When I was in third grade, I was playing with rectangles and diagonal lines, and discovered something very interesting with fractions. I've shown several math teachers and professors over the years, and never got an answer. Just a few, "Wow, that's neat!"
Draw a rectangle. Draw a line from the top left corner to the bottom right corner. Then draw a line from the top right corner to the bottom left corner. The intersection obviously becomes 1/2 units of the rectangle's width.
Now draw a line from the last intersection to the bottom line of the rectangle, and then from that point to the top right corner of the rectangle. The new intersection becomes 1/3 units of the rectangle's width.
Keep doing this and the denominator of the fraction increases by one each time to infinite. Why does this happen? I don't know how to prove why this happens, but it would be interesting if someone could. Can you? I never became a mathematician to prove it, but if it's easy, please forgive my mathematical ignorance. I tried this several years ago with AutoCAD and it does in fact work out.