Illusion and Forthrightness in Wikipedia
These somewhat off-topic remarks are at least almost as connected with category theory as the study of knots is, as they have their origin in my reliance on, and subsequent disillusionment with, the Wikipedia site for basic information about a certain knot. The knot? -- the familiar trefoil knot, aka (2,3)-torus knot. The offending Wikipedia page? -- http://en.wikipedia.org/wiki/Torus_knot . The problem? A reader might be forgiven for expecting that when a page of mathematical text offers a parametrization, in the form x = (2 + cos((q phi)/p))(cos(phi)) , y = (2 + cos((q phi)/p))(sin(phi)) , z = sin((q phi)/p) , of each (p,q)-torus knot, and then offers an illustration dubbed (2,3)-torus knot, that said illustration might have been produced by means of the p=2, q=3 instance of the parametrization given. But think just a moment, if you have that Torus_knot page open: for phi = 0, one has x = 3 and y = 0, which is *not* a point on the (2,3)-torus knot as shown *unless* one thinks of the "x-axis" as running vertically, counter to the usual expectation. Oh, but even then, taking the x-axis as vertical and the y-axis horizontal, if one uses PostScript to "draw" the curve with the parametrization given above (with p=2 and q=3, of course), the result is the "lumpy" figure I've put up, as .ps and .png files, respectively, here: http://tlvp.net/~tlvp/Trefoil/Wiki-2-3-Torus.ps , http://tlvp.net/~tlvp/Trefoil/Wiki-2-3-Torus.png . (You can "check the math" in the text of the .ps file, or let it be displayed as a graphic using, say, GhostView; a .png file is offered for those without any PostScript viewer.) Clearly the (2,3)-torus knot illustration is *not* obtained from the parametrization the Wiki page offers. Rather, it comes from a vertically oriented ellipse, with major and minor radii 3 and 1, respectively (approximately), drawn on a sheet of paper undergoing its own concurrent slow rotation as the ellipse is being drawn (in fact: an ellipse "with 30 degrees of precession for each 90 degrees of ellipse"), as shown, again in both .ps and .png files, here: http://tlvp.net/~tlvp/Trefoil/TrigTrefoilElliptic.ps , http://tlvp.net/~tlvp/Trefoil/TrigTrefoilElliptic.png . (Compare this precessional ellipse with the Wiki b/w illustration.) At first I was quite outraged that Wikipedia could be so utterly cavalier with mathematical accuracy. Then I thought, "Well, the (2,3)-torus knot as described in the text and the knot of the black/white illustration on that page, while clearly different from a curvature perspective (one has 6 points of zero curvature, the other has none), are at least equivalent as knots, so what's the harm? And finally I thought, "A reader who is informed of the parametrizations for each of a family of curves, and then sees displayed what is labeled as one of the curves in that family, has the right, if not explicitly informed otherwise, to suppose that the parametrization used for that displayed curve is the parametrization already given. For why else would the parametrization being used for the displayed curve not be mentioned? Only (presumably) because it should go *without saying*. So it's really rather dreadfully misleading -- if not downright dishonest (!) -- to lead the reader into temptation-to-err by omitting mention of the very different parametrization being used for that display." And that's why I write here now: How does one fix such a state of affairs? Or is there no better to be hoped for from Wikipedia? Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 9/12/2010 5:11 PM, Fred E.J. Linton wrote:
The knot? -- the familiar trefoil knot, aka (2,3)-torus knot. The offending Wikipedia page? -- http://en.wikipedia.org/wiki/Torus_knot . The problem? [...] if one uses PostScript to "draw" the curve with the parametrization given above (with p=2 and q=3, of course), the result is the "lumpy" figure I've put up, as .ps and .png files, respectively, here:
http://tlvp.net/~tlvp/Trefoil/Wiki-2-3-Torus.ps , http://tlvp.net/~tlvp/Trefoil/Wiki-2-3-Torus.png .
