Euclid's Construction of a Regular Icosahedron (XIII.16)

Let be a regular pentagon inscribed in circle with radius , and let , , , , and be the midpoints of the arcs , , , , and , respectively (step 2). is also a regular pentagon. The slider translates the pentagon in the direction normal to its plane up to the length of the radius (). Let and be the centers of and . The slider translates and , also in the direction normal to the plane , to its maximum value, which is the length of a side of the regular decagon inscribed in circle . The regular icosahedron is constructed by connecting corresponding points for maximum values of the sliders and .