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MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

israelhayom.com

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Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#41

Post by Jugurtha »

PhantomGremlin wrote: post #20

I loved the brief summary at the top. In contrast, many times there are "long form" articles which expect me to invest 10 minutes reading them before I even have a good idea what they're about. You know what I mean: Someone grew up privileged, or in the 'hood. Then had a plethora of tangential life experiences. Then maybe an epiphany. Then we begin to read something about the purported topic.

That was my thought, exactly. I'm rather into Bottom Line Up-Front and that website seems to provide a tight summary/outline for every story.

Writing well is something I'm shooting for because I'm always all over the place and I need more discipline.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#44

Post by IkmoIkmo »

hasenj wrote: post #29

Until someone brings a link from MIT's website, I'm calling bullshit on this one. This sounds like what pops up every other day in Egyptian newspapers about genius Egyptian kids who invent this or that. The theorem stated in the article is not a theorem at all. It's a direct consequence of the definition of a circle and is perfectly obvious to anyone who spends two minutes pondering the implications of that definitio…

> This sounds like what pops up every other day in Egyptian newspapers about genius Egyptian kids who invent this or that.

Haha, finally I come across someone saying this on the internet. But you're right it's exactly like that.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#45

Post by Houshalter »

dkopi wrote: post #39
hasenj wrote: post #29

Until someone brings a link from MIT's website, I'm calling bullshit on this one. This sounds like what pops up every other day in Egyptian newspapers about genius Egyptian kids who invent this or that. The theorem stated in the article is not a theorem at all. It's a direct consequence of the definition of a circle and is perfectly obvious to anyone who spends two minutes pondering the implications of that definitio…

A circle is defined as all the points in the same distance from a certain center point. But a point thats distanced from the circle circumference by R isn't necessarily the circles center. But if you can draw 3 (and hence more) lines from a point to the circles circumference that are all the same lenght, that is the circles center, and the distance is the radius.

I actually used this idea once. I had three known points which were approximately equidistant from an unknown center. I wanted to find the center.

So I used a hillclimbing algorithm to search for the center by guessing points and seeing how close they were. The fitness function was the difference between the proposed center and the three points. The idea being to minimize the distance between their. If the lines were exactly the same length, I would have found the center.

It didn't work at all though. It gave wildly incorrect answers, and sometimes even converged on infinity... Even when running it many many times to avoid local optima.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#46

Post by mixedmath »

catpolice wrote: post #17
eximius wrote: post #2

Hm, a clever little theorem. I'm surprised it isn't recorded somewhere. Perhaps it's just under a different name because of it's relative... simplicity? Not to belittle her accomplishment. This is something you'd expect Euclid to write about or something.

It follows from the statement that if two circles intersect in more than two points, they're identical. Which seems like a familiar theorem, that I can't seem to place. So it's more like a corollary. It would be remarkable if this hadn't come up before, but I want to believe because of how good a story it makes...

Your fact probably doesn't have a name. But a good statement would be that a quadratic polynomial has at most two roots.

To see this, suppose that the two circles are (x-a)^2 + (y-b)^2 = r and (x-c)^2 + (y-d)^2 = R. Subtracting gives an equation of the form (linear function in x and y) = r - R. This means that y is a linear function of x, and so we can use this to substitute in the equation for the first circle to get something of the form (quadratic function in x) = r.

Then as there are at most 2 solutions for x, and each gives the corresponding solution for y, we see that two distinct circles intersect in at most 2 points.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#47

Post by jschulenklopper »

db48x wrote: post #40

It's already fairly well established that any three points define a circle. Similarly, most circles have a center, and will therefore have radii which join those points to the center.

> Similarly, most circles have a center.

I think that all circles have a center.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#48

Post by jschulenklopper »

Houshalter wrote: post #45
dkopi wrote: post #39

Earlier quoted context omitted.

A circle is defined as all the points in the same distance from a certain center point. But a point thats distanced from the circle circumference by R isn't necessarily the circles center. But if you can draw 3 (and hence more) lines from a point to the circles circumference that are all the same lenght, that is the circles center, and the distance is the radius.

I actually used this idea once. I had three known points which were approximately equidistant from an unknown center. I wanted to find the center. So I used a hillclimbing algorithm to search for the center by guessing points and seeing how close they were. The fitness function was the difference between the proposed center and the three points. The idea being to minimize the distance between their. If the lines were…

> I had three known points which were approximately equidistant from an unknown center. I wanted to find the center.

See this: http://www.mathopenref.com/const3pointcircle.html

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#49

Post by randlet »

adt2bt wrote: post #25
randlet wrote: post #19

Kudos to the teacher who helped her develop her idea instead of scolding her for using ideas that weren't being taught in his classroom.

How common is that?

Pretty common (in North America anyways) I think. In speaking with teacher friends my impression is that math education has become quite formulaic with a large focus on learning algorithms to solve problems that occur in standardised tests.

Re: MIT Mathematician confirms: Israeli 10th-grader discovers new geometric theorem

#50

Post by 21 »

commenter23 wrote: post #37

It's a fluff piece. The real story here is that a 10th grader, after using a Theorem that wasn't taught in class, was encouraged to prove it - which she did, successfully. The teacher then sent it to a few academics who were thought that was a rather impressive accomplishment for a 10th grader, so they wrote her some encouraging words. That's it. The theorem and its proof are in Euclid's Elements, (Book 3 Proposition…

My memories are a bit faded, but when I was in 10th grade in an EU country we did similar geometry problems.

This doesn't seem considerably harder than the kinds of problems I remember doing. So I'm not sure how impressive this is, maybe a teacher should weigh in.

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