I am trying to make a sine wave with a straight edge and compass, is it even possible?
Can you make a Sine wave with a straight edge and compass
geometric-constructiongeometry
Related Solutions
No (to your first question). A constructible number must be algebraic. However $\pi$ is not an algebraic number ($\pi$ is a transcendental number). Also of interest with regards to this is the famous "squaring the circle" problem.
Your solution fails because you give up a move creating the center point of your bounding circle. To complete the challenge in eight moves, you need to use one of the preexisting intersections as your center point, and then rely on the fact that extending a line is not considered a move to create the fourth vertex of the square.
The following diagram illustrates these steps, where step 4b is the non-move extension of the line:
Best Answer
Since you liked my comment I will it into an answer. Any curve a compass makes has a constant curvature. There is no region along the sine curve that has constant curvature.