Screen a ray

Is it possible to place round mirror columns so that a ray with arbi­trary direc­tion parallel to the floor never reaches the walls? Columns with reflecting cylin­dric side surface may be of any diam­eter and can be placed anywhere with a single condi­tion: they shouldn't be tangent (if not the answer is obvious).

As we know, the law of reflec­tion is «the angle of inci­dence is equal to the angle of reflec­tion». If the mirror is not plane we should just measure the angles off the tangent plane to the point of reflec­tion.

How many columns do you need and how they should be placed so that a ray is shielded and doesn't reach the walls? Do we need finitely many mirrors or a infi­nite number of them? Or even that is not enough?

It's clear that one column is not enough. A ray can pass by the column, but even if it reflects it will reach the wall. Thus, a ray with any direc­tion will finally reach the wall.

It's obvious that two or three columns is not enough; the walls will remain visible in some direc­tions from the center of the hall. And so, if we radiate in this direc­tions we'll reach them.

Intu­ition tells us and an exper­i­ment shows that a small amount of columns is not suffi­cient to shield a ray.

Place rays reach the walls.

Well, an exper­i­ment is not a proof. Maybe we should rearrange the columns or take some more… It's still an open ques­tion for math­e­mati­cians if a finite number of columns (maybe a very large) is suffi­cient to shield a ray. If it is, what are their diam­e­ters and posi­tions? Or maybe even an infi­nite number of columns is not enough to solve the problem?

Maybe You will find out how to place the columns?

Other etudes in “Varios interesting subjects”