Zn=(Z(n-1))^2+C

where Z and C are complex numbers and C is constant, the mandelbrot set is the set of numbers Z0 for which Zn is bounded. that is, for any value n, including infinity, Zn is not infinite.

If you wish to calculate which points are part of the Mandelbrot set, it is reasonable to iterate a few hundred times and check if the absolute value of the current Z is more than some arbitrary number such as 2.

It is closely related to a julia set. the variable in a julia set is C, and Z0 is constant for any point on a julia set. If you take a point from a representation of the Mandelbrot set, you can create a Julia set for it by using its value of Z0 and plotting on the complex plane of C. There are programs that do this, such as the julia program in the Xscreensaver package.

Computer-generated representations of the Mandelbrot set are generally colorful, while the technical definition of the set allows for only two states: membership and non-membership. The colors typically come from either the number of iterations it takes for Zn exceed some value (indicating that the point is not a member) or the value of Zn with some arbitrary n. For example, if abs(Zn) is less than 2, assume it is a member and color it black. If it's between 2 and 3, color it red. 3-4, orange; 4-5, green; 5-6, blue; 6-100, indigo; 100-infinity, violet (numbers are pulled from the air and should not be taken as suggestions).

My uncle invited Benoit Mandelbrot over to his house once. He sent his son to pick Mandelbrot up at the airport, only he didn't know what Mandelbrot looked like. So he drew an approximation of the Mandelbrot set on a sign and held it up. Sure enough, the mathematician recognized it and was able to find his ride.

To code the mandelbrot set, make two nested loops ilterating from r = -1 to 1 and i = -1 to 1 by small increments, say 0.005. Map each (r,i) coordinate to a pixel (x,y) on the screen.

Let z be a complex number. Then, for each pixel, count how many times you can square z and add (r,i) to make a new z, before z's absolute value is larger than e.g. 2. Make sure to put an upper bound on this loop, because for some combinations of (r,i), z's absolute value never rises above 2.

Use these formulas:
zr = zr2 - zi2 + r
zi = 2zrzi + i
Make sure you use the old zr in the second formula, and not the one you just calculated.

Now, pick the color of the pixel in question based on how many times the loop ilterated, and put it on the screen. Make the color black if your loop hit the upper bound.

Voila! After waiting a bit, you'll have a beautiful mandelbrot fractal on your screen.

Okay, here's a general idea of it:

