Doubling the cube cube involve neusis, the cissoid of Diocles, the conchoid of Nicomedes, or the Philo line. Archytas solved the problem in the 4th century BC using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution. False claims of doubling the cube with compass and straightedge abound in mathematical crank literature (pseudomathematics). Origami may also be used to construct the cube root of two by folding paper. There is a simple neusis construction using a marked ruler for a length which is the cube root of 2 times another length. Then AG is the
Doubling the cube was the discovery by Hippocrates of Chios that it is equivalent to finding two mean proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths and , the duplication of the cube is equivalent to finding segments of lengths and so that In turn, this means that But Pierre Wantzel proved in 1837 that the cube root of 2 is not constructible; that is, it cannot be constructed with straightedge and compass. Menaechmus' original solution involves the intersection of two conic curves. Other more complicated methods of doubling the