The Math Forum



Search All of the Math Forum:

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


Math Forum » Discussions » sci.math.* » sci.math

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Robert Hill

Posts: 529
Registered: 12/8/04
Re: Random Triangle Problem
Posted: Aug 6, 1997 1:37 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply



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
Read Random Triangle Problem
Mike Housky
7/21/97
Read Re: Random Triangle Problem
Bill Taylor
7/22/97
Read Re: Random Triangle Problem
tony richards
7/24/97
Read Re: Random Triangle Problem
Brian M. Scott
7/23/97
Read Re: Random Triangle Problem
tony richards
7/23/97
Read Re: Random Triangle Problem
T. Sheridan
7/24/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Bill Taylor
7/25/97
Read Re: Random Triangle Problem
Ilias Kastanas
7/23/97
Read Re: Random Triangle Problem
Robert Hill
7/23/97
Read Re: Random Triangle Problem
tony richards
7/27/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
tony richards
7/30/97
Read Re: Random Triangle Problem
Bill Taylor
7/30/97
Read Re: Random Triangle Problem
tony richards
8/1/97
Read Re: Random Triangle Problem
Bill Taylor
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/24/97
Read Re: Random Triangle Problem
Robert Hill
7/25/97
Read Re: Random Triangle Problem
Robert Hill
7/30/97
Read Re: Random Triangle Problem
Bill Taylor
8/1/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/1/97
Read Re: Random Triangle Problem
John Rickard
8/1/97
Read Re: Random Triangle Problem
Chris Thompson
8/1/97
Read Re: Random Triangle Problem
John Rickard
8/4/97
Read Re: Random Triangle Problem
Bill Taylor
8/5/97
Read Re: Random Triangle Problem
John Rickard
7/25/97
Read Re: Random Triangle Problem
Charles H. Giffen
7/25/97
Read Re: Random Triangle Problem
Charles H. Giffen
7/28/97
Read Re: Random Triangle Problem
Hauke Reddmann
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/28/97
Read Re: Random Triangle Problem
Robert Hill
7/29/97
Read Re: Random Triangle Problem
tony richards
7/30/97
Read Re: Random Triangle Problem
Keith Ramsay
7/30/97
Read Re: Random Triangle Problem
tony richards
8/2/97
Read Re: Random Triangle Problem
Keith Ramsay
7/29/97
Read Re: Random Triangle Problem
tony richards
8/4/97
Read Re: Random Triangle Problem
Bill Taylor
8/5/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/6/97
Read Re: Random Triangle Problem
Terry Moore
8/7/97
Read Re: Random Triangle Problem (LONG summary)
Terry Moore
8/16/97
Read Re: Random Triangle Problem (LONG summary)
Kevin Brown
8/17/97
Read Re: Random Triangle Problem (LONG summary)
Kevin Brown
7/30/97
Read Re: Random Triangle Problem
Muad'dib
7/30/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
tony richards
8/6/97
Read Re: Random Triangle Problem
Terry Moore
7/31/97
Read Re: Random Triangle Problem
John Rickard
7/30/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
Robert Hill
7/31/97
Read Re: Random Triangle Problem
Robert Hill
8/1/97
Read Re: Random Triangle Problem
R J Morris
8/4/97
Read Re: Random Triangle Problem
Robert Hill
8/4/97
Read Re: Random Triangle Problem
Robert Hill
8/5/97
Read Re: Random Triangle Problem
Charles H. Giffen
8/6/97
Read Re: Random Triangle Problem
Robert Hill

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.