To employ the method, a diagram is drawn starting at the origin. A line segment is drawn rightwards by the magnitude of the leading coefficient, so that with a negative coefficient, the segment will end left of the origin. From the end of the first segment, another segment is drawn upwards by the magnitude of the second coefficient, then left by the magnitude of the third, then down by the magnitude of the fourth, and so on. The sequence of directions (not turns) is always rightward, upward, leftward, downward, then repeating itself. Thus, each turn is counterclockwise. The process continues for every coefficient of the polynomial, including zeros, with negative coefficients "walking backwards." The final point reached, at the end of the segment corresponding to the equation's constant term, is the terminus. A line is then launched from the origin at some angle , reflected off of each line segment at a right angle (not necessarily the "natural" angle of reflection), and
refracted at a right angle through the line through each segment (including a line for the zero coefficients) when the angled path does not hit the line segment on that line. The vertical and horizontal lines are reflected off or refracted through in the following sequence: the line containing the segment corresponding to the coefficient of , then of etc. Choosing so that the path lands on the terminus, is a root of this polynomial. For every real zero of the polynomial, there will be one unique initial angle and path that will land on the terminus. A quadratic with two real roots, for example, will have exactly two angles that satisfy the above conditions. For complex roots, one must also find a series of
similar triangles, but with the vertices of the root path displaced from the polynomial path by a distance equal to the imaginary part of the root. In this case, the root path will not be rectangular. If simultaneous folds are allowed, then any th-degree equation with a real root can be solved using simultaneous folds. In this example with , the polynomial's line segments are first drawn on a sheet of paper (black). Lines passing through reflections of the start and end points in the second and third segments, respectively (faint circle and square), and parallel to them (grey lines), are drawn. For each root, the paper is folded until the start point (black circle) and end point (black square) are reflected onto these lines. The axis of reflection (dash-dot line) defines the angled path corresponding to the root (blue, purple, and red). The negative of the gradients of their first segments, '''', yield the real roots , , and . == See also ==