Math Forum Discussions

James A Landau

Posts: 217
Registered: 12/3/04

Re: [HM] Sylvester's problem of Colllinear points
Posted: Nov 23, 2004 9:35 AM

Plain Text

In a message dated Sun, 21 Nov 2004 09:18:05 -0500, Samuel S Kutler
<reltuk@comcast.net> writes:

> Coxeter gives the formulation of Sylvester's problem as
>
> Prove that it is not possible to arrange any finite number [of points]
> so that a right line through every two of them shall pass through
> a 3rd, unless they all lie in the same right line.
>
> Then, on page 66 of the 2nd edition of Intro to Geo, a proof by L M Kelly.
>
> What is known of Kelly?

Leroy M. Kelly was a professor of mathematics at Michigan State University
His specialty was geometry. Interestingly, his bachelor's degree was in civil
engineering, but he switched to mathematics, apparently in graduate school..
While at Michigan State (1965-1969) I took a couple of geometry courses from
him. He was also at the time one of the compilers of the Putnam exam ("You may
ask me if a particular question is on the next Putnam exam and I'll tell you
yes or no.") As the preceding quote shows, he was a "character", and my
recollection was that he was quite a good lecturer.

At one point during my time at MSU, I discovered that he had an entry in
_American Men of Science_. This entry also gave away something that Dr. Kelly
tried to keep secret, namely what his middle initial stood for.

I'm afraid I don't have any other information about Dr. Kelly except for the
following quote from Howard Eves _A Survey of Geometry_ (Boston: Allyn and
Bacon, 1965, no ISBN) volume II page 242:

"In a piece of geometrical research concerning some covering problems in
distance geometry, done around 1942 by L. M. Kelly, it became important to know if
there exists a tetrahedron whose 16-point sphere has a radius equal to half
the circumradius of the tetrahedron. In an effor t to settle the matter, the
question was proposed as Problem E 540 in the October 1942 issue of _The
American Mathematical Monthly._ The following extremely simple solution appeared
int eh June-July issue of the same journal. <snip> Let R be the circumradius
and r the radius of the 16-pont sphere. If the tetrahedron is regular, then r =
R/3; and if it is trirectangular, then r = [infinity symbol], with R finite.
Since we may continuously deform a regular tetrahedron into a trirectangular
one, it follows that at some intermediate stage, we must have r = R/2."

[Eves does not state who supplied the above proof.]

A true anecdote about Sylvester's Problem: the math department at MSU kept a
notebook called the "Red Cedar Book" (named after the Red Cedar River, which
runs past Wells Hall in which the math department offices were) in which
faculty members wrote down problems they wanted to see answers to. Someone wrote in
the Red Cedar Book the three-dimensional analog of Sylvester's Problem (which
easily reduces to Sylvester's Problem by projecting each point onto the same
plane) and a little while later someone else added a note to the effect "you
should talk to Dr. Kelly".

- James A. Landau, writing from the Atlantic City, New Jersey area,
where one of the maple trees out front still has golden leaves on it.