There are really two trefoil knots: (i) the (2,3)-torus knot, which is a creature of topology, being defined up to homeomorphism of the complementary space, for which lumpiness is not an invariant; (ii) The trefoil arising as a a common motif in iconography and the visual arts. Thare are a number of variants of these, each defined up to similarity. They are dealt with in the section "Trefoils in religion and culture" on the Wikipedia page http://en.wikipedia.org/wiki/Trefoil_knot You seem to be interested in the second. Conceivably you could try modifying the topology article, but you will have to deal somehow with the objection that "lumpiness" is not a topological invariant, so why spoil a perfectly good parametrization by replacing it with a more complicated one that's solving a non-problem for this article? A better approach might be to expand the art section with a little mathematics giving a suitable parameterization of the kind you want. The ellipse-based one you suggest seems to work fine, and translates back from your postscript if I'm not mistaken as x = sin(3t)cos(t) + 3 cos(3t)sin(t) y = 3 cos(3t)cos(t) - sin(3t)sin(t) Here's another I came up with just now that also works. r = 4 - cos(3t) x = r sin(2t) - 3 sin(t) y = r cos(2t) + 3 cos(t) The parametrization in the torus knot article is this one with "4 - " replaced by "2 + " and the second half deleted from the expressions for x and y. Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Incidentally the following postscript program produces the trefoil at http://boole.stanford.edu/Trefoil.jpg =====clip here======== %!PS /step { .01 exch 1 1 1 setrgbcolor 0 setlinewidth {/t exch def /r 4 3 t mul cos sub def 2 t mul sin r mul t sin 3 mul sub 2 t mul cos r mul t cos 3 mul add lineto currentpoint stroke moveto t 360 div 1 1 sethsbcolor 2 setlinewidth} for } def 36 36 scale 8.5 11 translate 0 0 moveto 0 360 step 1 -1 moveto 60 80 step showpage =====clip here======= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Sun, Sep 12, 2010 at 5:11 PM, Fred E.J. Linton <fejlinton@usa.net> wrote:
And that's why I write here now: How does one fix such a state of affairs?
Click the "edit" button on the page and start typing.
Or is there no better to be hoped for from Wikipedia?
This is precisely the benefit of Wikipedia--people like you fix the errors. -- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
The parametrisation given corresponds to the symmetric placing of the trefoil on a torus, the fact that you get the `lumps' is an artifact of that. The knot parametrisation can neatly `turned around' changing the role of p and q. That is also a trefoil! Have fun, Take care. Don't tie yourself in knots :-) Tim On 13/09/2010 01:11, Fred E.J. Linton wrote:
These somewhat off-topic remarks are at least almost as connected with category theory as the study of knots is, as they have their origin in my reliance on, and subsequent disillusionment with, the Wikipedia site for basic information about a certain knot.
The knot? -- the familiar trefoil knot, aka (2,3)-torus knot. The offending Wikipedia page? -- http://en.wikipedia.org/wiki/Torus_knot . The problem?
A reader might be forgiven for expecting that when a page of mathematical text offers a parametrization, in the form
x = (2 + cos((q phi)/p))(cos(phi)) , y = (2 + cos((q phi)/p))(sin(phi)) , z = sin((q phi)/p) ,
of each (p,q)-torus knot, and then offers an illustration dubbed (2,3)-torus knot, that said illustration might have been produced by means of the p=2, q=3 instance of the parametrization given.
... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi, Peter,
P.S.: I am very impressed that you can write PostScript code by hand.
Turns out, now that I've gotten started on it, to be way easier than TeX; easier even than HTML.
I thought I was the only person in the universe who did this.
Nope: a whole lot of PostScript afficionados are to be found in the Usenet Newsgroup comp.lang.postscript; and there are more in some of the pdf newsgroups. But we are a pretty select minority :-) . Cheers, -- Fred
Fred E.J. Linton wrote: [a long complaint about a Wikipedia article]
And that's why I write here now: How does one fix such a state of affairs? Or is there no better to be hoped for from Wikipedia?
