Surfaces A and B still share a common umbilical point when viewed from above in the Argand plane, but this is now located at:
It is noted from Figure 14 that the positioning of the three complex roots no longer forms an isosceles triangle in the Argand plane. This observation is generally true for any cubic polynomial with complex coefficients, and is a consequence of there being no necessity for complex conjugate roots to occur.
The aim of this paper is to demonstrate visually the connection between the reduced polynomial y = [x.sup.n] - 1 in the Cartesian plane and the resulting n-roots which invariably appear in the Argand plane. There is no contradiction here: the reader will find a three-dimensional surface representation of Equation (2) provides the full link between both the Cartesian and Argand planes, and illustrates not only the location of the roots in relation to the original equation but also shows why they occur with conjugate pairings.
Each equation represents a three-dimensional surface (the actual classification of this, and the higher-order surfaces presented herein, is beyond the scope of this paper) in which the ordinate value A or B can be plotted over a grid of points in the Argand plane defined by the H and G axes.
It is noted that the square roots of 4 - 7.5i as plotted in Figure 1 are effectively translated by an amount -b/2a in the Argand plane as the final part of the solution process; the point -b/2a locates the new centre of the circle.
So, after rearrangement, [(x - 2).sup.3] = -4/3 and this problem reduces to finding the cube roots of -4/3, which by de Moivre's method are found to be -1.101, 0.55 - 0.953i, and 0.55 + 0.953i Now add 2 to these results to find the three roots of the original equation, viz., 0.899, 2.55 - 0.953i and 2.55 + 0.9537 It is observed that a general cubic will have three roots which when plotted in the Argand plane will form an isosceles triangle.