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Georgia Teen Stumbles On New Theorem
dread minerva writes "Proof that the kids are alright: The Atlanta Journal-Constitution published the following article about Josh Klehr, who discovered a math theorem while sitting in study hall one day in eigth grade. The theorem is now known as the Klehr-Bliss Theorem and a paper on it is being published in The American Mathematical Monthly."
And some of the most obscure set theory stuff, the Banach-Tarski Theorems, which were thought to be completely abstract actually help describe the Eightfold way of quark theory.
/.'ers: Number theory was thought to be the "queen of mathematics", unspeakably pure. Of course, now it's the workhorse of crypto.
Other "useless" stuff, but of more interest to
General Relativity: Space-time tells matter where to go; Matter tells space-time what shape to be.
Don Knuth solved a math problem at the beginning of the school year in High School. He got an A on the course and was excused from any extra work.
That article reads like a story on the onion:
Geekboy average Joe grocery-bagger astounds mathematician PhD's with a doodle from study hour.
Squeezed in the margin of his geography text book under a crude replica of a Limp Bizkit logo, a weird triangle with intersecting lines gives hope to millions of parents that their kid might actually do something meaningful.. even if the kid doesn't fully understand what they've done.
Speak truth to power.
Basically right, except you leave out the really important part, which is where Gauss explains his work and makes it accessible even to his teacher. How he did this was to argue that the sum of the numbers from 1 to n is half of twice that sum. Okay...we can go for that. But then he points out that this double sum can be written as n terms that combine the ascending and descending series like this:
(1+(n)), (2+(n-1), (3+(n-2), ... ((n-1)+2), ((n)+1)
Now, each of these terms has the sum (n+1), and there are n such terms since there are n terms in the original series. So the double sum is just n*(n+1), and the sum we want is just half of that.
And that is why he's Gauss, and you're not. :-)
Babar
Say what you will about this. But what amazes me about this story is that this kid took the initiative to check on whether the idea was novel or not. I think we can all learn a little from that.
The nine point circle includes:
o the midpoints of the three sides
o the feet of the three altitudes
o the midpoints of the lines joining the orthocenter (there the three altitudes meet) to the vertices.
The easiest way to find it is to simply take any of these three groups (say, the midpoints of the three sides) and find the circle that touches those three - the circle that circumscribes the triangle formed by those three.
--
Vidi, Vici, Veni
He was not suspended, expelled, or arrested for "Thinking While In Highschool"!
I AM, therefore I THINK!
Here [http://www.csm.astate.edu/Ninept.html] is a more visual definition of a nine-point circle for people like me who are much more visually oriented.
THIS is the kind of news that should be reported, not "some guy shot some clerk on the other side of this state" or "it might snow in [distant state] tonight."
Use my userscript to add story images to Slashdot. There's no going back.
First and foremost, I don't think the theorem is actually called the Klehr-Bliss theorem. AFAIK it's the van Lamoen theorem, since he was the first to furnish a full proof. Lou Talman had a quicker (and simpler) proof that was purely geometric, but I believe it was found to be flawed. I was working on a brute-force algebraic manipulation, but Floor van Lamoen carried essentially the same technique to its completion before I was able to. You can read about his proof here.
Josh's conjecture was pretty accurately summarized in the article. The point E mentioned is actually the circumcenter, the center of the only circle passing through the three vertices of the triangle. Also, it is not exactly correct to say that the lines through A, B, and C intersect in "a point" inside the triangle. The three lines are concurrent (they all pass through a common point, a rare thing for three lines to do), but Josh's slope-reciprocal construction is really just a reflection about the line y=x in the coordinate plane, and changing the orientation of the coordinate axes relative to the triangle makes the point of concurrence wander around inside the triangle. The kicker that I noticed is that as it wanders, it stays on the nine-point circle, or Feuerbach circle of the triangle. I've actually found that there's a lot more to be said along these lines, and to my knowledge none of it has been published.
For the public/private thread... I think that Josh was and is attending a private school (Paideia, an excellent school by the way) though I attended a public one (Collins Hill... not too bad as public schools go).
Not only does the theorem have little to no practical value, it also is of little interest to mathematicians. I've always thought of it as simply a little ditty in triangle geometry. I haven't yet read van Lamoen's article in the AMM, but I believe he mentions it only in passing.
And yes, it is vitally important to have an encouraging mentor. Steve Sigur, Josh's teacher, is a great guy and an excellent math teacher. I don't mean this to trivialize Josh's accomplishment--it's also vitally important to have a creative mind and be willing to explore--but Mr. Sigur deserves the real praise here.
I'd also like to take this opportunity to shamelessly plug The Geometer's Sketchpad. It's a great piece of software that dynamically creates geometric constructions. It's excellent for visualization. I used it to see the generalizations I was after, and I think Josh was using it when he first made his conjecture. If you've any interest in geometry--or are willing to have some anew--you should check this out. You can download a free sample version.
Anyway, I just wanted to post and settle a few things... If anyone has any questions, you can post them here or email me (I'm abliss at freeshell.org). Thanks for your attenton!
http://home.wxs.nl/~lamoen/wiskunde/concur.html
The extremely vague statements in the article look similar to what is presented there...
A slight correction. Emily Rosa did not prove that "theraputic touch" doesn't provide medical benefits. She proved that practitioners of it could not detect the proximity of another human due to the presence of their bio-electrical field (which definitly does exist, by the way). All her study showed was that the conscious human brain cannot reliable sense nearby electrical fields. It didn't prove or disprove that altering things in and near a human's electrical field have any other impacts on the human.
Think about this analogy. Even though I can't consciously tell how much Vitamin C is in the food I eat, the Vitamin C still affects my physical health. A study that shows people can't detect how many vitamins are in their food does not prove that vitamins are (or aren't) nutritionally helpful.
If people want to further study the bio-electrical field using scientific methods, great. Maybe we'll find better health that way and maybe we won't. This study just deals a blow to the nut-cases who don't use scientific backing for their therapy, but would they care about that study in the first place?
-Ted
I hope he gets an A in math for the rest of high school. How I would love to be in his math class and hear him say to his teacher "when you come up with law of mathematics on your own then you can tell me my math homework is wrong!"
I've hit Karma 50 and gotten a Score:5, Troll... I win!
The Monthly article in question appears to be "Morley Related Triangles on the Nine-Point Circle", by Floor van Lamoen, Amer. Math. Monthly vol. 107, no. 10, Dec. 2000, pages 941-945. The introduction says: "We identify two points M and H on Euler's nine point circle CN, found as intersections of three reflected lines. M and H each depend on the direction of a set of parallel lines. Posing the condition that M and H coincide for a certain direction, or that MH is a diameter of CN, we find two equilateral triangles in CN homothetic to Morley's famous trisector triangles."