One fixes a Wikipedia article by editing it. http://en.wikipedia.org/wiki/Wikipedia:How_to_edit_a_page Hope this helps. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi again, Peter, ------ Original Message ------ Received: Mon, 13 Sep 2010 02:48:23 PM EDT From: selinger@mathstat.dal.ca (Peter Selinger) To: fejlinton@usa.net Subject: Re: categories: Illusion and Forthrightness in Wikipedia
P.S.: I am very impressed that you can write PostScript code by hand.
Give me a reason to impress you twice over: help me learn to write SVG code by hand :-) .
I thought I was the only person in the universe who did this.
That'd be too lonely a position to be in, don'tcha think? Cheers, -- Fred
On Mon, Sep 13, 2010 at 8:11 AM, Fred E.J. Linton <fejlinton@usa.net> wrote:
And that's why I write here now: How does one fix such a state of affairs? Or is there no better to be hoped for from Wikipedia?
The way to fix it is: go to Wikipedia, click "Edit" and fix it. I was going to do this and say "look, it took just one minute!" - but it seems Peter Selinger beat me to it. (Just click on "View History" and you'll see someone named Selinger made this change on 1:52 UTC, September 14th, 2010.) Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 9/13/2010 12:02 PM, Timothy Porter wrote:
The parametrisation given corresponds to the symmetric placing of the trefoil on a torus, the fact that you get the `lumps' is an artifact of that. The knot parametrisation can neatly `turned around' changing the role of p and q. That is also a trefoil!
Granted a (3,2)-torus knot is also a trefoil knot topologically, but how does that help Fred? With the standard parametrization the plan view lacks the lumps bugging Fred but it has four crossings when Fred wants to keep it at three. What's the minimal modification to the standard parametrization x + iy = (cos(qt) + 2)exp(ipt) (sticking with z = sin(qt)) that gives a smooth trefoil with 3 crossings, for (p,q) either of (2,3) or (3,2)? I have this feeling there ought to be something slicker than (cos(qt) - 4)exp(ipt) - 3 exp(-it) with (p,q) = (2,3) (what I gave before, reflected about x+y = 0) but I can't see it. (I also don't know what -4 and -3 should generalize to for other than (2,3), but Fred hasn't asked for that yet.) Monoidal but not symmetric (trying to stay in scope here). Vaughan [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hi, Peter, You're right -- I *had* lost it:
z(t) = sin 3t
Oops, this should have been z(t) = sin 6t (in my parametrization and yours), or else it won't plot correctly in 3 dimensions.
Mmm ... I think "3t" remains correct in mine ... or have I lost it?
Not sure. Your parametrization of the (x,y)-coordinates is periodic with period pi. Your parameterization of z is periodic with period 2/3 pi. This seems to imply that for t in [0,pi], your curve traces out the same (x,y) coordinates as for [pi,2pi], but with the z coordinate negated.
Right, 6t it should have been, all along. Sorry 'bout that :-) . Cheers, -- Fred
On Tue, 14 Sep 2010 07:02:26 PM EDT John Baez <baez@math.ucr.edu> wrote:
... The way to fix it is: go to Wikipedia, click "Edit" and fix it.
I was going to do this and say "look, it took just one minute!" - but it seems Peter Selinger beat me to it. (Just click on "View History" and you'll see someone named Selinger made this change on 1:52 UTC, September 14th, 2010.)
Yep; certainly now my original objection no longer applies. Instead, I'm annoyed by the mealy-mouthed lack of content in the new "disclaimer" (?), | The illustrations on this page are derived from various different | parametrizations. Shouldn't a Wikipedia article be *providing* (rather than suppressing) information? Is it really informative to hide the fact that the color illustration (at top) for the (3,7)-torus knot *is* using the parametrization that the text displays, while the b/w (2,3)-torus knot (displayed next) is using rather a *different* style of parametrization, whose details are ... well, you see what I'm after? And if I change it, who next will change it away again? Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 9/17/2010 11:58 PM, Fred E.J. Linton wrote:
Is it really informative to hide the fact that the color illustration (at top) for the (3,7)-torus knot*is* using the parametrization that the text displays, while the b/w (2,3)-torus knot (displayed next) is using rather a*different* style of parametrization, whose details are ... well, you see what I'm after? And if I change it, who next will change it away again?
