Square root of 5 one, when traversing the cube "surface" (the shortest distance when traversing through the "inside" of the cube corresponds to the length of the cube diagonal, which is the square root of three times the edge). The number can be algebraically and geometrically related to and , as it is the length of the hypotenuse of a right triangle with catheti measuring and (again, the Pythagorean theorem proves this). Right triangles of such proportions can be found inside a cube: the sides of any triangle defined by the centre point of a cube, one of its vertices, and the middle point
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