FOTD 28-02-04 (Cellular Patterns [6])
FOTD -- February 28, 2004 (Rating 6) Fractal visionaries and enthusiasts: For some reason, today's image reminds me of the cells of a cellular telephone network. I named it "Cellular Patterns". It is actually a scene in the fractal that results when portions of Z^(-1.74) and Z^(-7.4) are combined, and (1/C) is added. Speaking of cell phones, they appear to be taking over the world. A simple 25-mile trip will reveal the network antennas everywhere -- many on their own cell-phone towers, or electric transmission towers, but also lurking in the least likely places such as on the roofs of schools and other buildings, hidden in church bell towers, perched on advertising billboards, water tanks, even on abandoned grain elevators. It appears to be only a matter of time before we will have cell-phone antennas on every light pole, on every telephone and utility pole in every neighborhood. We are living in an ever-increasing ocean of electromagnetic radiation. We can't escape it. We know that the natural electric energy in the human body helps control tissue growth. Tests have already demonstrated that the radiation from a hand- held cell phone causes measurable brain damage in laboratory animals, and does so at levels in the range of those now in use. Can anyone imagine permitting a substance to be added to food intended for human consumption if that substance has been demonstrated to cause brain damage in laboratory animals? If this were done, the outcry would be overwhelming. But where is the outcry about cell-phone radiation. All we hear are hints of concern and not very convincing assurances that all is well. I wonder whether cell phones will be the asbestos and cigarettes of the future, and the telephone companies will have a reputa- tion like the giant tobacco corporations have today. Or perhaps we will need to shield our homes in faraday cages to escape the sea of deadly radiation. . . . Hmmmm . . . look at the outburst that a simple fractal just caused. Today's image rates a nominal 6 and has an overall value of 84. It was rendered with the outside set to the 'tdis' option, which stands for 'total distance' -- the total distance the points travel before escaping beyond the escape radius. The scene is nearly as interesting when rendered with the traditional equal- iteration-band method. I chose the 'tdis' method by the flip of a coin. The render time of 7 minutes may be bypassed by downloading the finished image from Paul's FOTD web site at: <http://home.att.net/~Paul.N.Lee/FotD/FotD.html> With a temperature of 48F 9C, Friday was pleasant enough here at Fractal Central. At least the cats must have thought so. They spent over three hours on the porch, enjoying the sun, which is just now making it over the holly trees in the afternoons. My day was acceptable, and the fractal I found was acceptable. This is just about all one could ask for. Hopefully, today will be a repeat, and the fractal will be even better. Until next time, take care, and we survived for thousands of years without carrying telephones everywhere we go. Why has it now become almost impossible to go about our everyday lives without one? Jim Muth jamth@mindspring.com jimmuth@aol.com START 20.0 PAR-FORMULA FILE================================ Cellular_Patterns { ; time=0:07:09.25--SF5 on a P200 reset=2003 type=formula formulafile=allinone.frm formulaname=MandelbrotMix4 function=recip passes=1 center-mag=+1.88786870396801200/-0.000456232474488\ 38/6.944164e+011/1/-10.0008080291402823/0.00695999\ 736778776107 params=1/-1.74/0.4/-7.4/0/0 float=y maxiter=1200 inside=0 outside=tdis periodicity=10 colors=000pHEhRN``WTjdLtmMqjMohMlfMjdNhbNe`NcZNaXM\ cTLdPLeLKfIJgEJhAIi6Ij3PhHWgVafhYikVlnRoqOrtKuwHwz\ FuxDtvCstArr8qp7pn5ol4nj7ikAelCamFYmHUnKQoMMoNQjOT\ fOXaP_YPcTQfPQiL3gfKXX_MORBmL9nF8o97p46p6Dq8JqAPrC\ VrE`sGfsIlsHooGrkFugFwcJwbNwbQwbUwbYwb`wbdwbgwbhqc\ hiciadiVdjNejFej8eNJELKKJKQHKWFKaDKgBKm9Ks7KyBMoEN\ fIOXLPOOQFQUBSX8U_5heDvjLlmMcoNUrOLtPHuNDvL9wKDrOG\ mRJiUMdXQ__TWbWReZNhCkDGmPKo_OqjSsuNdWTcZZcadbcjbf\ paiuakpfikjgfneascXwaSw_NwZTwaZwddwgjwjpwm`5tljpjc\ riWsgPueIvdBw`FlXJaTMSYNQaNOeOMiOKmPIqPGuQEyQDjOAW\ N8HN63N44Q94TE4WI4ZN4aS4dWAfUGhTLiRRkQWlPTmQRrRPwS\ NzTLNUJLUIN_HOeGPkKQpPXjUbdZhZcoThuNmwHqwFuwDwwBzw\ 9zw7zw5zw3zw2zl9zaGzRMmGTh5ZcI`_UaWfbSrcjwrbwsWrtP\ muJitDes7ar1Yr2Wq2Vq3Tq3Sq3Rp4Pp4Op4Np9GqEAq6t2AjL\ E`bE`bF_bF_bFZaFZaFYaFYaFX`FX`FX`FW`FW_FV_FV_FU_FU\ _FTZFTZFSZFSZFRYFRYFQYy74 } frm:MandelbrotMix4 {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(-a*b*g*h)^j, k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END 20.0 PAR-FORMULA FILE==================================
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*************************************/
On Sat, 2004-02-28 at 16:51, Hiram Berry wrote:
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).