:::::::::::::::::::. ,X222222222SS25222222222222222222252h#@@@@#2rA9s29;s29@@@&2S5552222222222222222
::::::::::::::::::. G#92222225S22222222222222222222253H@@@S     :G@S.@GAi.   ;@@@AX5S5S2222222222222
:::::::::::::::::. HHX222252S222222222222222222225X&@@S.      i@HX @ ,@@@A@     ;G@HX22S522222222222
::::::::::::::::. #AX222SS2222222222222222222222h#@G2 .     hM@3i@X@3    r.@G   X  A@92222SS22222222
:::::::::::::::. #A222S5222222222222222222222X&@@M        .#@AX9@A@@. r ,@@;M@@     i@A52222S5222222
::::::::::::::. MA22Si2222222222222222222229M@@r         3@#HHsMr,   r@.  ;9@2M@@:    @@&22222i52222
:::::::::::::, AHX5S2222222222222222222252A@X          #@@HA9 r@9 ;@@@@@H, #r2r2H@@Ar.  @A522222S222
::::::::::::, 5MXS5222222222222222222229#@@r      : 3@@@@ , ;r2@r s@@@@@@; @@3@@.,h#@@;  @25222222S2
:::::::::::: ,#3SX2222222222222222222&@@3      .@@@@@@,&2:@@@5     ,M#@#:     .@@:2M@ @@;,@h5222222S
:::::::::::, @hS222222222222222225XH@@    . i@@@@@@.9B ::     #,@@@@9@#h@@@#  :  ;,   ;@M, @35222222
::::::::::: iB52222222222222252XGH#.   2@@@@#AA#@  .@@ .@@@ @@@@@@@@@@@@@@@@@@@3  @@#, &@A..@2222222
::::::::::, @222222222222523&A@2; S,H@@#GXX3G@3.  #@H.   2@@@@@@@@@@@@@@@@@@@@@@@#@@A @.B9X ##222222
:::::::::: iA22222252XhM@@@9;  ,r@H#B@5Mi9X;99 X,@@  i.H@@@@@@@@@@@@@@@@@@@@@@@@@@@   @3@@@  @h52222
:::::::::, #G2252GM@@@@M    3s@@G G@5  @r&@:   :@@s  9@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@    @@  :@22222
:::::::::.:M229H@@rS     rBHAA9#A  r:hh:   ;@@@@@, ,@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@: .@:  @X2222
::::::::: iB&@@25      r@@G2S5 r@# .;r   5.        s@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@  @@@,  @@5222
::::::::: B@@;       s@@GSii#@  @   :@@@@@@@@@,    @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@3 ;#M:   @2222
::::::::: @@       i@@@##B@@;#@@  ;A@@@@@@@@@@@@; @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ .@h@:   @&222
::::::::, @      &@22     5@      @@@@@@@@@@@@@@@ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@2 @ri@:   :#222
::::::::.;B#@@@@@;@@r5@@@Ss  .A@@.@@@@@@@@@@@@@@@h@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@::@ssX@:   ;@222
::::::::.;@r#235 r rr, .  , 2M@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@9  @  92@:   ;@222
::::::::.;29@@@@@@@@  :@; Mr  ,sr @@@@@@@@@@@@@@@,@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ @H@5@:   ;@222
::::::::, @       ;@@53 5&h@      S@@@@@@@@@@@@@@ @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ Br5@:   3B222
::::::::: @@        X@@HG29@@@@@i   A@@@@@@@@@@G  r@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ .@@#:   @3222
::::::::: 3#@H.       M@#XSss#A @r  G@s3G@@X@r     @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@; S@@,   @5222
::::::::: rB3B@#s       A@BG29;M@i       ,    ,,   ,@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@2 @ @:  @BS222
:::::::::,.@X2XA@@@@.    ;.3@##rs ,@9.h@  #@h3@@@M  5@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@:M  @@.  @22222
:::::::::: B&222222A@@@@B:.  .B@A;G , SM@@sh   ,3@#  @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@S ;#;@@  s@52222
::::::::::.;BX2222222223AB#@@B . 9@@@@&SrsSMX@  rs@,   :9@@@@@@@@@@@@@@@@@@@@@@@@@#.  @BrH#  @922222
::::::::::, #3222222222222252GA#@:, .AM@@MG3XA@#   @@X  .#@@@@@@@@@@@@@@@@@@@@@@@i@@@ 3:9MA @H522222
::::::::::: :#52222222222222222S23@#X   :G@@@@HH@@  3@ :H@@ .5@@@@@@@@@@@@@@@@@      : S@A BA2222222
:::::::::::, AH52222222222222222222XA@A;     #H@@@@s5S ,      , .. ,rM#i :      @@2S5 H@isS@22222222
::::::::::::. #G5222222222222222222252G@@.       rX@@@@sr@@@@##,   @@@@@@   M@@H r H@@@s  M2522222i5
::::::::::::: :@955222222222222222222225G@@s  .     ,#@@2   r  @# r@@@@@@S @@r: ##@@@H  ,@9222222222
:::::::::::::, s#3252222222222222222222222h#@@         ;h@@@@;G@  .,A@@A   G5HGB@@H    2@3222225S222
::::::::::::::, 5M322S2222222222222222222222XA@h9         M#M93r@Hs   @. HMX@&@@,    ,@#322225S22222
:::::::::::::::, 2M3222SS2222222222222222222223A@@S;.      :M@&iM r@H .  @@H#3;    ,@@&2222222222222
::::::::::::::::, 5#922225S2222222222222222222252h#@H,   ;   r@HXHG@. r5; M@r     B@G2222S5222222222
:::::::::::::::::, r@h2222255S2222222222222222222223H@@@.      @@@iS 3M#@@:     @@MX225S522222222222
::::::::::::::::::, .X222222225552222222222222222222222&@@@@#X   .#29M22.  AX@@@A2555522222222222222
(image by me, generated by Aros Fractals and ASCII-ed by ascgen.)
The Mandelbrot Set is generated by applying an iterated mathemathical formula to numbers in the complex number plane. e.g. Take complex number c = x + yi. Start with zero. With each iteration square the number and add c. If this value heads to infinity (if the modulous (distance from origin) gets bigger than 4 it will go to infinity) then c is not a member of the Mandelbrot set. If the number is still small after many iterations then the number could be a member of the set.

