Ooooh, I’ve thought of something. My applet finds the final vector length for each pixel, then goes back to see how long the series took to initially visit that value.
So, because the end value is chaotic, depending on the end count, it makes sense that the “first visit” count is chaotic.
But there is a unique pattern relating all those end values for all the r, i, coordinates in the set. Something like this is visible in the Xaos zmag or real/imag mode plot.
And there is a corresponding, unique pattern relating all the first-visit values for a given end state, as seen in my applet’s attractor plot.
So how we perceive the journey depends on where we stop.
Yeah, I know that conclusion doesn’t mean anything logically. But for me it has a sort of Zen-like feeling to it. FWIW.
Edit: Maybe I should go to sleep now. If I haven’t already!
I’ve thrown in a bit of code to display something like the Xaos zmag incolor, in addition to everything else. I apologize for the colors; the applet I started with uses the same color algorithm for everything, so it was hard to shoehorn in the new data. I’m not releasing this feature until/unless I redo the applet entirely to make it less awful.
What you see is part of the set around the base of the upper lobe, at the top of the main cardioid. Specifically:
Wedge on the right: Plain old Mandelbrot fractal, as rendered by my applet.
Orange/yellow gradients (with a little green): my applet’s guess at the Xaos-style zmag incolor mode. It looks a lot like what Xaos produces, except the count is off by one, and the color map is entirely different (I vary hue, while apparently Xaos varies brightness).
Multicolor tangled linear features: twice-visited values as normally plotted by my applet.
Black tangled linear features: A new mystery. At first glance I think these are points for which my algorithm neglected to color a previous-visit match; maybe as a consequence of the rounding you people kept harping about. For now I’m calling it the Gödel space.
did you consider adding your interior visualization mode to Xaos? At the least posting your visualisation method to the dev discussion group would likely get some more informed comments than you’ve got here: http://wmi.math.u-szeged.hu/xaos/doku.php?id=support:main
I probably should. Unfortunately I’m no more a real programmer than a real mathematician. Sure, I’ve played with everything from MIX assembly to LISP, and I periodically write some embedded C for an MPU I like to use. But real C development systems nowadays are too, ahem, feature laden for my taste.
When I was young, I habitually read the datasheet and register reference for every damned chip in a computer. (For years after that my job was writing them.) Nowadays I get turned off a language if the source and includes for hello world don’t fit on a single screen. That’s why for this thing I went with Java–it’s kind of lame, but my make file is only three lines long (including the shell invocation), and the graphics are all done by the browser (reminds me of Logo for some reason).
Thanks though, it would be good idea if I were up for it.
At the least I’ll try to clean up what I’ve got enough so I can release the source. Right now it’s not even all English symbology (the original appplet was German).
I know this is a couple of months old now, but I also have found some detail inside the set. What I found are similar looking to Pickover Stalks, but I used a completely different algorithm that preserves symmetry about the real axis. I was wondering if anyone else has seen something like this before (I cannot find similar pictures with Google). I have included a link to my wiki-commons page below, where I have uploaded a sample
Those are pretty interesting looking. Do they animate smoothly if you render out a zoom? And if you keep zooming in on where the concentric circles appear to converge is there more and more of them revealed?
Yes to both questions. There are in fact infinitely many as you zoom in on where they converge. You can tell that there are infinitely many loops by looking at the following image
See the description for some info. Basically, for the loop set in the middle right of the image, every second loop ‘repeats’ that is, it intersects other loop structures in exactly the same way that the loop two before it did. You can see for yourself. There are also loop structures at the end of the Mandelbrot filaments that repeat every cycle, instead of every other. This shows that there are infinitely many.
You can zoom in as far as your willies will take you, but the loops will begin to add up, so you will have to reduce the parameter so that there will not be an overload. I have attached a youtube video I made of an animation where I have continuously adjusted the parameter, and you can see the loops grow until they merge and are excluded from the mandelbrot set
Also, usually at the boundary of the mandelbrot set there are more and more of these contours impinging upon it, as this image shows
I am still studying what these mean, but they are cool.