Hiram, Your approach to specifying a slice of a 4D object is very interesting. I'm not really able to say whether it works or not - my geometry's a bit hazy when it comes to this sort of thing. However it seems to me that you don't have enough parameters to specify any view in R4. I think you have p1, p2 and (implicitly) #pixel = 6 real parameters. However it takes 6 real parameters to uniquely specify a location + an orientation in 3D (or R3, as you put it) (x,y,z,yaw,pitch,roll). So I can't see how 6 would be sufficient in R4 - surely more information is required. Does your C2-based disallow some orientations which are possible in R4? FWIW I specify an image in R4 with 10 real parameters: (x,y,z,w) coords or the image center and angles around the xy, xz, xw, yz, yw and zw planes. The angles aren't all that intuitive so I'd be interested to find a more intuitively graspable way to specify a view. PS http://research.microsoft.com/~hollasch/thesis/ lists another method of specifying a view of an R4 object, using 4D "from" and "to" points and 2 4D "up" and "over" vectors. Regards, -- Edwin
Edwin, I found your observations concerning the number of degrees of freedom in the different viewing spaces of the Julibrot to be _extremely_ thought provoking; I'll try to answer your queries based on what I found on closer examination of this issue, but I'm not yet certain. Also, having to deconstruct the mapping between these two spaces did lead me to speculate about what might be an important process when viewing fractal objects. I will respond in-line: [...snip my initial explanation of viewing the Julibrot object in C2 space instead of R4 space...]
Hiram,
Your approach to specifying a slice of a 4D object is very interesting. I'm not really able to say whether it works or not - my geometry's a bit hazy when it comes to this sort of thing.
I expect were all intuitively hazy on geometry in spaces with complex bases; from what I understand the human brain grasps depth perception through a "hologramatic" way, ie. we're wired to think about things as we perceive them-- arguably mostly in R3. I say "arguably" because color perception seems also to be normally three dimensional and I've wondered about the possibility of viewing mathematical objects with color and spatial dimensions treated as equivalent axes, ie. R6, and doing basis transformations in that space and then look at the object from the redefined view. Still, because we can't really visualize such spaces I think we have to fall back on mathematical tools to confirm or falsify the properties. My admittedly cursory look at this has shown that most geometrical properties hold when extending from R2 to C2, eg. a straight line can be parameterized by an anchor point, a direction and a parameter of variation, or that on that "line" (which we would probably think of as a plane in the C2 case because it has 2 degrees of freedom) there is exactly one point that either is the origin or has the property that a vector from the origin to the point is orthogonal to every vector generated by drawing from that point to any other point on the line (or "plane"). I obviously can't say "a single closest point" because the complex are unordered, but restated as orthogonality the relation holds, and according to my results all of the relations that matter for taking linear slices also hold.
However it seems to me that you don't have enough parameters to specify any view in R4.
