John, David, Do you have any idea what the thresholding is doing? http://homepages.slingshot.co.nz/~alpersj/mandelbrot/ When I was holding down the threshold "S" key, at one point the fractal does a kind of "inside out" transformation and then eventually becomes an M-set. This sudden transformation is very intriguing to me. The "inside out" transformation the fractal did reminded me of the Fractint feature: "Inversion" that I've read about. The concept is very simple, but the docs warn of such "bizarre effects" that you can need "something to hang on to." I put a copy of the Fractint docs for inversion below. I just Factory Restored my computer and can't try it on a fractal myself yet. Would someone try some inversion values (<Y> options screen) on an M-set and let me know if this is what David's post's web site is doing? I had the idea that when the inversion circle radius passed through the value of one, that this is where the fractal might be doing its inside-out flip/transformation... "Inverting through a circle not centered on the origin..." might be a fun thing to try the Fractint Evolver on! Thanks, - Hal Lane ######################## # hallane@earthlink.net ######################## Fractint Version 20.04 Page 92 3.5 Inversion Many years ago there was a brief craze for "anamorphic art": images painted and viewed with the use of a cylindrical mirror, so that they looked weirdly distorted on the canvas but correct in the [mirror's] distorted reflection. (This byway of art history may be a useful defense when your friends and family give you odd looks for staring at fractal images color-cycling on a CRT.) The Inversion option performs a related transformation on most of the fractal types. You define the center point and radius of a circle; Fractint maps each point inside the circle to a corresponding point outside, and vice-versa. This is known to mathematicians as inverting (or if you want to get precise, "everting") the plane, and is something they can contemplate without getting a headache. John Milnor (also mentioned in connection with the Distance Estimator Method (p. 91)), made his name in the 1950s with a method for everting a seven- dimensional sphere, so we have a lot of catching up to do. For example, if a point inside the circle is 1/3 of the way from the center to the radius, it is mapped to a point along the same radial line, but at a distance of (3 * radius) from the origin. An outside point at 4 times the radius is mapped inside at 1/4 the radius. The inversion parameters on the <Y> options screen allow entry of the radius and center coordinates of the inversion circle. A default choice of -1 sets the radius at 1/6 the smaller dimension of the image currently on the screen. The default values for Xcenter and Ycenter use the coordinates currently mapped to the center of the screen. Try this one out with a Newton (p. 49) plot, so its radial "spokes" will give you something to hang on to. Plot a Newton-method image, then set the inversion radius to 1, with default center coordinates. The center "explodes" to the periphery. Fractint Version 20.04 Page 93 Inverting through a circle not centered on the origin produces bizarre effects that we're not even going to try to describe. Aren't computers wonderful? --- --- This email has been checked for viruses by Avast antivirus software. https://www.avast.com/antivirus