If I had a circle inscribed in a square, touching the circle at 4 points, and then need to calculate the number of secant lines on it that can reflect off the square would it be $\frac{\infty}{4}$? For example if I had a string going from one of the points to the next point then to the next point, to the original point an diamond would form inside the circle. I used secant lines, because it is based off a physics design I am making where it appears (the limit would appear to be secant, but its tangent, since the line dies after it hits that one point) it disappears right after it hits a point. How many diamonds can I inscribe, $\frac{\infty}4$? I just suggest that as an answer since the diamond has 4 points. The angle of reflection is 45 degrees, however, it is 45 degrees for every "secant"(aka tangent line in reality) of the circle. Drawing

In case its confusing thanks to how I drew it, the outside is supposed to be a perfect square, and the oval a perfect circle, and the square inside the oval a perfect diamond. The circle is a path that an electron would follow, and the square (rectangle) is a containment array. And the diamond is to minimize the number of points being touched.

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    $\begingroup$ At what angle can it reflect off the square. With no restriction it'd be infinite. If it follows physics congruent angles the are only four $\endgroup$ – fleablood Sep 18 '16 at 0:47
  • $\begingroup$ To diagram this, just draw a diagram, take a photo and include that in your post. Or use some free geometry software. Personally I illustrate many posts using Cinderella, while others prefer other software. I think a diagram would help, since just as fleablood I don't really understand your conditions here. $\endgroup$ – MvG Sep 18 '16 at 9:26
  • $\begingroup$ @MvG Currently I am in school, I shall edit the post later. $\endgroup$ – Sigma6RPU Sep 19 '16 at 14:42
  • $\begingroup$ I am just in calculus 1, anything too complicated would require me to do research which I am willing to do. So don't hold back on your answers. $\endgroup$ – Sigma6RPU Sep 20 '16 at 0:12
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    $\begingroup$ Two questions. (a) does your definition of "diamond" require that side lengths are equal (rhombus)? Angles are equal (rectangle)? both (square)? (b) Are you trying to inscribe a "diamond" in the big square? If so what is the circle for? If you are trying to inscribe the a diamond in the circle, what is the big square for? $\endgroup$ – rikhavshah Sep 20 '16 at 1:23

The question is kind of vague, but I suspect you're asking the following:

Consider the billiard inside a square. How many orbits are $4$-periodic and consist of secant lines to the circle inscribed in the square?

Periodic orbits inside rectangles are a fun way to introduce yourself to the study of mathematical billiards. For a square table, an orbit returns to its starting point after $4$ reflections if and only if it starts at a $45^\circ$ angle. Here is what the clockwise orbits look like:

enter image description here

There are infinitely many such orbits, as many as there are real numbers! Your notation $\frac{\infty}4$ suggests as much, but it isn't clear what you mean by that. It would be better to ask questions with finite answers. Here's one way to do that:

Consider the billiard inside a square. Parametrize each clockwise $4$-periodic orbit with the point where it strikes the upper side of the square. What is the fraction of such orbits that consist of secant lines to the circle inscribed in the square?

With a little trigonometry, you can convince yourself that the answer is $\sqrt2-1\approx41\%$.

  • $\begingroup$ Your answer is very interesting, so basically as lets say as $n$ approaches infinity, $z$(the orbits) simultaneously approaches infinity. $\endgroup$ – Sigma6RPU Sep 22 '16 at 16:33
  • $\begingroup$ @Sigma6RPU What is $n$? $\endgroup$ – Chris Culter Sep 22 '16 at 17:37
  • $\begingroup$ Number of rectangles. $\endgroup$ – Sigma6RPU Sep 22 '16 at 23:29

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