Jim, I'm also alarmed at our escalating plunge into the EM sea. Your allusion to electromagnetic modulation of cell division is valid: I recently did a patent search for medical devices that employ pulsed EM fields to stimulate tissue healing for various traumas. There were dozens of patents! And several had statistically verified efficacy. The fields used in these devices are however periodic functions; I did a test a few years back to see what kind of EM fields we were exposing our biology to- a simple antenna pickup + assorted bandpass filters connected to o-scope in single-shot mode - then walked around my house and outside to see what kind of waveforms I'd get. In all bands except the ubiquitous power-line frequency the waveforms were _not_ peridiodic, rather they were decidedly fractal in nature, especially in corners and around large trees. I assume this is from nearly random reflections, refractions and attenuations provided by objects in the environment. Of course none of the biome including ourselves has any evolutionary experience with sustained chaotic fields at the strength being fed from our RF appliances. Also, don't forget the Persinger helmet. Quite apart from pathological effects, some EM fields demonstrably alter perception. Who knows what effects chronic subcritical levels may be having on the population as a whole? I thoroughly enjoyed your intrepid investigation of the Julibrot slices. One thing has puzzled me thoughout, however, why is the slice generating formulae (ie. SliceJulibrot1..3) so complicated? AFAICT it uses direction cosines to take a 2D slice of an R4 space. Isn't the {Z0,C} Julibrot space really just a two dimensional phase space of the process Z->Z^2+C? Your formalae appear to map C2 to R4, then take an R2 slice of it. It seems to me that it would be simpler to work natively in the C2 space, take a C1 slice and parameterize the viewing area directly by the complex number contained in the "pixel" variable-- IOW use complex quantities directly, both in the p1..p5 and in the calculations. It looks to me like most of the geometric properties of the Euclidean plane translate to the C2 plane, including the transcendental functions, so that taking an arbitrary slice of the Julibrot is just the drawing of a C1 "line" curve with the parameter of variation equal to "pixel". Each of these curves has a unique point which either is the Julibrot origin or has the property that a vector drawn from that origin to the point will be orthogonal to a vector drawn from that point to any other point on the curve, just as in the Euclidean plane, so the slices can be uniquely specified and I think this approach allows all rather than most slices to be drawn (within the bounds of computational range and precision). A .frm entry that I think does this is: /*********************FRM BEGINS*********************************************/ JulibrotSlice0+1i { ; attempt at a universal Julibrot linear slicer ; try to slice C2 {Z0,C} phase space w/a C1 parameterized "line" curve ; line origin's complex directed angle in p1, complex magnitude in p2 ((0+0i,0+0i) for M-set) ; the screen R2 plane is mapped onto part of the C1 line ; z and c used in the calc depend on C2 Cartesian position ismand = true ; allow to toggle orthogonal julibrots? Nuz = cos(p1) , Nuc = sin(p1) ; complex components of unit vector normal to (z0,c) C2 plane Nz = p2 * Nuz , Nc = p2 * Nuc ; normal vector at magnitude Muz = Nuc, Muc = -Nuz ; comps of unit tangential vector k = pixel ; map R2 screen pos to C1 curve variable parameter z = Nz + k * Muz, c = Nc + k * Muc ; find pos in (z0,c) plane lim = 9 ; guess-don't know what escape limit should be for oblique slices : z = sqr(z) + c |z| <= lim } /***********************FRM ENDS********************************************/ My abilities at image prospecting are miniscule compared to Jim's ample FOTD abilities, so it's with some trepidation that I include a par file using that formula at the end (color map file used included as attachment). Using a simple formula allows easily understood mutation strategies for the Julibrot slices: (1) stretch the parameter of variation "k" by some function to tile or distort the domain (aside: multiplication by a complex rotates and magnifies an image, which is already encompassed by the fractint controls, so some parameters can be elided), (2) change the equation of the slice to something nonlinear-- eg. spherical or Lisajous surface slices suggest themselves, or (3) mutate the iterated transformation z->z^2+C to some other z->F(z,C), as Jim did in today's FOTD. The exploration of the Julibrot slices and especially the implication that the M-set and the pure Julia slices are only special cases of a more general subsetting process has been disconcerting for me. I have been trying to understand some very basic, I guess you would call "Pythagorean", properties of the M-set, eg. measuring and finding ratios and patterns in the sizes and placement of the buds and minibrots, and trying to find an algorithm for enumerating them, that sort of thing. To me, finding those properties ascribes geometrical "meaning" to the set apart from its mundane definition as points which belong to attractors of a particular process. Also, the archipelago of minibrots seems to have the property of being "nearly-connected" that I think Osher Doctorow alluded to a few months ago-- between any two minibrots appearing on a particular tendril at a particular scale you can always find another minibrot at a smaller scale that is closer to either of the original ones than they are to one another, yet the ratio of brot-size to apparent curve length shrinks as the scale shrinks (the archipelago has dust qualities). But, slicing the Julibrot space obliquely seems to break these interesting patterns-- many of the slices have Julia character in one region, Mandelbrot character in another, and the objects lo ok decidedly unconnected. So I have a question: are the attractive objects (by objects I loosely mean the geometrical continuation of the buds and minibrots from C1 space into C2 space) of the Julibrot space "nearly-connected" like the M-set? Hiram Berry /************************PARS BEGIN*******************************************/ JulibrotSlice0+1i_init { ;C2 based slicing reset type=formula formulafile=julibrot.frm formulaname=JulibrotSlice0+1i corners=-4/-4/4/4 params=0/0/0/0 maxiter=2000 inside=0 colors=@chroma.map } PaisleyZipperJBS { reset=2000 type=formula formulafile=julibrot.frm formulaname=JulibrotSlice0+1i center-mag=-0.968336/0.1272/26.01933/1.0003/-85.015/-0.002 params=3/0.0312885046005249/1.35/0 cyclerange=0/23 colors=@sloshdun.map } TumbleweedJBS { reset=2000 type=formula formulafile=julibrot.frm formulaname=JulibrotSlice0+1i center-mag=+0.58460104500000000/+0.37746477100000000/49.04443/0.9997 params=4/0/0/0 cyclerange=0/23 colors=@sloshdun.map } FireflyDawnJBS { ; Fractint Version 2003 Patchlevel 1 reset=2003 type=formula formulafile=julibrot.frm formulaname=julibrotslice0+1i center-mag=-0.02349570395000000/-0.04868921628458539/205.3879/1/-177.500\ 000000000028/1.31075705844807544e-014 params=4.2/-3/0/0 float=y cyclerange=0/23 colors=@sloshdun.map } WaltzingWizardsJBS { reset=2000 type=formula formulafile=fractint.frm formulaname=JulibrotSlice0+1i center-mag=-0.07163941860000000/-3.98956436000000000/76.45664/0.9997/-95\ .013 params=3.5/-3/4/0 cyclerange=0/23 colors=@sloshdun.map } /********************************PARS END*************************************/