Apply the complex function/transform , with initially , many times – say 500 – to each point in an area near . In terms of the Cartesian plane,

Test, by measuring the distance , if the point is escaping to or staying near Maybe if distance , assume it’s escaping; colour it white, otherwise colour it black. Voila, the basic Mandelbrot set diagram:

[b&w M picture]

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```PROGRAM: Mandelbrot set
Ws, Hs = width, height of screen in pixels
Wr, Hr = actual width, height of desired visual field
Cx, Cy = centre of area to display (initially 0,0 usually)
LIMIT = 5 //Fairly arbitrary value to test for escape
Xinc = Wr/Ws //actual width represented by one pixel
Yinc = Hr/Hs
XLeft = Cx - Wr/2
XRight = Cx + Wr/2
YBottom = Cy - Hr/2
YTop = Cy + Hr/2
for xc=XLeft to XRight step Xinc
for yc=YBottom to YTop step Yinc
x=0
y=0
for i=1 to 500
xtemp=x*x-y*y+xc
y=2*x*y+yc
x=xtemp
if (x*x+y*y)>LIMIT break //quit i loop
end i
Draw(xc,yc,i)
end yc
end xc

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The value of is used for colouring the pixel.

Questions

Q. Testing for , not , like the program does here, and I usually do, because it’s faster, is different to the proper sqrt-distance. How different?

Q. There are many different ways of colouring the points besides just . Like what.

Q. Using discards all information except final distance from 0. How about direction, total distance travelled, x- and y-distance, how close to path got to an axis (Pickover stalks) etc etc etc.