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/ ]