Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2000 by Xah Lee. (http://xahlee.org/)

Cissoid of Diocles

This notebook proves by algebra that cissoid of Diocles can be used to solve the doubling the cube problem. i.e. Given a length a, find length x such that x^3==2*a^3.

Let there be a cissoid with circle radius a. Suppose a==1.

Let there be a cissoid y^2(2-x)==x^3, and a line y==-2x+4. They intersect at Q. The coordinate of Q is

Solve[{y^2 (2 - x) == x^3, y == -2 x + 4}, {x, y}]

First[{x, y} /. %]

Let O denote the origin. The intersection of the line OQ and the line x==1 is

Solve[{
        (4/5*(1  +  2*2^(1/3)  -  2^(2/3)))/
              (8/5  -  (4*2^(1/3))/5  +  (2*2^(2/3))/5)
                *x == y, x == 1}, {x, y}]

We verify that the y coordinate cubed is equal to 2 times the cube of the radius of the circle of the cissoid of Diocles.

(-
            ((2 (-1 - 2 (2^(1/3)) + (2^(2/3))))/(4 - 
                  2 (2^(1/3)) + (2^(2/3)))))^3 // N