Illusion and Forthrightness in Wikipedia

["Fred E.J. Linton" <fejlinton@usa.net>]

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/ ]