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Topic:
Re: [HM] Sylvester's problem of Colllinear points
Replies:
4
Last Post:
Nov 25, 2004 10:32 PM




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


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 (19651969) 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 16point 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 JuneJuly issue of the same journal. <snip> Let R be the circumradius and r the radius of the 16pont 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 threedimensional 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.



