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  • [Professor Masur]: But I thought I would mention one other

  • problem that arises that you can ask

  • about polygons. You could ask is there

  • what's called a periodic orbit? When I

  • say orbit, that might be "start off at

  • a point, you bounce off the walls and

  • after some long number of times; can you

  • always find a path that repeats itself?"

  • So let me give one very, very classical

  • example of that and that goes back to

  • the squareif you start, let's say on the

  • bottom, and move off at a right angle,

  • 90 degrees, you move to the top. Now when you

  • get to the top, because these are

  • parallel, this angle is also 90 degrees,

  • and so the billiard path says you

  • rebound where angle of reflection equals

  • angle of incidence, and that means that

  • when you come to the side you will

  • bounce off exactly the way that you came

  • in; you will come to the side and

  • bounce right back. You come back along

  • this line and then you will bounce and

  • repeat yourselfso that's what's called

  • periodic. There are other periodic lines.

  • So, for example, you could start at a

  • 45-degree angle, everything would be

  • 45 degrees, and you would come back right

  • where you started and then repeat

  • yourself. In fact, if you started any

  • point, at any angle that is a rational

  • number – a quotient of two integersif you started with an angle of 27 degrees

  • starting here, you will come back after a certain number of bounces.

  • That's true for a square; and, in fact, one can prove

  • for any rational polygon, were all the angles are fractions of 180 degrees,

  • or P over Q times 180 degrees,

  • again I like to write it like that,

  • you can always find a periodic orbit.

  • [Brady]: From any point? [Professor Masur]: No not from any point.

  • There's some periodic orbit, from somewhere.

  • Some periodic orbit from somewhere.

  • There might be; so there will be

  • some directSome point, in some

  • direction, where there will be a periodic orbit.

  • [Brady:] For example on this side there are none, but there is one sitting over here somewhere.

  • [Professor Masur:] Well usually what happens ifIt, it

  • In most cases, thatThat's possible;

  • in fact you can always find periodic

  • orbits that will hit every side. In most

  • cases a periodic orbit will hit a side.

  • There might be other points on that side,

  • where the orbit in that same direction, is not periodic.

  • So in other words, you start at a point on a side and move in a direction

  • it might close up, meaning periodic,

  • but if you move the point a little bit and headed off in the same direction,

  • what could happen is that it isn't periodic;

  • what might happen is it hits a corner

  • and then you don't know what to do.

  • That's different from the square. In the square no matter where I

  • started, if I went at 45 degrees, it would be periodic,

  • but in a general,

  • rational polygon that may not be the case.

  • I want to talk about triangles now

  • that are not necessarily rational, but just triangles.

  • If they are acute,

  • which means all of the angles are 90 degrees or less,

  • what you could do is: you could take the vertex of the triangle

  • and drop a perpendicular down to the opposite side to get a point,

  • and then drop this perpendicular down to the opposite side and get a point so these are right angles.

  • This triangle joining those points gives a periodic

  • orbit; meaning if I start here I go to

  • the line towards here this angle will

  • equal that angle and I'll bounce like that.

  • And then i'll get to this point and

  • this angle will equal this angle and

  • I'll bounce like that.

  • And I come back, this angle equal this angle, and I'll repeat myself.

  • This example was has been known for hundreds of years.

  • I don't think it was ever thought of as billiards, but anybody in high school or

  • even junior high could do this inIn trigonometry.

  • When you do obtuse triangles,

  • you can't drop a perpendicular because if I started at this vertex and drop the perpendicular,

  • it wouldn't be on that side.

  • That's because this angle is bigger than 90 degrees and, in fact,

  • for something as very simple as in a

  • triangle, obtuse triangle, one does not

  • know whether there are periodic orbits.

  • So this is a famous unsolved problem in

  • the subject of, what this is called is, dynamical systems.

  • Well so again if the obtuse triangle is rational,

  • then there are periodic orbits.

  • So I'm talking about obtuse triangles. I think I gave an example

  • 90 times square root of twothat particular one I would

  • not be able to tell you if it had a periodic orbit.

  • There are some

  • Some reason where people, by very, very hard

  • work, have shown where if this angle is less than a hundred degrees,

  • then there are periodic orbits. But that was very,

  • very hard work and if it was a hundred and ten degrees;

  • again I'm assuming not rational;

  • then we don't know the answer.

  • If it's rational we always know, if all the

  • angles, if this is a hundred and eleven,

  • and this is 36.

  • What's left? 33 degrees there will be.

  • But if one of the angles is an irrational

  • number then we don't know.

  • [Brady:] So what, does greatness await if someone can crack that?

  • [Professor Masur:] Well, it's… It's um

  • There is a very prominent, one of the most prominent

  • mathematicians in the United States or

  • in the world in the subject of

  • dynamical systems whose name is

  • Professor Katok at Penn State has

  • listed this as one of the five

  • outstanding problems in dynamical systems.

  • But really there's nothing you

  • could do. You can try desperately to

  • solve it, but if it hasn't been solved

  • for a hundred years, you probably aren't going to.

  • And you know it's only given to one

  • person, so to speak, to solve a particular

  • one of these problems. So we're used to

  • it and here's an atmosphere of resignation.

[Professor Masur]: But I thought I would mention one other

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B1 periodic angle orbit rational bounce triangle

Problems with Periodic Orbits - Numberphile

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    林宜悉 posted on 2020/03/27
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