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This section is intended only as a quick introduction to the types; more detail on the mathematics of types other than the Mandelbrot set can be found in Appendix A, and some notes on how Fractint calculates them in chapter 9. Fractint starts by default with the Mandelbrot set. You can change that by firing it up with the command-line argument "TYPE=" followed by one of the names in parentheses below, or within the program by hitting <T> and typing in the name. If parameters are needed, you will be prompted for them. In the text that follows, due to the limitations of the ASCII character set, "a*b" means "a times b", and "a^b" means "a to the power b". The Mandelbrot Set (type=mandel) This set is the classic: the only one implemented in many plotting programs, and the source of most of the printed fractal images published in recent years. Like most of the other types in Fractint, it is simply a graph: the x (horizontal) and y (vertical) coordinate axes represent ranges of two independent quantities, with various colors used to symbolize levels of a third quantity which depends on the first two. So far, so good: basic analytic geometry. Now things get a bit hairier. The x axis is ordinary, vanilla real numbers. The y axis is an imaginary number, i.e. a real number times i, where i is the square root of -1. Every point on the plane -- in this case, your PC's display screen -- represents a complex number of the form: x-coordinate + i * y-coordinate If your math training stopped before you got to imaginary and complex numbers, this is not the place to catch up. Suffice it to say that they are just as "real" as the numbers you count fingers with (they're used every day by electrical engineers) and they can undergo the same kinds of algebraic operations. OK, now pick any complex number -- any point on the complex plane -- and call it C, a constant. Pick another, this time one which can vary, and call it Z. Starting with Z=0 (i.e., at the origin, where the real and imaginary axes cross), calculate the value of the expression Z^2 + C Take the result, make it the new value of the variable Z, and calculate again. Take that result, make it Z, and do it again, and so on: in mathematical terms, iterate the function Z(n+1) = Z(n)^2 + C. For certain values of C, the result "levels off" after a while. For all others, it grows without limit. The Mandelbrot set you see at the start -- the solid-colored lake (blue by default), the blue circles sprouting from it, and indeed every point of that color -- is the set of all points C for which the value of Z is less than 2 after 150 iterations (default setting). All the surrounding "contours" of other colors represent points for which Z exceeds 2 after 149 iterations (the contour closest to the M-set itself), 148 iterations, (the next one out), and so on. Some features of interest: 1. Use the <X> options screen to increase the number of iterations. Notice that the boundary of the M-set becomes more and more convoluted (the technical terms are "wiggly," "squiggly," and "utterly bizarre") as the Z-values for points that were still within the set after 150 iterations turn out to exceed 2 after 200, 500, or 1200. In fact, it can be proven that the true boundary is infinitely long: detail without limit. 2. Although there appear to be isolated "islands" of blue, zoom in -- that is, plot for a smaller range of coordinates to show more detail -- and you'll see that there are fine "causeways" of blue connecting them to the main set. As you zoomed, smaller islands became visible; the same is true for them. In fact, there are no isolated points in the M-set: it is "connected" in a strict mathematical sense. 3. The upper and lower halves of the first image are symmetric (a fact that Fractint makes use of here and in some other fractal types to speed plotting). But notice that the same general features -- lobed discs, spirals, starbursts -- tend to repeat themselves (although never exactly) at smaller and smaller scales, so that it can be impossible to judge by eye the scale of a given image. 4. In a sense, the contour colors are window-dressing: mathematically, it is the properties of the M-set itself that are interesting, and no information about it would be lost if all points outside the set were assigned the same color. If you're a serious, no-nonsense type, you may want to cycle the colors just once to see the kind of silliness that other people enjoy, and then never do it again. Go ahead. Just once, now. We trust you. Julia Sets (type=julia) These sets were named for mathematician Gaston Julia, and can be generated by a simple change in the iteration process described above. Start with a specified value of C, "C-real + i * C-imaginary"; use as the initial value of Z "x-coordinate + i * y-coordinate"; and repeat the same iteration, Z(n+1) = Z(n)^2 + C. There is a Julia set corresponding to every point on the complex plane - - an infinite number of Julia sets. But the most visually interesting tend to be found for the same C values where the M-set image is busiest, i.e. points just outside the boundary. Go too far inside, and the corresponding Julia set is a circle; go too far outside, and it breaks up into scattered points. In fact, all Julia sets for C within the M-set share the "connected" property of the M-set, and all those for C outside lack it. Fractint's spacebar toggle lets you "flip" between any view of the M-set and the Julia set for the point C at the center of that screen. You can then toggle back, or zoom your way into the Julia set for a while and then return to the M-set. So if the infinite complexity of the M-set palls, remember: each of its infinite points opens up a whole new Julia set. Historically, the Julia sets came first: it was while looking at the M- set as an "index" of all the Julia sets' origins that Mandelbrot noticed its properties. The relationship between the Mandelbrot set and Julia set can hold between other sets as well. Many of Fractint's types are "Mandelbrot/Julia" pairs (sometimes called "M-sets" or "J-sets". All these are generated by equations that are of the form z(k+1) = f(z(k),c), where the function orbit is the sequence z(0), z(1), ..., and the variable c is a complex parameter of the equation. The value c is fixed for "Julia" sets and is equal to the first two parameters entered with the "params=Creal/Cimag" command. The initial orbit value z(0) is the complex number corresponding to the screen pixel. For Mandelbrot sets, the parameter c is the complex number corresponding to the screen pixel. The value z(0) is c plus a perturbation equal to the values of the first two parameters. See discussion of "type=mandellambda". This approach may or may not be the "standard" way to create "Mandelbrot" sets out of "Julia" sets. Some equations