In a certain sense that's true. There are several implicit questions in such a statement that have to be separated before it can be addressed. First, do you mean any view, or any linear view, and if the latter, linear in which space? Second, if we answer the first question, why is it necessary to view the slices of the R4 image of the Julibrot in C2 in order to say that we can view all linear slices of the Julibrot-- is R4 (rather, should it be) the natural or canonical domain for the Julibrot object? And third, most interestingly, regardless of whether we should map between C2 and R4, what happens to the slice when we do? Concerning the first question, if we don't constrain the slices to be linear then it's trivial that every slice in one of the spaces can map one-to-one with a slice in the other, since there are 4 real components uniquely associated with any given point in either space. A slice is nothing other than an arbitrary set of points in one of the spaces. The number of families of slices is infinite, so I assume we're talking about linear slices only, though curved slices generated by well known formulae are reasonable too. Concerning the second, my contention is, and maybe I didn't make the point clearly enough before, that it is not necessary to map all the R4-linear slices to C2 to say that we can view all the linear slices of the Julibrot object. The quality of linearity has to be associated with some particular space before it has meaning, and I am saying that the canonical space for viewing the Julibrot object is C2, for the simple reason that its generating process is complex in two variables, not real in four variables. IOW, if we want to look at linear slices the quality of linearity has to be defined on some space; in some sense that is a subjective choice, but I maintain that the canonical choice for that is same space as the generating process itself, in this case C2 and not R4. It is not meaningless to require such a choice: I can imagine several different kinds of space with four degrees of freedom in components where linearity of a slice would not be preserved when mapping from one to the other. It's just that the images might be nonlinear curves, plausibly even disconnected, on the other end. Regarding the third question, I don't know precisely what the mapping does at this point. Definitely some of the planes in R4 map into nonlinear curves in C2. Their form (product terms of the coordinates) suggests to me they are a subset of the hyperboloids in C2. If we take C2 as the natural space for the Julibrot object, its image in R4 is stretched and distorted so that some curved surfaces are flattened into planes. Unfortunately the only approach I know to pin down the mapping is write an equivalent (I mean use same general method of generating slice) general slicing algorithm in each and then trying to figure out appropriate changes of variable to go from one to the other. The C2 case is easy, but the general R4 planes are tedious. I did manage to come up with one for fractint, copied below, though it is not efficient (I mean by this that it uses more than the minimum number of degrees of freedom, or that two different sets of parameters can generate the same slice). I haven't yet figured out how to write the C2 version generating the same real coefficients (associating (x,y,z,w)<->(x+yi,z+wi)).
I think you have p1, p2 and (implicitly) #pixel = 6 real parameters. However it takes 6 real parameters to uniquely specify a location + an orientation in 3D (or R3, as you put it) (x,y,z,yaw,pitch,roll). So I can't see how 6 would be sufficient in R4 - surely more information is required. Does your C2-based disallow some orientations which are possible in R4?
That's what I suspected at first too. But it turns out to be more prosaic than that. As I tried to state above, planes in one space have curvature when mapped into the other. It just looks like a stretching for some parts of the slices, though topology is _apparently_ maintained. I include a .par with this message that shows what looks very much like stretching of a mostly M-like section of the Julibrot on a hyperboloid surface (apparent stretching in more than 2 directions). This was rendered from a planar slice in R4, which I think adds some slight validity to my notion that C2 should be the canonical space for manipulating Julibrots; I've not seen that kind of distortion in the C2 based slices.
FWIW I specify an image in R4 with 10 real parameters: (x,y,z,w) coords or the image center and angles around the xy, xz, xw, yz, yw and zw planes. The angles aren't all that intuitive so I'd be interested to find a more intuitively graspable way to specify a view. Well, the way I look at the R4 slicing and d.o.f. is like this: a plane can be specified by 3 noncolinear points whether it's in R3 or R4, right? So we've got 4dof for each point = 12 dof. Now the plane allows 2dof of movement for each point, so the definition of the plane is redundant by 3*2 = 6dof; end result is it should be possible to specify a plane in R4 with 12-6 = 6 coefficients. Every plane has a single point closest to the R4 origin and we can project some axis of R4 onto the plane to serve as an orientation, so formally there are only 2dof in choosing a point's position (but practically could be a problem since any particular axis might be perpindicular to the plane) and those are the parameters of variation. Like you though, I found it a whole lot easier to understand and to use with 10 parameters for the plane definition, though I didn't use direction cosines to the planes. The views are of course highly redundant with the extra 4dof, ie. lots of skewed, magnified and translated pictures of the same plane, but the method seems easily graspable. My approach was to define the plane by an arbitrary anchor point (x,y,z,w) and two unit directions (angle,angle,angle). You might find this approach more intuitive, or there might be holes in it that I didn't see, but I reasoned that when specifying directions in R4 one could extend the idea of {longitude,latitude} to {longitude,latitude,ternitude} (name chosen arbitrarily for the third angle. The idea is that any arbitrary point on the sphere of radius 1 in R4 could be specified by some combination of the three angles by scaling up the polar coordinate relations: w = sin(alpha) z = cos(alpha)*sin(phi) y = cos(alpha)*cos(phi)*sin(theta) x = cos(alpha)*cos(phi)*cos(theta), theta=longitude, phi=latitude, alpha=ternitude and that an arbitrary point is just the unit vector (x,y,z,w) times its (real) magnitude. So for the plane the points are the anchor point plus linear combinations of the unit vectors generated from each angle triplet.
