The different regions of the fractal correspond to different kinds of stable, closed orbits. Neat!!
The largest white region is a 1-cycle (fixed point) orbit, the big circle to the left of it is the 2-cycle orbit region. Can you find the three different 3-cycle regions? The six different 4-cycle regions? Can you predict the cycle length and orbit shape of a region beforehand? As an orbit settles into its pattern, what does the shape of the dying off transient say about nearby regions? It's a fun puzzle!
See Help section below for much more info. For info on downloading the Tcl/Tk application version (it has a bigger image and a few more features) click here .
mandelstep v0.4 Copyright (c) 1997 Karl J. Runge (The following is taken verbatim from the online help text) ======================== How to use "mandelstep": ======================== "mandelstep" IS NOT a fractal generator or zoomer. Excellent programs for doing generation can be found. For example: Fractint: http://spanky.triumf.ca/www/fractint/FRACTINT.HTML Xmandel: http://chem.leeds.ac.uk/ICAMS/people/zsolt/xmandel.html XaoS: http://www.paru.cas.cz/~hubicka/XaoS/ and there are others. Fractal generation programs want to do as many pixel calculations as fast as possible, "mandelstep", on the other hand, wants to do as few pixel calculations as slowly as possible!!!! =============== Huh? Why Slow?? =============== Fractals are beautiful and mysterious things to look at, but if you want to at least start to try understand a little bit about them, the fractal generation programs (out of necessity) hide too much information. They basically give the bottom line: is a point in the fractal set or not, and if not, color it to show how in a crude sense how "far away" it is from the set. Since there can easily be half a million pixels, each requiring 100-200, or more, iterations each, it is very useful that the fractal generation programs do not show all the information contained in the iteration "trajectories". The idea behind "mandelstep" is that by letting a person select starting positions and looking at a handful of iteration "trajectories" or "orbits" the person can begin to understand more about the different regions of the Mandelbrot set. I.e., not just whether a point is inside the set, but what sort of "dance" it does to be a member. It is not long before one is using one's intuition about what sort of "dance" a point will do before one runs the actual iteration. It's a puzzle. It's fun! ========= Mappings: ========= For many fractals, the X-Y plane describes a control parameter of an iterated mapping function. If, for a fixed value of the control parameter, repeated iteration leads to the coordinates diverging towards infinity that point is said to not be in the fractal set. How fast it diverges to infinity is usually used to determine what color is assigned to it. This gives beautiful images! A point that does not lead to divergence toward infinity is said to be in the fractal set (in practice, if it has not diverged after some pre- defined "large enough" value, i.e. N_MAX = 128, then for practical purposes it is said that the point is in fractal set, although who knows exactly what will happen in the next N_MAX iterations!). Typically a point declared to be in the set would be colored black or white. They are white for "mandelstep". For the classical Mandelbrot set the iterated mapping function is: x' = x^2 - y^2 + Q_x y' = 2xy + Q_y or in complex numbers notation, (z = x + iy): z' = z^2 + Q The prime, e.g. x', means the updated value after one iteration. Perhaps x <== x^2 - y^2 + Q_x, y <== 2xy + Q_y is a better notation. Q = (Q_x, Q_y) is the control parameter of the iterated mapping function. A Q value is in the Mandelbrot set if (starting the first iteration at z = (0,0)) after infinitely many iterations the z variable has not diverged to infinity. It is not hard to show that if |z| > 2 then z will quickly diverge to infinity. ======== Running: ======== When "mandelstep" starts up (it even does that slowly!) it will show you the complex Q plane with the classical Mandelbrot fractal (derived via XaoS mentioned above) superimposed on the background. A little black dot denotes Q = (0,0). white regions denote the points inside the set (i.e. iteration does not diverge for those points). OK, here are the steps to actually run something: Click the mouse somewhere to select a point Q. To do a single iteration step from starting a z = Q, click the "Step" button. Or to run a whole bunch, say 30, in succession, click on "Run". That is basically it. For a Q-point to be in the Mandelbrot set it seems (to me) that it must converge to a stable n-cycle orbit. That is to say, an orbit that repeats after n iterations. The biggest round white region (the one containing 0,0) is where the iteration evolves to a 1-cycle. Try it and see! The next largish section that is a circle to the left of the 1- cycle region is where 2-cycles are stable. Crossing over a boundary inside the Mandelbrot set usually means an n-cycle region has become unstable and an m-cycle has become stable. It's fun predicting (at least I find it fun!) what the cycle lengths will be for a given region. Also, the shape of the orbit will sometimes be surprising. Run in the 1-cycle region close to another region and see how the shape of slowest dying transient (as the orbit collapses to a point) lets you predict the cycle length and orbit shape of the nearby region without even running in it! "mandelstep" does not (yet) let you get into the fascinating "super deep zooms" regions