At first I thought there must have been some race condition here, until I transformed all times including those on Fred's email header into UTC (Universal Coordinated Time). The relevant events in chronological order are then 01:52, 14 September: User Selinger "clarified that images don't agree with formula," by pointing out in the Wikipedia article that "Other parametrizations are also possible, because knots are defined up to continuous deformation. The illustrations on this page are derived from various different parametrizations." 20:29, 17 September: User Vaughan Pratt "illustrated point about different parametrizations" by appending to Peter's second sentence ": for example (leaving z unchanged) the (3,8)-torus knot uses r = cos(qφ) + 4 for a smoother effect while the (2,3)-torus knot avoids inflexion points altogether by taking x + iy = r e^{pφi} − 3 e^{−φi} where r = cos(qφ) − 4." 06:58, 18 September: Fred posted to the categories list as above, namely that the 14 September edit was insufficient and that the (2,3)-torus knot is using a different *style* of parametrization. Now Wikipedia edits are seen essentially instantaneously by all users. Hence Fred must have based his complaint on the state of the article 10.5 hours before his post to this list, without checking whether the current version still had that defect (e.g. by refreshing the page). Since x + iy = re^{pφi} − 3e^{−φi} is merely the Euler-De Moivre abbreviation for x = r cos(pφi) - 3 cos(−φi), y = r sin(pφi) - 3 sin(−φi), I wouldn't call this a change in *style* from the original parametrization of x = r cos(pφi), y = r sin(pφi). (Now that I think of it, the abbreviation is not short enough to offset the loss of clarity so I unabbreviated it just now. All these changes can be tracked with the article's history page, accessed via the History tab at the top.) Fred's variable-ellipse representation of the "aflective" (inflexion-point-free) trefoil that he posted to this list the other day is certainly a change in style, regardless of whether it's equivalent to some version of the original style. But his claim that his is *the* representation is called into question by the above parametrization, which can be seen at http://boole.stanford.edu/Trefoil.jpg (actually I changed it very slightly by decreasing "- 4" to "- 4.3", increasing "- 3" to "- 2.93", and rotating the figure 90 degrees clockwise, for a closer match to the figure, but perhaps the figure should be changed instead). But whether the Wikipedia figure is a better match to my parametrization or to Fred's is I would say moot given that there is no need to change parametrization *style* to get an aflective trefoil, as should be clear from http://boole.stanford.edu/Trefoil.jpg . Even more puzzling to me is that prior to Fred's posting to this list on this topic he had sent me 32 emails regarding his problem of drawing a trefoil-shaped logo in Postscript, and I'd answered him 25 times, to his complete satisfaction I thought. That he then turned to the categories list for further help with trefoils would seem to indicate that I'd been less helpful than I thought. I guess I can justify whatever he found unhelpful with the observation that with free consulting you get what you pay for. Vaughan Pratt [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
In response to jb, John Baez <baez@math.ucr.edu>, who wrote:
Instead, I'm annoyed by the mealy-mouthed lack of content in the new "disclaimer" (?), Then change it, or argue about this issue on the talk page for that
... >Yep; certainly now my original objection no longer applies. article.
Fair enough; thanks for the tip.
Every Wikipedia has a talk page, accessible with a single mouse click, where people discuss that article and how to improve it. Registering your complaints here rather than there merely makes it much less likely that they will have any impact.
But if you're saying you just don't like Wikipedia, fine.
If that's all you think I'm saying, I'm sorry. But no matter; ... .
Best, jb
Cheers, -- Fred [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (6)
-
Fred E.J. Linton -
John Baez -
Mike Stay -
Timothy Porter -
Toby Bartels -
Vaughan Pratt