In article <email@example.com>, firstname.lastname@example.org (Bill Taylor) writes: > email@example.com (Robert Hill) writes: > > |> To say the same thing > |> slightly differently, it was not for me obvious a priori that it was right > |> to assume the vertices to be independently and uniformly distributed on the > |> circumcircle. Why should there not be a tendency for them to be a bit > |> bunched, or a bit dispersed? > > Indeed, good point. A priori it is not at all obvious they should be > independent. But closer inspection reveals that it is merely a somewhat > disguised form of the angles-at-the-centre-and-at-the-circumference result.
What is a disguised form of what depends on what you are taking as obvious. (Is that sentence comprehensible?) It seems a long time since I wrote the words which Bill is commenting on here, and I can't be bothered to dig out the larger passage from which they are snipped, but IIRC what I was trying to say was that
(a) It's not initially obvious [and, I might now add, after the Cauchy results, maybe not true] that it's a good idea to replace the notion of UID vertices by UID orientations of sides. Though the side orientations individually should obviously be uniformly distributed, it might be that, if we could find some "right" way of thinking about UID vertices, we would see that they imply some form of dependence among the side orientations.
[UID = uniformly and independently distributed. Nothing to do with IUD or UDI.]
(b) Fairly obviously, via the angle-at-centre stuff, UID orientations of sides are equivalent to UID vertices on the circumcircle, among triangles with a given circumcircle.
(c) Translating (a) via (b), we get the statement quoted by Bill. When one is looking for intuitions, it is often worthwhile to try replacing a statement with a logically equivalent but superficially different one. If one has no intuitions about one version, one may have some about the other. (Or one may, of course, feel that "A is obviously true" and that "B is obviously false" and be embarrassed by the discovery that A and B are equivalent!)
As I said elsewhere, I'm getting more and more dubious about the idea that we can have any worthwhile intuitions about the mathematically meaningless uniform distribution of vertices. Hence I am now very dubious about the idea in the second sentence of (a). But if I wanted to defend that sentence, I'd say that it's not very different from what some set theorists do, when, having proved the independence of (say) CH, they start looking for new "obviously true" axioms which would enable CH to be decided one way or the other.
> |> Another question: is there some interesting shape of bounded set > |> such that the probability is 3/4 for vertices independently and > |> uniformly distributed within that set? > > What an excellent question! I hope someone finds one. Meanwhile we > need those n-gon results...
Charles Giffen has now pointed out that there must be a rectangle. I'd rather lost interest in the n-gons after a mystery was solved by the discovery that my initial circle results were really for a quadrant. I'll try to grab a few minutes to look at that program some time.
-- Robert Hill
University Computing Service, Leeds University, England
"Though all my wares be trash, the heart is true." - John Dowland, Fine Knacks for Ladies (1600)