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`******* Mandelbrot Set by Ric *******
`August 2004

remstart
The Mandelbrot Set is a fractal - a pattern with an infinite level of detail.
It is generated using a very simple mathematical equation, but the result is
a stunningly intricate, unpredictable, and non-repeating 'chaotic' pattern.
remend

sync on
s=1  `initial scale factor
limit#=200.0 `maximum number of iterations

zoom: `restart point for zooming in

`c is the input complex number (which has a 'real' part and an 'imaginary' part.)
`z is the output complex number.
for cr#=(-3+y#) to (3+y#) step 0.016  `set a range of values for c(real part
for ci#=(-3+x#) to (3+x#) step 0.016  `and imaginary parts).

cr#=cr#/s `scale the values of c by the scale factor
ci#=ci#/s
zr#=cr#  `feed new value of c into initial value of z
zi#=ci#  `same for imaginary parts

it#=0.0  `reset number of iterations
repeat

remstart
Now use the equation for the Mandelbrot set, z=z^2+c, expanded using
complex algebra to give:  Z(real)=[Z(r)^2-z(i)^2)+C(r)], and
Z(imaginary)=[(2*Z(r)*Z(i))+C(i)], and repeat the process for it# iterations
to allow z the chance to increase.
remend

zrnew#=((zr#^2)-(zi#^2)+cr#)
zinew#=((2*zr#*zi#)+ci#)
it#=it#+1
zr#=zrnew#  `note that zi(new) relies on old value of zr, not zr(new).
zi#=zinew#

remstart
Now test whether the amplitude of z (the hypontenues of its
real and imaginary parts) has increased above a certain limit
(ie an unbound/divergent number). If all iterations have
been completed then z is bound/non divergent.
remend

until sqrt((zr#^2)+(zi#^2))>4 or it#>limit#

cr#=cr#*s `unscale c back to normal ready to print pattern
ci#=ci#*s

remstart
draw a dot at a position which represents c, and in a shade which
represents how many iterations it took for z to go over limit
remend

if it#>limit# then it#=0 ` black if it was straight away (no iterations)
ink rgb (it#*12,it#*4,it#*7),0
box ((ci#-x#)*60+(screen width()/2)),((cr#-y#)*60+(screen height()/2)),1+((ci#-x#)*60+(screen width()/2)),1+((cr#-y#)*60+(screen height()/2))

next ci#  `repeat for next value of c
sync
if escapekey()=1 then exit
next cr#

`zoom in and reposition the fractal by offseting c by x,y

do
if mouseclick()=1
  x#=(x#+(mousex()-(0.5*screen width()))/(60))*2
  y#=(y#+(mousey()-(0.5*screen height()))/(60))*2
s=s*2
cls
goto zoom
endif
loop

`have a nice day.

box (ci#*s*6+(screen width()/2)),int(cr#*s*6+(screen height()/2)),1+(ci#*s*6+(screen width()/2)),1+int(cr#*s*6+(screen height()/2))