread_title:  Please use standard <ref> citations  

message: 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)
replies: text: 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)
replies: text: 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)
replies: text: 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)
replies: text: Already there are uncountably many linear functions on a unit interval. —David Eppstein (talk) 19:19, 3 April 2024 (UTC)
replies: text: 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)
thread_title:  Article fails to address the topic of curves filling space