Mandelbrot set and Julia
sets
an applet made by António
Miguel de Campos (Portugal)
Start by clicking in a point in screen  you
see the evolution (the orbits) resulting from the
iteration of z_{n+1}=
z_{n}^{2}+c,
50 times, using that point as c, and z_{0}=0.
Each point z_{n+1 }is
shown in yellow and is connected to the previous point z_{n
}by a red line.
Press
and drag the mouse to select an area for zooming, and
then press c.
If you click keys 1 through 7
 you see some selected details in the
Mandelbrot set.
The key z makes «zoom in»
and Z
«zoom out» by a factor of 2, to the full
window. Use UPand DOWN keyboard
keys to change the number of iterations (counter
cnt shown at right in window) and then press
the c key to have more
detail when zoom is large. If you press u, i, o or
p , you
can see the different coloring possibilities. Use LEFT
and RIGHT  keyboard keys to change the «color
number» (cor shown at right in window)
and then press the c . (Colors used are
related to the number of executed iterations n
by the expression n^{0,1 v} in which v
is the value cor indicated on screen.)
Then try other commands shown in the following table.
When some command does not work, try reseting the applet
using m, or/and clicking in a point
inside the window.
z ,
Z

zoom in , zoom out 
b

bifurcations (cnt=300) 
m , M

Mandelbrot
, Mandelbrot with/ Julia for clicked point 
e , E

orbits
(last points cnt=300, slow evolution cnt=50) 
j

Julia
(for clicked point in Mandelbrot) 
t, T

trajectories
(clearing cnt=300, accumulating cnt=50) 
c

compute
(inside area selected using mouse) 
UP, DOWN

change
number of iterations 
1 a 7

some
selected details in Mandelbrot set 
LEFT,
RIGHT

change
color variable 
F1,..,F6

z_{n+1}=
z_{n}^{3}+c, z_{n+1}= z_{n}^{4}+c
,..., z_{n+1}= z_{n}^{8}+c 
S

change
sign 
F7,..,F12

z_{n+1}=
z_{n}^{3}+z_{n}^{2}+c,
...., z_{n+1}= z_{n}^{8}+z_{n}^{2}+c 
s , ESC

show with
successive zooms (centered in point clicked with
mouse), stop the show

k,u,i,o,p

color
palette, random colors, default colors,
alternative colors, black&white 
I

limit/don't
limit number of points to 50, in orbits drawn
after clicked point 
Detailed
information about the commands
If you prefer to use a 771
by 539 applet window, click here
(slower computations, no detailed information)
Higher
order Mandelbrot sets
F1 to F6  z_{n+1}=
z_{n}^{3}+c_{ }, z_{n+1}=
z_{n}^{4}+c_{ }, z_{n+1}=
z_{n}^{5}+c_{ }, z_{n+1}=
z_{n}^{6}+c_{ }, z_{n+1}=
z_{n}^{7}+c_{ }, z_{n+1}=
z_{n}^{8}+c.
F7 to F12  z_{n+1}=
z_{n}^{3}+z_{n}^{2}+c_{
}, z_{n+1}= z_{n}^{4}+z_{n}^{2}+c_{
}, z_{n+1}= z_{n}^{5}+z_{n}^{2}+c_{
}, z_{n+1}= z_{n}^{6}+z_{n}^{2}+c_{
}, z_{n+1}= z_{n}^{7}+z_{n}^{2}+c_{
}, z_{n+1}= z_{n}^{8}+z_{n}^{2}+c.
Zoom show
After choosing a point, if you press s , you start
zoom ins centered in that point.
To stop zooming, press
ESC. If, during the zoom you press u, i, o or
p , you
can see the different coloring possibilities. (Other
keyboard keys also work during zooming).
Julia Sets
If you press M (Mandelbrot
with/ Julia), you will see also
a brief presentation of the Julia set for the (previously
mouse) selected c. If you press m , you go
back to the default mode.
If you press j (after
m and clicking), you will see
the Julia set for the (previously mouse) selected
c. Press m , to go
back to the Mandelbrot set.
Orbits
e trajectories
e  shows
last 50 points of orbits for a series of initial
complex points. A grid with spacing of 5 is used
and only points corresponding to complex numbers
with a modulus < 1,41 (i.e. a^{2}+b^{2}
< 2) are shown. Use a number of iterations (cnt
shown at right in window) greater than 300. (Use
UP, DOWN to
change number of iterations) NOTE: in the bottom
right you can read «aguarde até ao fim», which
stands for «wait until the end». In the end,
you can read «FIM», which stands for «END» (in
portuguese).
E  a more
slow and detailed evolution of some of the points
(grid spacing of 40). (Use a cnt=50,
so that the iteration does not take too long). The
initial point of each evolution is shown by the
small line segments in the window border. Notice
that for some points you have periodic cycles.
T  see
evolution trajectories for a series of initial
complex points (grid with spacing of 4, i.e.,
column and line values of 0,4,8, 12, etc.). (Use a cnt=50,
so that the iteration does not take too long).
t  see
evolution trajectories more clearly. After each
line, screen is cleared. The numbers being tested
are drawn in white (see the white horizontal line
which goes down in the screen) the points
obtained from them, up to selected cnt, are in
yellow. (Use a cnt>300)
NOTE: For e and t,
if you use a greater value for cnt (e.g., 700),
it takes longer but you get drawings which are
much more interesting.
Bifurcations in Mandelbrot
map
b  The
orbits computed from real values are composed of
real points and their trajectories are easy to
visualize because they all lie in the real axe.
This command shows the orbit points for each real
c value, projected in the vertical of each point,
i.e., the real points are presented as if they
were complex values (e.g.,  1
is shown as if it was  i ).

l
 Clicking in a point in the
Mandelbrot set  we see the evolution (orbits)
resulting from 50 iterations using that point as c,
and z_{0}=0.
This command allows the visualization of the
evolution for the number of iterations selected
in cnt. Press again to go back
to normal mode.
S
 In each iteration the sign is
inverted, i.e., z_{n+1}= (z_{n}^{N}+..+c)*
= Re {z_{n}^{N}+..+c}  Im {z_{n}^{N}+..+c}
is used. The sets obtained are graphically very
interesting. Press again to go back to normal
mode.
Fractal
figures generated using Lsystems
Wall
papererrors due to finite precision
A
fractal leaf and the Sierpinski triangle
If you liked, send me a
messageto the address :to.campos1@clix.pt