e.g. c = 0 + 0i
i=0, n=0, n=n^2 + c => n=0
i=1, n=0, n=n^2 + c => n=0
this is a member, it is never going to go to infinity.

e.g. c = -1 + 0i
i=0, n=0, n=n^2 + c => n=-1
i=1, n=-1, n=n^2 + c => n=0
i=2, n=0, n=n^2 + c => n=-1
i=3, n=-1, n=n^2 + c => n=0
this is a member, it is never going to go to infinity.

e.g. c = 1 + 0i
i=0, n=0, n=n^2 + c => n=1
i=1, n=1, n=n^2 + c => n=2
i=2, n=2, n=n^2 + c => n=5
this is NOT a member, it is going to go to infinity.

Pretty pictures are generated when looking at the interesting areas around the edge of fractals. Colours are chosen depending on how fast the exiting numbers go to infinity. You can generate an ascii Mandelbrot Set picture with the following code.

A quick foundation course in complex number arithmetic:
i is the square root of -1. So i^2 = -1.
let c = x + yi, then c^2 = x^2 + 2xyi - y^2.

#include < stdio.h >

#define SWIDTH 79
#define SHEIGHT 47

#define MINX -2.0
#define MINY 0
#define MAXX 0.5
#define MAXY 1.5
#define DIFX 2.5
#define DIFY 1.5

#define MAXIT 64

char shades [ 38 ] =" ..^^:://II&&@@@***%%%$$$###";

int iter(double r, double i)
{
int foo=0;
double rr,ii,zr,zi;
rr=ii=zi=zr=0.0;
while(foo< MAXIT && rr+ii< 4){
  foo++;
  zi=2*zr*zi+i;
  zr=rr-ii+r; 
  rr=zr*zr;
  ii=zi*zi;
/*printf("%d %f %f %f %f %f %f\n",foo,r,i,zr,zi,rr,ii);
*/
  }
return foo;
}

main()
{
int x,y,i;
double xp,yp;

for(y=0;y<=SHEIGHT;y++){
  xp=(double)y*DIFX/(double)SHEIGHT+MINX;
  for(x=0;x< SWIDTH;x++){
    yp=(double)x*DIFY/(double)SWIDTH+MINY;
    i=iter(xp,yp);
    printf("%c",shades [ i%32 ] );
    }
  printf("\n");
  }
}

The following is some Visual Basic code I came up with for drawing the Mandelbrot Set.

fractalmodule.vb:

Imports System.ComponentModel
Imports System.Drawing

Public Class MainForm
    Inherits System.Windows.Forms.Form

    Private RealMax, RealMin As Double
    Private ImaginaryMax, ImaginaryMin As Double
    Private MandelPicture As Bitmap

    Private IsDragging As Boolean
    Private DragRect As Rectangle
    Private Distance As Integer
    Private MouseTick As Integer

    Private CalcTime As Single

    Public Sub New()
        MyBase.New()
        MainForm = Me
        InitializeComponent()

        RealMax = 0.75
        RealMin = -2.25
        ImaginaryMax = 1.5
        ImaginaryMin = -1.5

        Show()
        Calc()
    End Sub

    Protected Overloads Overrides Sub Dispose(ByVal disposing As Boolean)
        If disposing Then
            If Not (components Is Nothing) Then
                components.Dispose()
            End If
        End If
        MyBase.Dispose(disposing)
    End Sub

