In 1525, Albrecht D¨urer published a treatise on geometry, featuring, among many items, an algorithm for purportedly constructing a regular pentagon by starting with a regular hexagon, whose consecutive vertices, V0, V1, V2, V3, V4, V5, lie among a unit circle C with center O. Let r be the edge-lengths of the hexagonal sides. Here’s the algorithm. Let A be the point C lying on a ray from O through the midpoint of a segment V1V2. With V1 and V2 as two vertices of D¨urer’s pentagon, let a third vertex B be the intersection of ray V0A and a circle, center V2 and radius r. Angle̸ V 1V2B should be 108◦; but is it? We contrast this algorithm with that of Eudoxus during the time of Plato’s Academy, who used the golden triangle.