that give the beautiful images. But many things can be learned from looking at the global iteration picture. Can your intuition learned at the global level be applied way down in the deep zoom regions? It is, afterall, a self similar fractal! ----------------------------------------------------------------------------- Now to touch upon all the features, buttons, etc. ================= Menu Bar Buttons: ================= Step: ----- On the main menubar the "Step" button will perform one iteration of the mapping function. Run: ---- The next button over "Run", will do a number (default 30) of iterations in succession. The starting point is blue, the latest point is red, and the rest are green. Follow the red dot for the orbit. Stop: ----- As the name implies, this button will stop the current "Run". Hit "Run" again to resume, or switch to a new set of run parameters if you like. Clear: ------ This will remove all trajectories/orbits from the display. (See "Clear Last Orbit" below) Help: ----- Shows the current help text. X: and Y: --------- As the mouse is moving around the fractal background, the Q_x and Q_y of the mouse position are shown in these entry boxes. If you click the mouse button the Q-value at point of the click will be "frozen" in the display boxes. You can then, say, move the mouse over to the "Run" button. You can also type values directly into the X: and Y: entry boxes. Hit
in either one to "freeze/set" the values, then go over to "Step" or "Run" to start iterating the mapping function for this Q. ======== Actions: ======== More things one can do. Long Run: --------- Do a run 5 times the normal length. X-Y Trace: ------------ Pops up a text window with the history of x and y values for the latest run. Clear Last Orbit: ---------- Remove the latest trajectory/orbit from the display. Exit: ----- I forget what this one does :-) ====================== Options: ====================== Set the parameters. Connect Points: --------------- Checkbox to set mode to draw lines between the iteration points or not. Set N iterations: ----------------- Set the number of iterations in one "Run" click. An entry box will appear on the Menu Bar. Type the value in the entry box and then hit to set it. Delay: ------ Set the time delay between iterations in the "Run" action. Not that "mandelstep" is blazingly fast or anything, but you may want to slow it down a bit to watch some complicated orbit. Enter the delay time in milliseconds. There are 1000 of them in 1 second. As in the previous paragraph, the entry box appears on the Menu Bar, Hit to set the value to what you typed in. ============================================================================= Well, that's all for now except for some misc. notes below. Please send suggestions, comments, etc to me at email@example.com The main thing on the TODO list is: Let user drag out a region of the plane with the mouse, a second canvas appears where the region is zoomed (maybe call xaos or something in "filter" mode, or just scale?) Then the user can click on the expanded view to select the starting point with higher resolution. Iterations should be shown on both the main and zoomed canvases. mandelstep v0.4 Copyright (c) 1997 Karl J. Runge ============================================================================= Misc. Notes: ------------ One can solve for the two main regions of the classical Mandelbrot set by solving using some simple math: If you denote the mapping by f(z) = z^2 + Q, then look for fixed point solutions z = f(z), this will give a quadratic equation for the 1-cycle orbit z's. By examining the derivative df/dz at the fixed point one finds it is a stable orbit when |df/dz| < 1 (hint expand about the solution z + dz and look for the condition that |dz| shrinks). The formula |df/dz| < 1 gives exactly the big cardioid shape that encloses the point (0,0). Seriously cool! What about the stable 2-cycle region? Look at fixed points z = f(f(z)) this will give you a quartic equation in z, but you can factor out the 1-cycle solutions found above (since a 1-cycle is clearly also a 2-cycle), this leaves another quadratic equation to solve for the "real" 2-cycles. The condition that is it stable ( |d(f(f(z))/dz| < 1) gives the equation for the circle (centered at (-1,0)) immediately to left of the 1-cycle cardioid. The circle's radius is 1/4. Well, the math starts getting hard at z = f(f(f(z))), (3-cycles) and above, but you can let "mandelstep" do all the work for you! Start a run inside a given region and what what sort of n-cycle it settles into. In my non-rigorous view of these things, all of the infinitely many sections of the mandelbrot fractal can be thought as some sort of stable n-cycle. As n goes up the region of stability gets smaller and smaller giving rise to smaller still regions with longer period orbits. And this continues downward to smaller scales without limit. Amazing! Another way to proceed is to look for Super Stable orbits. These are ones obeying the stability criterion: d(f(...(f(z))...)/dz = 0. It turns out this requires one of the points on the orbit be z = 0. The control parameters corresponding to these super stable orbits tend to be near the center of its region, i.e. farthest from the region boundary (where |df(...)/dz| = 1). Anyway, it is not too hard to get a symbolic manipulator work out f(f(...f(0)...)) = Q, then solve numerically for the roots Q of this polynomial (it is all Q, no z for this approach). Sure enough, the roots are right near the centers of the corresponding n-cycle regions of the set.