    Private Sub Calc()
        Dim p, q As Integer
        Dim pMax, qMax As Integer

        Dim MandelValue As Integer
        Dim MandelColour As Color
        Dim ColourMultiplier As Integer
        Dim StopWatch As DateTime = DateTime.Now

        MandelPicture = New Bitmap(Picture.Width, Picture.Height)

        pMax = Picture.Width
        qMax = Picture.Height
        ColourMultiplier = CInt(256 / Iterations)

        For p = 0 To pMax - 1
            For q = 0 To qMax - 1
                MandelValue = (ColourMultiplier * GetMandel(ScaleIt(pMax, p, _
		        RealMin, RealMax), ScaleIt(qMax, q, ImaginaryMin, _
                    ImaginaryMax))) - 1
                MandelColour = Color.FromArgb(MandelValue, MandelValue, _
                    MandelValue)
                MandelPicture.SetPixel(p, q, MandelColour)
            Next
            Status.Text = Int(p / pMax * 100) & "%"
        Next

        Picture.Image = MandelPicture

        CalcTime = CSng(DateDiff(Microsoft.VisualBasic.DateInterval.Second, _
            StopWatch, DateTime.Now))
        PutViewportInStatus(RealMin, RealMax, ImaginaryMin, ImaginaryMax, _
            CalcTime)
    End Sub

    Private Sub Picture_MouseMove(ByVal sender As Object, ByVal e As _
        System.Windows.Forms.MouseEventArgs) Handles Picture.MouseMove
        Dim TempPic As Image
        Dim MyGfx As Graphics
        Dim MyPen As New Pen(Color.FromKnownColor( _
            Drawing.KnownColor.IndianRed), 1)

        If IsDragging Then
            If e.X - DragRect.Left > e.Y - DragRect.Top Then
                Distance = e.X - DragRect.Left
            Else
                Distance = e.Y - DragRect.Top
            End If

            If MouseTick > 5 Then 'only draw every 5 pixels of movement
                MouseTick = 0

                TempPic = MandelPicture.Clone( _
                    System.Drawing.Rectangle.FromLTRB(0, 0, MandelPicture.Width, _
                    MandelPicture.Height), Drawing.Imaging.PixelFormat.DontCare)

                MyGfx = Graphics.FromImage(TempPic)
                MyGfx.DrawRectangle(MyPen, DragRect.Left, DragRect.Top, _
                    Distance, Distance)

                Picture.Image = TempPic
            End If

            MouseTick += 1
            PutViewportInStatus(ScaleIt(Picture.Width, DragRect.Left, RealMin, _
                RealMax), ScaleIt(Picture.Height, DragRect.Top, ImaginaryMin, _
                ImaginaryMax), ScaleIt(Picture.Width, DragRect.Left + Distance, +
                RealMin, RealMax), ScaleIt(Picture.Height, DragRect.Top + _
                Distance, ImaginaryMin, ImaginaryMax))
        Else
            Status.Text = "Location: (" & Format(ScaleIt(Picture.Width, e.X, _
                RealMin, RealMax), "0.00 ") & Format(ScaleIt(Picture.Height, _
                e.Y, ImaginaryMin, ImaginaryMax), "+ 0.00;- 0.00") & ")"
        End If
    End Sub

    Private Sub PutViewportInStatus(ByVal pMin As Double, ByVal qMin As Double, _
        ByVal pMax As Double, ByVal qMax As Double, Optional ByVal Time _
            As Single = 0)
        Status.Text = "Viewport: (" & Format(pMin, "0.00;-0.00") & _
            Format(qMin, " + 0.00; - 0.00") & ") - (" & Format(pMax, _
            "0.00;-0.00") & Format(qMax, " + 0.00; - 0.00") & ")" & _
            Format(Time, "\; (0.00 seconds);;#")
    End Sub

    Private Sub Picture_MouseDown(ByVal sender As Object, ByVal e As _
        System.Windows.Forms.MouseEventArgs) Handles Picture.MouseDown
        Select Case e.Button
            Case MouseButtons.Left
                IsDragging = True
                DragRect.X = e.X
                DragRect.Y = e.Y
        End Select
    End Sub

