Impossible constructions Compass-and-straightedge construction

1 impossible constructions

1.1 squaring circle
1.2 doubling cube
1.3 angle trisection

impossible constructions

the ancient greeks thought construction problems not solve obstinate, not unsolvable. modern methods, however, these compass-and-straightedge constructions have been shown logically impossible perform. (the problems themselves, however, solvable, , greeks knew how solve them, without constraint of working straightedge , compass.)

squaring circle

the famous of these problems, squaring circle, otherwise known quadrature of circle, involves constructing square same area given circle using straightedge , compass.

squaring circle has been proven impossible, involves generating transcendental number, is, √π. algebraic numbers can constructed ruler , compass alone, namely constructed integers finite sequence of operations of addition, subtraction, multiplication, division, , taking square roots. phrase squaring circle used mean doing impossible reason.

without constraint of requiring solution ruler , compass alone, problem solvable wide variety of geometric , algebraic means, , solved many times in antiquity.

a method comes close approximating quadrature of circle can achieved using kepler triangle.

doubling cube

doubling cube construction, using straight-edge , compass, of edge of cube has twice volume of cube given edge. impossible because cube root of 2, though algebraic, cannot computed integers addition, subtraction, multiplication, division, , taking square roots. follows because minimal polynomial on rationals has degree 3. construction possible using straightedge 2 marks on , compass.

angle trisection

angle trisection construction, using straightedge , compass, of angle one-third of given arbitrary angle. impossible in general case. example, though angle of π/3 radians (60°) cannot trisected, angle 2π/5 radians (72° = 360°/5) can trisected. general trisection problem solved when straightedge 2 marks on allowed (a neusis construction).