PS http://research.microsoft.com/~hollasch/thesis/ lists another method of specifying a view of an R4 object, using 4D "from" and "to" points and 2 4D "up" and "over" vectors.
I look forward to reading that; there would reasonably be several different ways to specify general planar slices in R4. Now that I think about it, your description sounds similar to what I did in the .frm; it may be the same method. My next step is to use the same approach in C2, then look at the code of both and see if a transformation of parameters can be made in the C2 formula to make its parameters match the R4 code in number and role. A further point I'd like to make is that it's obvious from this process that the seemingly trivial mapping from one space to another with the exact same number of real dof, even with a one-to-one mapping between components of the dimensions, has distinctly nontrivial consequences. We take each complex dimension in C2 where two subcomponents are associated (say it has o rder 2), and break it into two independent dimensions, each with one component (say those dimensions have order 1). An innocent operation without many interesting results, right? And yet this very simple act of dimension breaking generates curvature in the target image where once there was linearity. I suspect that dimensions can be broken in ways where the resultant dimensions have non-integer order-- which ought to be a fallow field of inquiry for generating fractals. Regards, Hiram /*********************FRM for general R4 slices possibly follows*******************************/ JulibrotSlice_R4 { ;p1: origin_x, origin_y; Mset init 0,0 ;p2: origin_z, origin_w; Mset init 0,0 ;p3: vector N longitude(deg), latitude(deg); Mset init 90,90 ;p4: vec N ternitude(deg), vec M longitude(deg): Mset init 90,90 ;p5: vec M latitude(deg), vec M ternitude(deg): Mset init 90,0 ; this does all R2 planar slices of the R4 mapping ; via dimension-splitting of the C2 Julibrot process phase space ; the origin is arbitrarily chosen, as are two unit vectors {N,M} that define ; the plane P = Origin + a * N + b * M; {a,b} = "pixel", parm of variation. ; unit vectors specified by 4-space direction:{longitude,latitude,ternitude} ; algorithm is not efficient, extraneous dof allows skewing ; in the dimension splitting, arbitrarily associate Z0=x+yi,C=z+wi IF (initflag==0) initflag=1, degtorad = pi/180.0, N_long = real(p3 * degtorad), N_lat = imag(p3 * degtorad), N_tern = real(p4 * degtorad), M_long = imag(p4 * degtorad), M_lat = real(p5 * degtorad), M_tern = imag(p5 * degtorad), N_w = sin(N_tern), N_z = cos(N_tern) * sin(N_lat), N_y = cos(N_tern) * cos(N_lat) * sin(N_long), N_x = cos(N_tern) * cos(N_lat) * cos(N_long), M_w = sin(M_tern), M_z = cos(M_tern) * sin(M_lat), M_y = cos(M_tern) * cos(M_lat) * sin(M_long), M_x = cos(M_tern) * cos(M_lat) * cos(M_long) ENDIF a = imag(pixel), b = real(pixel), z = p1 + a * N_x + b * M_x + flip(a * N_y + b * M_y), C = p2 + a * N_z + b * M_z + flip(a * N_w + b * M_w), : z=sqr(z)+C, |z|<=1024 } /*********************FRM ENDS*****************************************************/ /*********************PAR showing hyperboloid distortion in R4 slice***************************/ show_hyperboloid { ; Fractint Version 2003 Patchlevel 1 reset=2003 type=formula formulafile=exprmntl.frm formulaname=JulibrotSlice_R4 passes=1 center-mag=-1.11223/-0.214471/0.2264147 params=0/0/0/0/135/-90/135/-135/0/135 outside=tdis logmap=yes colors=000m00eee0000000L00e0<3>eL0SSLllLssLzzLzzz000555<3>HHHKKKOOO<3>cc\ chhhmmmsssU0cxzw<3>xjUxfNxbFxi8wU0<4>ZfQUiV0e0Koe<3>0zz<2>0Gz<3>rVzzVzzV\ rzVjB`0zVV<3>zzV<3>VzV<3>VzzVrzVjzLW0hhz<3>zhz<3>zhh<3>zzhvzhqzhWQ0hzh<2\
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Jim Muth wrote:
FOTD -- February 28, 2004 (Rating 6)
Fractal visionaries and enthusiasts:
For some reason, today's image reminds me of the cells of a cellular telephone network. I named it "Cellular Patterns". It is actually a scene in the fractal that results when portions of Z^(-1.74) and Z^(-7.4) are combined, and (1/C) is added.
