Many years ago, I happened across an account by a biologist --- citation alas long-forgotten --- who as a student had dutifully absorbed the well-known fact, immortalised in all the best textbooks, that the pancreas secretes amylase (or some other enzyme) in every mammalian species --- with the sole exception of the rhinoceros. As time went by, he prospered in his profession, and reached at last that pinnacle at which he could command the acquisition of a genuine rhinoceros pancreas (by then unattached, presumably, to its previous owner) in order to satisfy a persistant though entirely natural curiosity concerning whatever physiological peculiarity might be responsible for such an idiosyncrasy. And whaddya know? It did secrete amylase (or whatever it was) after all! [I don't suppose anyone can remind me of my long-forgotten source for this history?] Since then, I have encountered numerous rhinoceros' pancreases (or whatever the plural is). Here's the latest. In books on (fairly) elementary geometry will be commonly be found a construction credited to Klein, in which the usual hyperbolic, elliptic, Euclidean, etc. flavours are specialised from (say 3-space) projective geometry by choosing an "absolute" quadric K --- say c w^2 + x^2 + y^2 + z^2 = 0 with c constant, [x/w, y/w, z/w] Cartesian coordinates of a general finite point. The desired symmetry subgroup is selected by the constraint that K remains invariant. A standard account then proceeds along the lines of an associated polarity, quadric degenerate when Euclidean, winding up with "when c < 0, hyperbolic; c = 0, Euclidean; c > 0, elliptic". 'Ang abaht --- when c = 0, K represents a single (real) point at the origin --- that's not invariant under translation ??!! It seems that what the original author must have had in mind was rather the polarity, for which the quadric was simply a shorthand. In the Euclidean case the polarity turns out to be [w,x,y,z] -> [0,x,y,z]; that is, actually c = oo --- obviously enough, since the other geometries approach flatness as their radius increases. A recent transcription of this momentary lapse of XIX-th century concentration --- from Klein himself, perhaps? --- turns up in Judith N. Cederberg "A course in modern geometries" (2001), in section 4.12 on page 299 --- there in 2-space rather than 3-space. This particular pachyderm will doubtless still be trotting around, in 2-space and 3-space, for another 100 years. A little depressing, innit? Fred Lunnon
On 3/4/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
Since then, I have encountered numerous rhinoceros' pancreases (or whatever the plural is). ...
However, some more specific clarification now seems to called for --- I must admit I'm royally confused about this entire issue, and it would appear that I may not be alone. My absolute quadric will be c w^2 + x^2 + y^2 + z^2 = 0 . Whether the ground field is real or complex is essentially irrelevant. Reference (A) is Judith N. Cederberg "A course in modern geometries" (2001), section 4.12 on page 299 She discusses 2-space rather than 3-space, and her c corresponds to my 1/c (ouch!); translated, she actually claims that c = oo gives _affine_ (oof!), rather than Euclidean geometry. Now I had belatedly noticed that fixing the (degenerate) quadric c = oo permits dilations, as well as rotations; but she seems not to have realised that it also permits translations, which are not affine! Reference (B) is H.S.M.Coxeter "Non-Euclidean Geometry" sect 10.94 page 212 He says more cautiously that as c -> 0, the associated polarity degenerates [sect 9.5 p186 ff.], and there is a continuous transition from elliptic to hyperbolic via Euclidean; however as far as I can tell, he nowhere actually claims that fixing this (distinct) quadric determines the symmetry group. While fixing c = 0 obviously excludes translations, it's harder to establish whether this behaviour is a continuous limit ... The more I try to get to grips with what initially appeared to be a simple and elegant concept, the more unsatisfactorily slippery it appears to become! Fred Lunnon
On 3/6/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
... Now I had belatedly noticed that fixing the (degenerate) quadric c = oo permits dilations, as well as rotations; but she seems not to have realised that it also permits translations, which are not affine! ...
A well-wisher who prefers to remain anonymous has politely pointed out that I don't know my affine from my elbowine. Now I think about it, I have actually complained on several past occasions of not knowing what other people meant by this term (privately suspecting that they didn't know either) --- this time I carelessly assumed it must refer to the Cartesian linear group. But now he refers me to the fount of all wisdom, at which I may bathe and finally become cleansed (perhaps) --- http://en.wikipedia.org/wiki/Affine_transformation [I have been previously been taken sniffily to task for citing wikipedia --- but have to aver that they seem pretty reliable, if often unpolished and on the terse side, regarding mathematical topics I feel competent to judge.] An "affine transformation", it sez 'ere, is x -> A x + b where A, b are fixed matrix, vector and x is a (finite dimensional) vector variable. Ah: I knew there was something dodgy about this notion --- these transformations are not closed under composition! So that has to be patched up by immediately abandoning the Cartesian coordinates motivating it, and moving homogeneous coordinates instead. Which perhaps we should have been using all along. But then, of course, we'd never have even considered coming up with such a clunky definition in the first place. What conceivable application does it have? Has anybody ever come across a problem which could be solved using affine geometry, rather than Cartesian, Euclidean, projective, elliptic, hyperbolic ... you know, all the intuitively reasonable options? A brave attempt is made to refute such intemperate criticism at http://en.wikipedia.org/wiki/Affine_geometry which quotes in particular the theorem about the concurrency of the medians of a triangle. I find this example unconvincing: surely it is essentially (a special case of) a theorem of projective geometry? I advise all you youngsters out there to cultivate a deep suspicion of anyone using this term in the future! Including me. Especially me. Fred Lunnon
On 3/6/11, Fred lunnon <fred.lunnon@gmail.com> wrote:
--- this time I carelessly assumed it must refer to the Cartesian linear group.
As I recall, Einstein was rather fond of the term in discussing relativity, but I don't remember why. Weyl, too. Perhaps the term "linear" means "first degree" rather than "vector space homomorphism." In any event, the constant term can be given a nominal vector coefficient and that is the equivalent of using projective space. Altogether you get the semidirect product of rotations (or suchlike) and translations. Isn't Wikipedia wonderful? "The Father of all Lies!" -hvm
On Monday 07 March 2011 01:58:27 Fred Lunnon wrote:
An "affine transformation", it sez 'ere, is x -> A x + b where A, b are fixed matrix, vector and x is a (finite dimensional) vector variable. Ah: I knew there was something dodgy about this notion --- these transformations are not closed under composition! So that has to be patched up by immediately abandoning the Cartesian coordinates motivating it, and moving homogeneous coordinates instead.
A(Cx+d)+b = (AC)x + (Ad+b). In what sense are these transformations not closed under composition? -- g
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