Below one may plot (approximations of) the set of all complex numbers of the form Σ d_n cⁿ 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 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.