XMANDEL(N)XMANDEL(N)NAMExmandel - window based interface to Mandelbrot sets and Julia sets
SYNOPSISxmandel [-display display]
DESCRIPTION
Xmandel is a user friendly interface for generating Mandelbrot sets and
Julia sets. It initially comes up with with several command buttons,
which are described below, for controlling the execution. A Mandelbrot
set is drawn in the window of the initial form when the mandel button
is selected. A separate window is created for drawing the Julia sets.
THEORY
Let z0 be a number in the complex plane (x + yi). Choose a complex
constant C. Calculate z1 = z0 ** 2 + C. Repeat this recursively, so
that z2 = z1 ** 2 + C, z3 = z2 ** 2 + C and so on. z[n] will either
tend to infinity or zero, depending on its initial value and the con‐
stant C. Specifically if the absolute value of z[n], expressed as |z|
= sqrt(x**2 + y**2) is greater than 2, then the recursive formula will
diverge.
So, to calculate a Julia set, take each point near (0,0i), and use the
formula z = z**2 + C recursively. The Julia set is the set of points
for which z = z**2 + C would iterate indefinitely for the constant C.
Pixels, which represent numbers in the complex plane, are set to the
number of iterations before |z| exceeds 2. This then becomes an index
into the hardware colormap. Each color then represents the number of
iterations before divergence is detected.
To calculate a Mandelbrot set, again take each point near (0,0i), use
the same formula z = z**2 + C recursively. This time let C be the ini‐
tial value of the point itself (C = z0). Rather than having the same C
for every point in the complex plane as in Julia set calculations, C is
different for each point in the plane. Again let the pixel value be
the number of iterations before |z| exceeds 2.
On monochrome displays, the pixel value is set to 1 if the iteration
count is 64, otherwise 0.
Mandelbrot sets and Julia sets are obviously closely related as can be
seen from the similarity of their respective formulas. If the constant
C is chosen from the interior of the Mandelbrot set, then the Julia set
calculated from that constant C will be connected, that is have no gaps
or discontinuities. If the constant C is chosen from outside the Man‐
delbrot set, the Julia set will be disconnected, more like grains of
dust (Fatou clouds). If the constant C is chosen from the border of
the Mandelbrot set, then the Julia set will be more convoluted. Given
this relationship between points in the Mandelbrot set and the Julia
set generated, Xmandel provides user selection of the constant C by
mouse selection in the Mandelbrot window.
BUTTONS
To control execution of the calculations, various buttons are provided.
The buttons are:
mandel -
Calculates a Mandelbrot set from (-2.0, -1.5) to (1.0, 1.5) and
display it in the window provided.
mandelzoom -
In order to zoom in on a given area in the Mandelbrot set, a
zoom button is provided. The area to be zoomed in on is
selected with the left mouse button. Left button down begins
the selection, dragging with left button down draws a rubber
banded box to show the zoom area, and left button up begins the
calculation. You can zoom in on a zoomed in area until you
reach the limits of the precision of your hardware. Selecting a
zoom area that crosses a window border doesn't work.
unzoom -
Return to previous zoom. Note that you can zoom all the way out
by selecting the mandel button.
redo - Because the Mandelbrot calculations are CPU intensive, xmandel
does not restart the calculation automatically on receipt of an
exposure event. This is left up to user control. The redo but‐
ton will simply recalculate the current zoom level and display
it in the Mandelbrot window. This is also useful for seeing new
detail when the iteration count is increased.
julia -
Calculates a Julia set. The user is required to select a point
inside the Mandelbrot window using the left mouse button as the
constant C for the Julia set calculation. It will open a new
window if needed. The Julia set is centered around (0,0), going
from (-1.5, -1.5) to (1.5, 1.5). Julia set points can be
selected from zoomed in Mandelbrot windows as well. Beware of
selecting points outside the Mandelbrot window.
clear -
Clears the Mandelbrot window.
quit - Exit the xmandel program.
increate iterations -
On color displays, the iteration count (sometimes called dwell)
is initially set to 256, on monochrome, 64. The increate itera‐
tion button will increase the interation count
by 256 on color or 64 on monochrome. This is useful for seeing
more detail when zoomed in.
reset iterations -
Will reset the iteration count to its default value of 256 or
64.
LABELS
hostname -
The name of the host is displayed in the topmost pane. This is
handy when comparing the performance of multiple copies of xman‐
del.
iteration count -
The current iteration count is displayed in the second pane.
current view -
The region of the Mandelbrot being displayed is given in the
bottommost pane, as a range of x and y values in real coordi‐
nates.
Julia set constant -
Julia sets are displayed in a separate window, and the value of
the constant used for the Julia set calculation is given to the
window manager to be displayed in the title bar.
BUGS
Xmandel uses hard coded values for button colors, assuming a 256 color
colormap.
Xmandel deliberately does not handle exposure events.
Selecting a zoom area that crosses a window border doesn't work.
Performance is slow on workstations, especially workstations without
floating point hardware.
AUTHOR
John L. Freeman
jlf@cray.com
4th Berkeley Distribution 07 March 1989 XMANDEL(N)