    Private Sub Picture_MouseUp(ByVal sender As Object, ByVal e As _
        System.Windows.Forms.MouseEventArgs) Handles Picture.MouseUp
        Dim Input As String

        Select Case e.Button
            Case MouseButtons.Left

                If IsDragging = True Then
                    IsDragging = False
                    Picture.Image = MandelPicture

                    If Distance > 0 Then
                        RealMin = ScaleIt(Picture.Width, DragRect.Left, _
                            RealMin, RealMax)
                        ImaginaryMin = ScaleIt(Picture.Height, DragRect.Top, _
                            ImaginaryMin, ImaginaryMax)

                        RealMax = ScaleIt(Picture.Width, DragRect.Left + _
                            Distance, RealMin, RealMax)
                        ImaginaryMax = ScaleIt(Picture.Height, DragRect.Top + _
                            Distance, ImaginaryMin, ImaginaryMax)

                        Calc()
                    End If
                End If
            Case MouseButtons.Right
                IsDragging = False
                'Input = InputBox("0 < Iterations <= 256", "64").Trim
                'If IsNumeric(Input) Then
                'If CInt(Input) > 0 And CInt(Input) <= 256 Then
                '    Iterations = CInt(Input)
                '    Calc()
                'Else
                '    Beep()
                'End If
                'End If
        End Select
    End Sub

    Private Sub Picture_MouseLeave(ByVal sender As Object, ByVal e As _
        System.EventArgs) Handles Picture.MouseLeave
        PutViewportInStatus(RealMin, RealMax, ImaginaryMin, ImaginaryMax, _
            CalcTime)
    End Sub

#Region " Windows Form Designer generated code "

    'Required by the Windows Form Designer
    Private components As System.ComponentModel.Container
    Private WithEvents Status As System.Windows.Forms.StatusBar
    Private WithEvents Picture As System.Windows.Forms.PictureBox
    Dim WithEvents MainForm As System.Windows.Forms.Form

    'NOTE: The following procedure is required by the Windows Form Designer
    'It can be modified using the Windows Form Designer.  
    'Do not modify it using the code editor.
    Private Sub InitializeComponent()
        Me.Picture = New System.Windows.Forms.PictureBox()
        Me.Status = New System.Windows.Forms.StatusBar()
        Me.SuspendLayout()
        '
        'Picture
        '
        Me.Picture.Anchor = (((System.Windows.Forms.AnchorStyles.Top Or _
            System.Windows.Forms.AnchorStyles.Bottom) _
                    Or System.Windows.Forms.AnchorStyles.Left) _
                    Or System.Windows.Forms.AnchorStyles.Right)
        Me.Picture.BorderStyle = System.Windows.Forms.BorderStyle.Fixed3D
        Me.Picture.Cursor = System.Windows.Forms.Cursors.Cross
        Me.Picture.Name = "Picture"
        Me.Picture.Size = New System.Drawing.Size(397, 373)
        Me.Picture.TabIndex = 0
        Me.Picture.TabStop = False
        '
        'Status
        '
        Me.Status.Location = New System.Drawing.Point(0, 375)
        Me.Status.Name = "Status"
        Me.Status.Size = New System.Drawing.Size(397, 20)
        Me.Status.TabIndex = 1
        Me.Status.Text = "Please wait"
        '
        'MainForm
        '
        Me.AutoScaleBaseSize = New System.Drawing.Size(5, 13)
        Me.ClientSize = New System.Drawing.Size(397, 395)
        Me.Controls.AddRange(New System.Windows.Forms.Control() {Me.Status, _
            Me.Picture})
        Me.Font = New System.Drawing.Font("Tahoma", 8.0!)
        Me.MaximizeBox = False
        Me.MinimizeBox = False
        Me.Name = "MainForm"
        Me.StartPosition = System.Windows.Forms.FormStartPosition.CenterParent
        Me.Text = "Mandelbrot Set"
        Me.ResumeLayout(False)

