This post explores the relationship between the Riemann sphere representation of the extended complex plane and the complex plane. The 'north pole' is shown as a blue point. The points of the sphere are shown in red. Three points represent: z, 1/conjugate(z), 1/z. The 2D slider moves point on sphere. The black points are the stereographic projections of the points on the sphere onto the(complex) plane z=0 On the sphere the first mapping corresponds to reflection through the plane z=0. The second mapping is complex conjugation and corresponds to reflection in the plane y=0. The composite of these two reflections corresponds to an 180 degree rotation around the real axis(illustrated by the brown arc, noting the green line intersections the real axis).
Newton's method: basins of attraction
This post is inspired by Invitation to Mathematics. The graphic is a ListDensityPlot of the number of iterations of Newton's method to find root till convergence (using length of the list from FixedPointList function) for points in the complex plane (the unit square). The polynomial is a cubic with roots zero, 1 and the parameter point. The roots are seen as the left and right lower corners and the point in the Slider2D. Gradient color schemes are used to represent the convergence. The basins of attraction are,therefore, illustrated as well as the set of points whose orbit diverges. In future posts, I aim to color the basins for the particular root.
Newton's method: basins of attraction colored by root
In this post, I have modified the previous code to color points in the basins by the root they converge to and diverging points differently. The contrast (and consequent clarity) varies by color scheme.