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** d_{n} c^{n} where the sum is from 1 to infinity,
d_{n} 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 C
_{0}intersect C_{1}is nonempty or empty where C_{0}is the set of numbers in C where d_{0}= 0 and C_{1}is the set of numbers in C where d_{0}= 1. - C has self-similarity: C is the disjoint union of C
_{0}and C_{1}. We readily see C_{0}= cC and C_{1}= c + cC are each scaled and translated copies of C. Continuing, C_{0}is the disjoint union of c^{2}C and c^{2}+ c^{2}C, etc.

Use the sliders to select a value for c = a + bi. Click the color palettes to select colors for
C_{0} and C_{1}. Beware; it is not easy to see when C is connected, that is, when C_{0}
and C_{1} 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.