    End Sub

#End Region

End Class

mainform.vb:

Imports System.ComponentModel
Imports System.Drawing

Public Class MainForm
    Inherits System.Windows.Forms.Form

    Private RealMax, RealMin As Double
    Private ImaginaryMax, ImaginaryMin As Double
    Private MandelPicture As Bitmap

    Private IsDragging As Boolean
    Private DragRect As Rectangle
    Private Distance As Integer
    Private MouseTick As Integer

    Private CalcTime As Single

    Public Sub New()
        MyBase.New()
        MainForm = Me
        InitializeComponent()

        RealMax = 0.75
        RealMin = -2.25
        ImaginaryMax = 1.5
        ImaginaryMin = -1.5

        Show()
        Calc()
    End Sub

    Protected Overloads Overrides Sub Dispose(ByVal disposing As Boolean)
        If disposing Then
            If Not (components Is Nothing) Then
                components.Dispose()
            End If
        End If
        MyBase.Dispose(disposing)
    End Sub

    Private Sub Calc()
        Dim p, q As Integer
        Dim pMax, qMax As Integer

        Dim MandelValue As Integer
        Dim MandelColour As Color
        Dim ColourMultiplier As Integer
        Dim StopWatch As DateTime = DateTime.Now

        MandelPicture = New Bitmap(Picture.Width, Picture.Height)

        pMax = Picture.Width
        qMax = Picture.Height
        ColourMultiplier = CInt(256 / Iterations)

        For p = 0 To pMax - 1
            For q = 0 To qMax - 1
                MandelValue = (ColourMultiplier * GetMandel(ScaleIt(pMax, _
                    p, RealMin, RealMax), ScaleIt(qMax, q, ImaginaryMin, _
                    ImaginaryMax))) - 1
                MandelColour = Color.FromArgb(MandelValue, MandelValue, _
                    MandelValue)
                MandelPicture.SetPixel(p, q, MandelColour)
            Next
            Status.Text = Int(p / pMax * 100) & "%"
        Next

        Picture.Image = MandelPicture

        CalcTime = CSng(DateDiff(Microsoft.VisualBasic.DateInterval.Second, _
            StopWatch, DateTime.Now))
        PutViewportInStatus(RealMin, RealMax, ImaginaryMin, ImaginaryMax, _
            CalcTime)
    End Sub

    Private Sub Picture_MouseMove(ByVal sender As Object, ByVal e As _
        System.Windows.Forms.MouseEventArgs) Handles Picture.MouseMove
        Dim TempPic As Image
        Dim MyGfx As Graphics
        Dim MyPen As New Pen(Color.FromKnownColor(Drawing.KnownColor.IndianRed _
            ), 1)

        If IsDragging Then
            If e.X - DragRect.Left > e.Y - DragRect.Top Then
                Distance = e.X - DragRect.Left
            Else
                Distance = e.Y - DragRect.Top
            End If

            If MouseTick > 5 Then 'only draw every 5 pixels of movement
                MouseTick = 0

                TempPic = MandelPicture.Clone( _
                    System.Drawing.Rectangle.FromLTRB(0, 0, MandelPicture.Width, _
                    MandelPicture.Height), Drawing.Imaging.PixelFormat.DontCare)

                MyGfx = Graphics.FromImage(TempPic)
                MyGfx.DrawRectangle(MyPen, DragRect.Left, DragRect.Top, _
                    Distance, Distance)

                Picture.Image = TempPic
            End If

            MouseTick += 1
            PutViewportInStatus(ScaleIt(Picture.Width, DragRect.Left, RealMin, _
                RealMax), ScaleIt(Picture.Height, DragRect.Top, ImaginaryMin, _
                ImaginaryMax), ScaleIt(Picture.Width, DragRect.Left + Distance, _
                RealMin, RealMax), ScaleIt(Picture.Height, DragRect.Top + _
                Distance, ImaginaryMin, ImaginaryMax))
        Else
            Status.Text = "Location: (" & Format(ScaleIt(Picture.Width, e.X, _
                RealMin, RealMax), "0.00 ") & Format(ScaleIt(Picture.Height, _
                e.Y, ImaginaryMin, ImaginaryMax), "+ 0.00;- 0.00") & ")"
        End If

