Sylvester's line problem, known as the Sylvester-Gallai theorem in proved form, states that it is not possible to arrange a finite number of points so that a line through every two of them passes through a third unless they are all on a single line. This problem was proposed by Sylvester (1893), who asked readers to "Prove that it is not possible to arrange any finite number of real points so that a right line through every two of them shall pass through a third, unless they all lie in the same right line."
Woodall (1893) published a four-line "solution," but an editorial comment following his result pointed out two holes in the argument and sketched another line of enquiry, which is characterized as "equally incomplete, but may be worth notice." However, no correct proof was published at the time (Croft et al. 1991, p. 159), but the problem was revived by Erdős (1943) and correctly solved by Grünwald (1944). Coxeter (1948, 1969) transformed the problem into an elementary form, and a very short proof using the notion of Euclidean distance was given by Kelly (Coxeter 1948, 1969; Chvátal 2004). The theorem also follows using projective duality from a result of Melchior (1940) proved by a simple application of Euler's polyhedral formula (Chvátal 2004).
Additional information on the theorem can be found in Borwein and Moser (1990), Erdős and Purdy (1991), Pach and Agarwal (1995), and Chvátal (2003).
In September 2003, X. Chen proved a conjecture of Chvátal that, with a certain definition of a line, the Sylvester-Gallai theorem extends to arbitrary finite metric spaces.
T. Gallai's proof has been outlined by P. Erdös in his submission of the problem to The American Mathematical Monthly in 1943.
Solution
Given the set Π of noncollinear points, consider the set of lines Σ that pass through at least two points of Π. Such lines are said to be connecting. Among the connecting lines, those that pass through exactly two points of Π are called ordinary.
R. Steinberg's was actually the first published solution to Syvester's problem,
Solution
Given the set Π of noncollinear points, consider the set of lines Σ that pass through at least two points of Π. Such lines are said to be connecting. Among the connecting lines, those that pass through exactly two points of Π are called ordinary. We consider the configuration in the projective plane.
The Sylvester–Gallai theorem in geometry states that, given a finite number of points in the Euclidean plane, either
* all the points lie on a single line; or
* there is a line which contains exactly two of the points.
It is named after James Joseph Sylvester, who posed it as a problem in 1893, and Tibor Gallai, who published one of the first proofs of this theorem in 1944.
A line that contains exactly two of a set of points is known as an ordinary line. According to a strengthening of the theorem, every finite point set (not all on a line) has at least a linear number of ordinary lines. There is an algorithm that finds an ordinary line in a set of n points in time proportional to n log n in the worst case.
Last night was the first lecture of Dr. Miller’s Gems And Astonishments of Mathematics: Past and Present class at UCLA Extension. There are a good 15 or so people in the class, so there’s still room (and time) to register if you’re interested. While Dr. Miller typically lectures on one broad topic for a quarter (or sometimes two) in which the treatment continually builds heavy complexity over time, this class will cover 1-2 much smaller particular mathematical problems each week. Thus week 11 won’t rely on knowing all the material from the prior weeks, which may make things easier for some who are overly busy. If you have the time on Tuesday nights and are interested in math or love solving problems, this is an excellent class to consider. If you’re unsure, stop by one of the first lectures on Tuesday nights from 7-10 to check them out before registering.
Lecture notes
For those who may have missed last night’s first lecture, I’m linking to a Livescribe PDF document which includes the written notes as well as the accompanying audio from the lecture. If you view it in Acrobat Reader version X (or higher), you should be able to access the audio portion of the lecture and experience it in real time almost as if you had been present in person. (Instructions for using Livescribe PDF documents.)
We’ve covered the following topics:
Class Introduction
Erdős Discrepancy Problem
n-cubes
Hilbert’s Cube Lemma (1892)
Schur (1916)
Van der Waerden (1927)
Sylvester’s Line Problem (partial coverage to be finished in the next lecture)
Over the coming days and months, I’ll likely bookmark some related papers and research on these and other topics in the class using the class identifier MATHX451.44 as a tag in addition to topic specific tags.
Course Description
Mathematics has evolved over the centuries not only by building on the work of past generations, but also through unforeseen discoveries or conjectures that continue to tantalize, bewilder, and engage academics and the public alike. This course, the first in a two-quarter sequence, is a survey of about two dozen problems—some dating back 400 years, but all readily stated and understood—that either remain unsolved or have been settled in fairly recent times. Each of them, aside from presenting its own intrigue, has led to the development of novel mathematical approaches to problem solving. Topics to be discussed include (Google away!): Conway’s Look and Say Sequences, Kepler’s Conjecture, Szilassi’s Polyhedron, the ABC Conjecture, Benford’s Law, Hadamard’s Conjecture, Parrondo’s Paradox, and the Collatz Conjecture. The course should appeal to devotees of mathematical reasoning and those wishing to keep abreast of recent and continuing mathematical developments.
Suggested Prerequisites
Some exposure to advanced mathematical methods, particularly those pertaining to number theory and matrix theory. Most in the class are taking the course for “fun” and the enjoyment of learning, so there is a huge breadth of mathematical abilities represented–don’t not take the course because you feel you’ll get lost.
I’d complained to the UCLA administration before about how dirty the windows were in the Math Sciences Building, but they went even further than I expected in fixing the problem. Not only did they clean the windows they put in new flooring, brand new modern chairs, wood paneling on the walls, new projection, and new white boards! I particularly love the new swivel chairs, and it’s nice to have such a lovely new environment in which to study math.
The newly renovated classroom space in UCLA’s Math Sciences Building
Category Theory for Winter 2019
As I mentioned the other day, Dr. Miller has also announced (and reiterated last night) that he’ll be teaching a course on the topic of Category Theory for the Winter quarter coming up. Thus if you’re interested in abstract mathematics or areas of computer programming that use it, start getting ready!
