Timeline for Proofs without words
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Nov 2, 2020 at 14:57 | comment | added | Pietro Majer | well, it is clear that if the curves are parametrized with constant velocity they converge uniformly | |
| Jun 26, 2014 at 7:45 | comment | added | Ian Morris | Contrary to gowers, I don't think it's clear that we are defining a sequence of continuous functions which converges uniformly. We aren't defining a sequence of functions at all, only a sequence of images of the interval under a function. If we choose the "wrong" sequence of parameterisations of this sequence of curves then we do not get a limit function. The sequence of parameterisations (which is not illustrated) is crucial to proving the existence of the limit object: without some indication of which parameterisations we must choose there is no proof. | |
| Jun 17, 2014 at 3:00 | history | edited | senshin | CC BY-SA 3.0 |
rehost to imgur to prevent linkrot
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| Feb 15, 2012 at 3:02 | comment | added | Gerry Myerson | @Michael Hardy, I think it has been done: youtube.com/watch?v=4RQmLNa5ZNo&feature=related | |
| Nov 17, 2011 at 14:14 | comment | added | Pietro Majer | Remarkably, no picture nor mention to it was made in Peano's article, the construction being completely based on ternary expansions. The picture of a sequence converging to a square-filling curve appeared one year later in the paper by Hilbert. | |
| Apr 10, 2011 at 20:18 | comment | added | gowers | If you look at the picture in detail you can see that you are defining a sequence of continuous functions that converge uniformly. It's also clear from the picture that the image is dense. Therefore the limiting function exists and its image (being dense and compact) is the whole square. Of course, this proof isn't 100% visual but the non-visual part -- the basic facts about uniform convergence and compactness -- can be regarded as background knowledge. So I think it's a nice example. | |
| Nov 16, 2010 at 21:51 | comment | added | Michael Hardy | Project: Fill the square one pixel at a time by following (an approximation to) this curve; then find some suitable baroque music accompaniment; then upload it to youtube. | |
| Sep 14, 2010 at 9:22 | comment | added | Johannes Hahn | Existence of the limits object is something that is very often forgotten. For example most Introductions to fractals give geometric descriptions of Koch's snowflake etc. via such an iteration but don't prove that there exists a limit of this iteration. | |
| Sep 14, 2010 at 8:47 | comment | added | Michael Burge | How can you be sure that you're eventually covering all the points with irrational or transcendental coordinates? And giving a sequence of curves which fill more and more of the plane isn't the same as giving a single curve that does it all at once - it's not clear that such a limiting curve exists just looking at the pictures. | |
| Sep 14, 2010 at 8:24 | history | answered | Alexis Monnerot-Dumaine | CC BY-SA 2.5 |