Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Random Triangle Problem
Replies: 57   Last Post: Aug 17, 1997 10:51 PM

 Messages: [ Previous | Next ]
 Robert Hill Posts: 529 Registered: 12/8/04
Re: Random Triangle Problem
Posted: Aug 6, 1997 1:37 PM

In article <5s63dg\$pgj\$3@cantuc.canterbury.ac.nz>, mathwft@math.canterbury.ac.nz (Bill Taylor) writes:
> eclrh@sun.leeds.ac.uk (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)

Date Subject Author
7/16/97 Mike Housky
7/21/97 Bill Taylor
7/22/97 tony richards
7/24/97 Brian M. Scott
7/23/97 tony richards
7/23/97 T. Sheridan
7/24/97 Bill Taylor
7/24/97 Bill Taylor
7/25/97 Ilias Kastanas
7/23/97 Robert Hill
7/23/97 tony richards
7/27/97 Bill Taylor
7/24/97 Robert Hill
7/28/97 tony richards
7/30/97 Bill Taylor
7/30/97 tony richards
8/1/97 Bill Taylor
7/24/97 Robert Hill
7/24/97 Robert Hill
7/24/97 Robert Hill
7/25/97 Robert Hill
7/30/97 Bill Taylor
8/1/97 Charles H. Giffen
8/1/97 John Rickard
8/1/97 Chris Thompson
8/1/97 John Rickard
8/4/97 Bill Taylor
8/5/97 John Rickard
7/25/97 Charles H. Giffen
7/25/97 Charles H. Giffen
7/28/97 Hauke Reddmann
7/28/97 Robert Hill
7/28/97 Robert Hill
7/28/97 Robert Hill
7/29/97 tony richards
7/30/97 Keith Ramsay
7/30/97 tony richards
8/2/97 Keith Ramsay
7/29/97 tony richards
8/4/97 Bill Taylor
8/5/97 Charles H. Giffen
8/6/97 Terry Moore
8/7/97 Terry Moore
8/16/97 Kevin Brown
8/17/97 Kevin Brown
7/30/97 Robert Hill
7/31/97 tony richards
8/6/97 Terry Moore
7/31/97 John Rickard
7/30/97 Robert Hill
7/31/97 Robert Hill
7/31/97 Robert Hill
8/1/97 R J Morris
8/4/97 Robert Hill
8/4/97 Robert Hill
8/5/97 Charles H. Giffen
8/6/97 Robert Hill