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. 
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.
The question is kind of vague, but I suspect you're asking the following:
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:
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:
With a little trigonometry, you can convince yourself that the answer is $\sqrt2-1\approx41\%$.