Folding the Regular Pentagon Using Bisections and Perpendiculars Summary: Folding the Regular Pentagon Using Bisections and Perpendiculars Roger C. Alperin There are numerous origami constructions for the pentagon, see for example [3]. Dureisseix [2] provided constructions for optimal polygons inscribed in a square using origami folds. The optimal pentagon has four vertices on the four edges of the square and one vertex on the diagonal so that the pentagon has that diagonal as an axis of symmetry. However, the construction is known to be a `Pythagorean' construction [1], that is, we can achieve it using only the origami bisection of angles or the construction of a perpendicular to a line through a given point. We accomplish this for a maximal inscribed pentagon in the square by modifying Dureisseix' construction (notably in his second step). Suppose the side length of our square OPQR is one. Fold the hoizontal mid- line MN of the square parallel to RO. From the vertex P construct the diagonal PM and the angle bisector PA of MPO with A on the edge RO. Let = APO, x = tan() = OA, then 2 = MPO and tan(2) = 2 = MN P N ; hence x satisfies 2x 1-x2 = 2 so x = Collections: Mathematics