    End Sub

    Private Sub PutViewportInStatus(ByVal pMin As Double, ByVal qMin As Double, _
        ByVal pMax As Double, ByVal qMax As Double, Optional ByVal Time As _
        Single = 0)
        Status.Text = "Viewport: (" & Format(pMin, "0.00;-0.00") & _
            Format(qMin, " + 0.00; - 0.00") & ") - (" & Format(pMax, _
            "0.00;-0.00") & Format(qMax, " + 0.00; - 0.00") & ")" & _
            Format(Time, "\; (0.00 seconds);;#")
    End Sub

    Private Sub Picture_MouseDown(ByVal sender As Object, ByVal e As _
        System.Windows.Forms.MouseEventArgs) Handles Picture.MouseDown
        Select Case e.Button
            Case MouseButtons.Left
                IsDragging = True
                DragRect.X = e.X
                DragRect.Y = e.Y
        End Select
    End Sub

    Private Sub Picture_MouseUp(ByVal sender As Object, ByVal e As _
        System.Windows.Forms.MouseEventArgs) Handles Picture.MouseUp
        Dim Input As String

        Select Case e.Button
            Case MouseButtons.Left

                If IsDragging = True Then
                    IsDragging = False
                    Picture.Image = MandelPicture

                    If Distance > 0 Then
                        RealMin = ScaleIt(Picture.Width, DragRect.Left, RealMin, _
                            RealMax)
                        ImaginaryMin = ScaleIt(Picture.Height, DragRect.Top, _
                            ImaginaryMin, ImaginaryMax)

                        RealMax = ScaleIt(Picture.Width, DragRect.Left + _
                            Distance, RealMin, RealMax)
                        ImaginaryMax = ScaleIt(Picture.Height, DragRect.Top + _
                            Distance, ImaginaryMin, ImaginaryMax)

                        Calc()
                    End If
                End If
            Case MouseButtons.Right
                IsDragging = False
                'Input = InputBox("0 < Iterations <= 256", "64").Trim
                'If IsNumeric(Input) Then
                'If CInt(Input) > 0 And CInt(Input) <= 256 Then
                '    Iterations = CInt(Input)
                '    Calc()
                'Else
                '    Beep()
                'End If
                'End If
        End Select
    End Sub

    Private Sub Picture_MouseLeave(ByVal sender As Object, ByVal e As _
        System.EventArgs) Handles Picture.MouseLeave
        PutViewportInStatus(RealMin, RealMax, ImaginaryMin, ImaginaryMax, _
            CalcTime)
    End Sub

#Region " Windows Form Designer generated code "

    'Required by the Windows Form Designer
    Private components As System.ComponentModel.Container
    Private WithEvents Status As System.Windows.Forms.StatusBar
    Private WithEvents Picture As System.Windows.Forms.PictureBox
    Dim WithEvents MainForm As System.Windows.Forms.Form

