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.
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.
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.
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.
It took me a moment to realize what it was exactly since I hadn’t yet added a field to indicate it, but since the IndieWeb chat doesn’t send webmentions by itself, I’m glad I support refbacks to be aware of comments on my posts. The avatar didn’t come through quite like it should, but it’s nice to be able to treat refbacks like any other type of mention.
The chat has some reasonable microformats markup, so I suppose the parser could do a more solid job, but this is a pretty great start. Sadly, Refback isn’t as real-time as Webmention, but it’s better than nothing.
My first mention (aka refback) from the IndieWeb chat. Click on the photo to see the UI display on my site.
I suppose we could all be posting chats on our own sites and syndicating into places like IRC to own our two directional conversations, but until I get around to the other half… (or at least for WordPress, I recall having gotten syndication to IRC for WithKnown working a while back via plugin.)
Former national security adviser H.R. McMaster on Tuesday defended his ex-White House colleague Gary Cohn for removing a trade-related document from President's Trump.
If it isn’t already, the learning management system will soon be obsolete, Jonathan Rees argues. Let’s replace it in ways that treat professors like the professionals they are.
This article has the flavor of IndieWeb about it…
Highlights, Quotes, Annotations, & Marginalia
The real internet is structured by myriad people with different aesthetics and different needs. Online course design decisions should reflect the instructor’s individuality in the same way that everyone else’s webpages do. ❧
September 26, 2018 at 05:22PM
Third, the post-LMS world should protect the pedagogical prerogatives and intellectual property rights of faculty members at all levels of employment. This means, for example, that contingent faculty should be free to take the online courses they develop wherever they happen to be teaching. Similarly, professors who choose to tape their own lectures should retain exclusive rights to those tapes. After all, it’s not as if you have to turn over your lecture notes to your old university whenever you change jobs. ❧
Own your pedagogy. Send just like anything else out there… September 26, 2018 at 05:27PM
President Donald Trump has grown increasingly dissatisfied with the way Brett Kavanaugh has defended himself in wake of sexual assault allegations that have threatened to derail his Supreme Court nomination, multiple sources tell CNN.
Trump, however, insisted later in the day that they weren't laughing at him, saying, "People had a good time with me. We were doing it together, we had a good time."
This past week, I finally pulled the plug and deleted my Facebook account. Or, rather, I scheduled my Facebook to be deleted in two weeks since, apparently, the company doesn’t allow users to delete accounts without giving itself a grace period. A lot of friends have pointed out real things I might miss by not being on the platform, but I feel so much lighter without the Facebook echo chamber in my head. I don’t judge people who stay on the platform–I’m not an evangelist–but it just wasn’t working for me.
Terence Tao (YouTube)
WordCamp: Publishers (Twitter)
