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.

Dim Z As Complex = Z0
Dim C As New Complex(ReaC, ImaC)
clr = 1
Do While clr < MaxIterations And Z.MagnitudeSquared < _
    MAX_MAG_SQUARED
    ' Calculate Z(clr).
    Z = Z * Z + C
    clr = clr + 1
Loop

' Set the pixel's value.
m_Bm.SetPixel(X, Y, m_Colors(clr Mod NumColors))

The following code shows the parts of the Complex class that are most important to this example.

Public Class Complex
    Public Re As Double = 0
    Public Im As Double = 0
    ' ...

    ' Binary +.
    Public Shared Operator +(ByVal c1 As Complex, ByVal c2 _
        As Complex) As Complex
        Return New Complex(c1.Re + c2.Re, c1.Im + c2.Im)
    End Operator

    ' Binary *.
    Public Shared Operator *(ByVal c1 As Complex, ByVal c2 _
        As Complex) As Complex
        Return New Complex( _
            c1.Re * c2.Re - c1.Im * c2.Im, _
            c1.Re * c2.Im + c1.Im * c2.Re)
    End Operator

    ' ...
End Class

Compare this example to Use a Complex class to draw the Mandelbrot set easily in Visual Basic 6. The .NET version is much faster.