Cantor-like Sets in the Complex Plane

Based on
"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
http://www.jstor.org/stable/2325031

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.

• C is uncountable.
• C is compact.
• C is either connected or totally disconnected depending, respectively, upon whether C0 intersect C1 is nonempty or empty where C0 is the set of numbers in C where d0 = 0 and C1 is the set of numbers in C where d0 = 1.
• C has self-similarity: C is the disjoint union of C0 and C1. We readily see C0 = cC and C1 = c + cC are each scaled and translated copies of C. Continuing, C0 is the disjoint union of c2C and c2 + c2C, etc.

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.