    'NOTE: The following procedure is required by the Windows Form Designer
    'It can be modified using the Windows Form Designer.  
    'Do not modify it using the code editor.
    Private Sub InitializeComponent()
        Me.Picture = New System.Windows.Forms.PictureBox()
        Me.Status = New System.Windows.Forms.StatusBar()
        Me.SuspendLayout()
        '
        'Picture
        '
        Me.Picture.Anchor = (((System.Windows.Forms.AnchorStyles.Top Or _
            System.Windows.Forms.AnchorStyles.Bottom) _
                    Or System.Windows.Forms.AnchorStyles.Left) _
                    Or System.Windows.Forms.AnchorStyles.Right)
        Me.Picture.BorderStyle = System.Windows.Forms.BorderStyle.Fixed3D
        Me.Picture.Cursor = System.Windows.Forms.Cursors.Cross
        Me.Picture.Name = "Picture"
        Me.Picture.Size = New System.Drawing.Size(397, 373)
        Me.Picture.TabIndex = 0
        Me.Picture.TabStop = False
        '
        'Status
        '
        Me.Status.Location = New System.Drawing.Point(0, 375)
        Me.Status.Name = "Status"
        Me.Status.Size = New System.Drawing.Size(397, 20)
        Me.Status.TabIndex = 1
        Me.Status.Text = "Please wait"
        '
        'MainForm
        '
        Me.AutoScaleBaseSize = New System.Drawing.Size(5, 13)
        Me.ClientSize = New System.Drawing.Size(397, 395)
        Me.Controls.AddRange(New System.Windows.Forms.Control() {Me.Status, Me.Picture})
        Me.Font = New System.Drawing.Font("Tahoma", 8.0!)
        Me.MaximizeBox = False
        Me.MinimizeBox = False
        Me.Name = "MainForm"
        Me.StartPosition = System.Windows.Forms.FormStartPosition.CenterParent
        Me.Text = "Mandelbrot Set"
        Me.ResumeLayout(False)

    End Sub

#End Region

End Class
And here's another implimentation in some -very- ugly Python. It is the result of trying to compose Mandelbrot Set entirely of List Comprehensions. Almost got there, but I couldn't manage to shoehorn the core iteration into a list comprehension, so I was forced to use reduce there.

As I said, this is ugly; it looks more like scheme than python. There's some whitespace added, but (excluding the import) the whole program is a single statement.

This requires a python newer than 2.1, due to the use of nested scopes. With 2.2 or newer, the import statement is unneeded.
thx to jamesc for his shades string.


#!/usr/bin/env python
from __future__ import nested_scopes

(lambda ITERS, BOUND, SCALE, RMIN, RMAX, IMIN, IMAX, STEP, SHADES,WRITE,LOG:
 [r for r in range( RMIN*SCALE, RMAX*SCALE, STEP) if
    WRITE('\n') or
    [i for i in range(IMIN*SCALE, IMAX*SCALE, STEP) if
        (lambda j : WRITE(SHADES[int(LOG(j+1))%len(SHADES)]))
        ((lambda k : reduce( lambda x, y : (abs(x[1]) <= BOUND) and
                                           (x[0]+1, (x[1])**2 + y[1]) or
                                            x,
                            [(0,k)]*ITERS))
            (complex(r/SCALE, i/SCALE))[0])]])(
10000, 4, 150.0, -2.25, 1.5, -1.25, 1.25, 5, ".^:/I&@*%%$$###,",
__import__('sys').stdout.write, __import__('math').log)

Some thoughts on the form Zn=(Z(n-1))^2+C:

^2 can be replaced with ^P, where P is any number greater than 1. Whole numbers produce sets with P-1 buds, whereas other numbers produce extremely interesting chaotic sets (P=1.5 is my personal favorite). P can also be complex, which produces even cooler fractals; these don't even have the characteristic set centered around the origin!

Multiple dimensional fractals can be made by having any combination of Zr, Zi, Cr, Ci, Pr, and Pi be dependent on any number of axis (four is the most human minds can usually handle ;). I have not yet tested quaternions or octonions.

If any point Z0 is found to be not contained within the set, then any of the Zn's encountered during iteration are also not contained. The opposite is theoretically true, but only if you iterate infinitely.

I wrote this particularly fugly bit of perl once as a joke for a friend... (needs a terminal with ANSI colour).


#!/usr/bin/perl
($x,$xi,$w,$cmax)=(shift||-2,shift||-1.5,shift||3,9+shift||300);
for ($ai=$xi+$w;$ai>=$xi;$ai-=$w/22){
for ($a=$x;$a<=$x+$w;$a+=$w/75){
($c,$b,$bi)=0;for (;$c<$cmax;$c++){
$b=$b**2-$bi**2+$a+$w/150;$bi=2*($b*$bi)+$ai+$w/44;last if $b**2+$bi**2>100000;}
printf $c==$cmax?"@":"\033[1;".($c%7+31)."m#\033[0m";}printf "\n";}

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