Cantor-like Sets in the Complex Plane
"Number Systems With a Complex Base: a Fractal Tool for Teaching Topology," Daniel Goffinet
The American Mathematical Monthly, Vol. 98, No. 3 (Mar., 1991), pp. 249-255
Below one may plot (approximations of) the set of all complex numbers of the form Σ dn cn where the sum is from 1 to infinity, dn is 0 or 1, and c is a fixed complex number with |c|<1.
When c = 1/3 we obtain a standard Cantor middle-thirds set in the interval [1, 1/2]. Non-real values of b give more exotic sets C that share the following properties with the familiar Cantor sets.
Use the sliders to select a value for c = a + bi. Click the color palettes to select colors for C0 and C1. Beware; it is not easy to see when C is connected, that is, when C0 and C1 intersect. See D. Goffinet's interesting article for details.
If you have Flash 10 or higher, you can save the screen image as a PNG file by clicking on the screen.