You can draw a Mandelbrot set by iterating the equation:

    Zn = Zn-12 + C

Where Z_n and C are complex numbers. For a point (X, Y) in a particular area of the set, the program sets Z₀ = 0 and C = X + Y * i. The program iterates this equation until the magnitude of Z_n is at least 2 or the program performs a maximum number of iterations.

This example uses a Complex class to manage the complex numbers easily and intuitively. The following code shows the program's main loop.

Set Z = New Complex
Z.Initialize Z0.Real, Z0.Imaginary
        
Set C = New Complex
C.Initialize ReaC, ImaC

clr = 1
Do While clr < MaxIterations And Z.MagnitudeSquared() < _
    MAX_MAG_SQUARED
    ' Calculate Z(clr).
    Set Z = Z.Times(Z).Plus(C)
    clr = clr + 1
Loop

' Set the pixel's value.
picCanvas.PSet (X, Y), QBColor(clr Mod 16)

The following code shows the most important parts of the Complex class.

Public Real As Double
Public Imaginary As Double

Public Sub Initialize(ByVal new_Real As Double, ByVal _
    new_Imaginary As Double)
    Real = new_Real
    Imaginary = new_Imaginary
End Sub

Public Function MagnitudeSquared() As Double
    MagnitudeSquared = Real * Real + Imaginary * Imaginary
End Function

Public Function Magnitude() As Double
    Magnitude = Sqr(Real * Real + Imaginary + Imaginary)
End Function

Public Function Times(ByVal c2 As Complex) As Complex
Dim result As Complex

    Set result = New Complex
    result.Real = Me.Real * c2.Real - Me.Imaginary * _
        c2.Imaginary
    result.Imaginary = Me.Real * c2.Imaginary + Me.Imaginary _
        * c2.Real
    Set Times = result
End Function

Public Function Plus(ByVal c2 As Complex) As Complex
Dim result As Complex

    Set result = New Complex
    result.Real = Me.Real + c2.Real
    result.Imaginary = Me.Imaginary + c2.Imaginary
    Set Plus = result
End Function

Compare this example to Use a Complex class to draw the Mandelbrot set easily in Visual Basic .NET. That example has a few extra features that I haven't gotten aroundd to adding to this example (such as letting you pick the colors the program uses), but something you should look at is their performances. The .NET version is much faster.