# Fractals

The Sierpinski carpet is a straightforward example of a fractal that you can code conveniently using recursion. We converted the c code in Wikipedia to Pascal. Program SierpinskiCarpet recursively divides each blank square into a grid of nine squares and colours the central square. It produced the following output.

Output from program SierpinskiCarpet

Program SierpinskiCarpet uses the graph unit which is available in Lazarus but not in Delphi.

program SierpinskiCarpet; {$mode objfpc}{$H+} uses SysUtils, Graph; var Gd, Gm : smallint; CountX, CountY, MaxX, MaxY : integer; function PixelToBeFilled (x, y, Width, Height : integer) : Boolean; var x2, y2 : integer; begin // First stopping condition if x < 1 then result := False else begin //If the grid was split in 9 parts, what part(x2,y2) would x,y fit into? x2 := (x * 3) DIV Width; //an integer from 0..2 inclusive y2 := (y * 3) DIV Height; //an integer from 0..2 inclusive //Second stopping condition //if in the centre square, it should be filled. if (x2 = 1) and (y2 = 1) then result := True else begin // General case {Offset x and y so it becomes bounded by 0 .. width DIV 3 and 0.. height DIV 3 and prepares for recursive call.} x := x - (x2 * Width) DIV 3; y := y - (y2 * Height) DIV 3; //Recursive call Result := PixelToBeFilled(x, y, Width DIV 3, Height DIV 3); end; end; end; begin write('When the graphics window opens, you may need to restore'+ ' down or minimise'#13#10'the window then maximise it' + ' to see all of the graphic.'#13#10 + 'Please press Return to continue.'); readln; Gd := Detect; Gm := 0; InitGraph(Gd, Gm, ''); //For best results length of side should be a power of 3. MaxY := 3; repeat MaxY := MaxY * 3; until MaxY * 3 > GetMaxY; MaxX := MaxY; //Draw carpet for CountX := 1 to MaxX do for CountY := 1 to MaxY do if PixelToBeFilled (CountX, CountY, MaxX, MaxY) then putPixel(CountX, CountY, Red); readln; end.

Program SierpinskiCurve provides good examples of mutual recursion. The code is based on the outline given in Recursion via Pascal by J. S. Rohl. The code was designed to enable continuous drawing by a plotter rather than for the console, which had poor resolution back in 1984! We show the curves of order 1 and 3. Try to predict the output of other orders before running them.

Sierpinski Curve (order 1)

Sierpinski Curve (order 3)

Lines and procedures are named by the letters of the compass directions that the output is following. For example, LineSE draws a diagonal line from top left to bottom right and procedure N draws a sequence of lines starting one diagonal line above bottom of the graphic and finishing one diagonal line below the top. The following graphic shows the increasing complexity of the output of procedure N as the order increases from 1 to 3.

Output from procedure N with orders 1, 2 and 3

We include `forward` declarations of procedures S, E and W because they call each other and otherwise the calls to procedures not yet coded would not be accepted by the compiler.

program SierpinskiCurve; {$mode objfpc}{$H+} uses SysUtils, Graph, Math; type order = 1..7; var SleepTime, LeftMargin, BottomMargin, GraphicWidth, x, y, h: integer; Gd, Gm : smallint; UserOrder : order; procedure S(i: order); forward; procedure E(i: order); forward; procedure W(i: order); forward; procedure LineN; begin y := y - 2 * h; lineTo(x, y); end; procedure LineS; begin y := y + 2 * h; lineTo(x, y); end; procedure LineE; begin x := x + 2 * h; lineTo(x, y); end; procedure LineW; begin x := x - 2 * h; lineTo(x, y); end; procedure LineNW; begin x := x - h; y := y - h; lineTo(x, y); end; procedure LineNE; begin x := x + h; y := y - h; lineTo(x, y); end; procedure LineSE; begin x := x + h; y := y + h; lineTo(x, y); end; procedure LineSW; begin x := x - h; y := y + h; lineTo(x, y); end; procedure N(i: order); begin if i = 1 then begin LineNE; LineN; LineNW; sleep(SleepTime); end else begin N(i - 1); LineNE; E(i - 1); LineN; W(i - 1); LineNW; N(i - 1); end; end; procedure E(i: order); begin if i = 1 then begin LineSE; LineE; LineNE; sleep(SleepTime); end else begin E(i-1); LineSE; S(i-1); LineE; N(i-1); LineNE; E(i-1); end; end; procedure S(i: order); begin if i = 1 then begin LineSW; LineS; LineSE; sleep(SleepTime); end else begin S(i - 1); LineSW; W(i - 1); LineS; E(i - 1); LineSE; S(i - 1); end; end; procedure W(i: order); begin if i = 1 then begin LineNW; LineW; LineSW; sleep(SleepTime); end else begin W(i - 1); LineNW; N(i - 1); LineW; S(i - 1); LineSW; W(i - 1); end; end; procedure Sierpinski(i: order); begin x := LeftMargin; y := GetMaxY - (h + BottomMargin); moveTo(x, y ); N(i); LineNE; E(i); LineSE; S(i); LineSW; W(i); LineNW; end; begin writeln('When the graphics window opens, you may need to restore'+ ' down or minimise'#13#10'the window then maximise it' + ' to see all of the graphic.'#13#10); write( 'Enter complexity integer (1 for basic shape, 7 for most complex). '); readln(UserOrder); if UserOrder < 4 then SleepTime := 600 DIV UserOrder else SleepTime := 0; Gd := Detect; Gm := 0; InitGraph(Gd, Gm, ''); //GraphicWidth is number of units GraphicWidth := trunc(power(2, UserOrder + 2) - 2); //h is size of 1 unit h:= GetMaxY DIV GraphicWidth; LeftMargin := (GetMaxX - h * GraphicWidth) DIV 2; BottomMargin := (GetMaxY - h * GraphicWidth) DIV 2; Sierpinski(UserOrder); writeln('Please press Return to exit.'); readln; end.

To run the program in Delphi 7, download and add wingraph.pas to the project as described in a section of our graphics tutorial. The first four lines should be as follows, with the rest of the program as above for Lazarus.

program SierpinskiCurve; {$APPTYPE CONSOLE} uses SysUtils, Math, wingraph in 'wingraph.pas';

## Experimenting with Fractals

You should be tempted to ring the changes on these programs, for example by changing line colours, fill colours, and line drawing procedures. Try to superimpose fractals with different orders. You can find plenty of other fractals to code and join the growing number of fractal artists.