<P> Archimedes, Nicomedes and Apollonius gave constructions involving the use of a markable ruler . This would permit them, for example, to take a line segment, two lines (or circles), and a point; and then draw a line which passes through the given point and intersects three lines, and such that the distance between the points of intersection equals the given segment . This the Greeks called neusis ("inclination", "tendency" or "verging"), because the new line tends to the point . In this expanded scheme, we can trisect an arbitrary angle (see Archimedes' trisection) or extract an arbitrary cube root (due to Nicomedes). Hence, any distance whose ratio to an existing distance is the solution of a cubic or a quartic equation is constructible . Regular polygons with solid constructions, like the heptagon, are constructible; and John H. Conway and Richard K. Guy give constructions for several of them; . </P> <P> The neusis construction is more powerful than a conic drawing tool, as one can construct complex numbers that do not have solid constructions . In fact, using this tool one can solve some quintics that are not solvable using radicals . It is known that one cannot solve an irreducible polynomial of prime degree greater or equal to 7 using the neusis construction, so it is not possible to construct a regular 23 - gon or 29 - gon using this tool . Benjamin and Snyder proved that it is possible to construct the regular 11 - gon, but did not give a construction . It is still open as to whether a regular 25 - gon or 31 - gon is constructible using this tool . </P> <P> In 1998 Simon Plouffe gave a ruler and compass algorithm that can be used to compute binary digits of certain numbers . The algorithm involves the repeated doubling of an angle and becomes physically impractical after about 20 binary digits . </P>

Regular polygon of 17 sides could be constructed by ruler and compass alone