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Title Draw Mandelbrot sets Mandelbrot, fractal, chaos Graphics, Algorithms

See my book Visual Basic Graphics Programming for full details. This book also explains many other fractals including Julia sets, Hilbert curves, and Sierpinski curves.

The heart of this program is subroutine DrawMandelbrot. The idea is to repeatedly calculate Z(n+1) = Z^2 + C where Z(n) and C are complex numbers. For each point (X, Y) in the picture, the program sets C = X + i*Y (where i is the imaginary square root of -1). The program evaluates Z(n) repeatedly until it has performed a set limit of steps or the value has a magnitude greater than 2 (it can be shown that the value grows infinitely if it ever exceeds 2). The program assigns the point a color depending on how quickly its magnitude exceeded 2. See the book for more details.

```Private Sub DrawMandelbrot()
Const MAX_MAG_SQUARED = 4  ' Work until the magnitude
' squared > 4.

Dim clr As Long
Dim i As Integer
Dim j As Integer
Dim ReaC As Double
Dim ImaC As Double
Dim dReaC As Double
Dim dImaC As Double
Dim ReaZ As Double
Dim ImaZ As Double
Dim ReaZ2 As Double
Dim ImaZ2 As Double
Dim wid As Integer
Dim hgt As Integer

DrawMode = vbCopyPen
Line (0, 0)-(ScaleWidth, ScaleHeight), BackColor, BF
MousePointer = vbHourglass
DoEvents

' dReaC is the change in the real part
' (X value) for C. dImaC is the change in the
' imaginary part (Y value).
wid = ScaleWidth
hgt = ScaleHeight
dReaC = (VisibleXmax - VisibleXmin) / (wid - 1)
dImaC = (VisibleYmax - VisibleYmin) / (hgt - 1)

' Calculate the values.
ReaC = VisibleXmin
For i = 1 To wid
ImaC = VisibleYmin
For j = 1 To hgt
ReaZ = 0
ImaZ = 0
ReaZ2 = 0
ImaZ2 = 0
clr = 1
Do While clr < MaxIterations And _
ReaZ2 + ImaZ2 < MAX_MAG_SQUARED
' Calculate Z(clr).
ReaZ2 = ReaZ * ReaZ
ImaZ2 = ImaZ * ImaZ
ImaZ = 2 * ImaZ * ReaZ + ImaC
ReaZ = ReaZ2 - ImaZ2 + ReaC
clr = clr + 1
Loop
PSet (i, j), QBColor(clr Mod 16)
ImaC = ImaC + dImaC
Next j
DoEvents
ReaC = ReaC + dReaC
Next i

Refresh
Picture = Image

MousePointer = vbCrosshair
DrawMode = vbInvert
End Sub```