Re: Illusion and Forthrightness in Wikipedia

[Vaughan Pratt <pratt@cs.stanford.edu>]

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

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