Speaking of cell phones, they appear to be taking over the world. A simple 25-mile trip will reveal the network antennas everywhere -- many on their own cell-phone towers, or electric transmission towers, but also lurking in the least likely places such as on the roofs of schools and other buildings, hidden in church bell towers, perched on advertising billboards, water tanks, even on abandoned grain elevators. It appears to be only a matter of time before we will have cell-phone antennas on every light pole, on every telephone and utility pole in every neighborhood.
We are living in an ever-increasing ocean of electromagnetic radiation. We can't escape it. We know that the natural electric energy in the human body helps control tissue growth. Tests have already demonstrated that the radiation from a hand- held cell phone causes measurable brain damage in laboratory animals, and does so at levels in the range of those now in use. Can anyone imagine permitting a substance to be added to food intended for human consumption if that substance has been demonstrated to cause brain damage in laboratory animals? If this were done, the outcry would be overwhelming. But where is the outcry about cell-phone radiation. All we hear are hints of concern and not very convincing assurances that all is well.
Some of the artificial sweeteners are known to be carcinogenic in lab animals. The effects of EM radiation have not been determined, studies done have never conclusively shown that there is a problem, in fact a few studies show the opposite, reduction of certain cancers.
I wonder whether cell phones will be the asbestos and cigarettes of the future, and the telephone companies will have a reputa- tion like the giant tobacco corporations have today. Or perhaps we will need to shield our homes in faraday cages to escape the sea of deadly radiation. . . .
Hmmmm . . . look at the outburst that a simple fractal just caused.
Today's image rates a nominal 6 and has an overall value of 84. It was rendered with the outside set to the 'tdis' option, which stands for 'total distance' -- the total distance the points travel before escaping beyond the escape radius. The scene is nearly as interesting when rendered with the traditional equal- iteration-band method. I chose the 'tdis' method by the flip of a coin.
The render time of 7 minutes may be bypassed by downloading the finished image from Paul's FOTD web site at:
<http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
With a temperature of 48F 9C, Friday was pleasant enough here at Fractal Central. At least the cats must have thought so. They spent over three hours on the porch, enjoying the sun, which is just now making it over the holly trees in the afternoons.
My day was acceptable, and the fractal I found was acceptable. This is just about all one could ask for. Hopefully, today will be a repeat, and the fractal will be even better. Until next time, take care, and we survived for thousands of years without carrying telephones everywhere we go. Why has it now become almost impossible to go about our everyday lives without one?
Jim Muth jamth@mindspring.com jimmuth@aol.com
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On Sat, 28 Feb 2004, Jim Muth wrote: (...)
We are living in an ever-increasing ocean of electromagnetic radiation. We can't escape it. We know that the natural electric energy in the human body helps control tissue growth.
Before I become a shill for the LCD monitor makers, you are probably sitting in front of the strongest source of radiation, as the intensity of most radiation sources decreases at the inverse square of distance. In a point source, you can work out that relation from the geometry of a sphere: 4/3*pi*r^2 and 4*pi*r^3 (notice that one is the differential of the other, which aids memorization).
Can anyone imagine permitting a substance to be added to food intended for human consumption if that substance has been demonstrated to cause brain damage in laboratory animals?
You haven't met Nancy Markle. She's listed on quackwatch, so beware of the Federal Drug Administration.
Ifthis were done, the outcry would be overwhelming.
Her list has a lot of traffic, but three congressional hearings is nothing to shake a stick at. It's just that it's hard to pin-point the most serious and definite problem that Aspartame causes. It is _not_ hard to point fingers for malfeasance, but I see little point in that. As the Japanese say, fix the problem, not the blame. (...)
Hmmmm . . . look at the outburst that a simple fractal just caused.
I think caffeine might hav more to do with it. Try bitter (Seville or Marmalade) orange juice or Yerba Mate. They seem to be in season. (...)
I chose the 'tdis' method by the flip of a coin.
That takes longer than the first pass at default iteration counts on this machine, and not much longer on yours, so I would try many of the views before you raise the iteration count to what would take nine months on an 8086 to display about thirty more pixels. My fractals tend to hav the vast majority of pixels escape in the first sixteen iterations. The gray areas on the default map are routinely the more interesting, but only on elaborately contrived equations do I get a lot showing up in the dark region of the default palette. The black at the top of the palette, that might make stripes or discordant gaps or high-frequency noise in the fringe of an imaje rarely comes into play.
participants (5)
-
Edwin -
Hiram Berry -
Jim Muth -
Kevin Sexton -
SherLok Merfy