have additional parameters. These values is entered as the third for fourth params= value for both Julia and Mandelbrot sets. The variables x and y refer to the real and imaginary parts of z; similarly, cx and cy are the real and imaginary parts of the parameter c and fx(z) and fy(z) are the real and imaginary parts of f(z). The variable c is sometimes called lambda for historical reasons. NOTE: if you use the "PARAMS=" argument to warp the M-set by starting with an initial value of Z other than 0, the M-set/J-sets correspondence breaks down and the spacebar toggle no longer works. Newton domains of attraction (type=newtbasin) The Newton formula is an algorithm used to find the roots of polynomial equations by successive "guesses" that converge on the correct value as you feed the results of each approximation back into the formula. It works very well -- unless you are unlucky enough to pick a value that is on a line BETWEEN two actual roots. In that case, the sequence explodes into chaos, with results that diverge more and more wildly as you continue the iteration. This fractal type shows the results for the polynomial Z^n - 1, which has n roots in the complex plane. Use the <T>ype command and enter "newtbasin" in response to the prompt. You will be asked for a parameter, the "order" of the equation (an integer from 3 through 10 -- 3 for x^3-1, 7 for x^7-1, etc.). A second parameter is a flag to turn on alternating shades showing changes in the number of iterations needed to attract an orbit. Some people like stripes and some don't, as always, Fractint gives you a choice! The coloring of the plot shows the "basins of attraction" for each root of the polynomial -- i.e., an initial guess within any area of a given color would lead you to one of the roots. As you can see, things get a bit weird along certain radial lines or "spokes," those being the lines between actual roots. By "weird," we mean infinitely complex in the good old fractal sense. Zoom in and see for yourself. This fractal type is symmetric about the origin, with the number of "spokes" depending on the order you select. It uses floating-point math if you have an FPU, or a somewhat slower integer algorithm if you don't have one. Newton (type=newton) The generating formula here is identical to that for newtbasin, but the coloring scheme is different. Pixels are colored not according to the root that would be "converged on" if you started using Newton's formula from that point, but according to the iteration when the value is close to a root. For example, if the calculations for a particular pixel converge to the 7th root on the 23rd iteration, NEWTBASIN will color that pixel using color #7, but NEWTON will color it using color #23. If you have a 256-color mode, use it: the effects can be much livelier than those you get with type=newtbasin, and color cycling becomes, like, downright cosmic. If your "corners" choice is symmetrical, Fractint exploits the symmetry for faster display. There is symmetry in newtbasin, too, but the current version of the software isn't smart enough to exploit it. The applicable "params=" values are the same as newtbasin. Try "params=4." Other values are 3 through 10. 8 has twice the symmetry and is faster. As with newtbasin, an FPU helps. Complex Newton (type=complexnewton/complexbasin) Well, hey, "Z^n - 1" is so boring when you can use "Z^a - b" where "a" and "b" are complex numbers! The new "complexnewton" and "complexbasin" fractal types are just the old "newton" and "newtbasin" fractal types with this little added twist. When you select these fractal types, you are prompted for four values (the real and imaginary portions of "a" and "b"). If "a" has a complex portion, the fractal has a discontinuity along the negative axis - relax, we finally figured out that it's *supposed* to be there! Lambda Sets (type=lambda) This type calculates the Julia set of the formula lambda*Z*(1-Z). That is, the value Z[0] is initialized with the value corresponding to each pixel position, and the formula iterated. The pixel is colored according to the iteration when the sum of the squares of the real and imaginary parts exceeds 4. Two parameters, the real and imaginary parts of lambda, are required. Try 0 and 1 to see the classical fractal "dragon". Then try 0.2 and 1 for a lot more detail to zoom in on. It turns out that all quadratic Julia-type sets can be calculated using just the formula z^2+c (the "classic" Julia"), so that this type is redundant, but we include it for reason of it's prominence in the history of fractals. Mandellambda Sets (type=mandellambda) This type is the "Mandelbrot equivalent" of the lambda set. A comment is in order here. Almost all the Fractint "Mandelbrot" sets are created from orbits generated using formulas like z(n+1) = f(z(n),C), with z(0) and C initialized to the complex value corresponding to the current pixel. Our reasoning was that "Mandelbrots" are maps of the corresponding "Julias". Using this scheme each pixel of a "Mandelbrot" is colored the same as the Julia set corresponding to that pixel. However, Kevin Allen informs us that the MANDELLAMBDA set appears in the literature with z(0) initialized to a critical point (a point where the derivative of the formula is zero), which in this case happens to be the point (.5,0). Since Kevin knows more about Dr. Mandelbrot than we do, and Dr. Mandelbrot knows more about fractals than we do, we defer! Starting with version 14 Fractint calculates MANDELAMBDA Dr. Mandelbrot's way instead of our way. But ALL THE OTHER "Mandelbrot" sets in Fractint are still calculated OUR way! (Fortunately for us, for the classic Mandelbrot Set these two methods are the same!) Well now, folks, apart from questions of faithfulness to fractals named in the literature (which we DO take seriously!), if a formula makes a beautiful fractal, it is not wrong. In fact some of the best fractals in Fractint are the results of mistakes! Nevertheless, thanks to Kevin for keeping us accurate! (See description of "initorbit=" command for a way to experiment with different orbit intializations). Plasma Clouds (type=plasma) Plasma clouds ARE real live fractals , even though we didn't know it at first. They are generated by a recursive algorithm that randomly picks colors of the corner of a rectangle, and then continues recursively quartering previous rectangles. Random colors are averaged with those of the outer rectangles so that small neighborhoods do not show much change, for a smoothed-out, cloud-like effect. The more colors your video mode supports, the better. The result, believe it or not, is a fractal landscape viewed as a contour map, with colors indicating constant elevation. To see this, save and view with the "3" command (see section on 3D), and your "cloud" will be converted to a mountain! You've GOT to try palette cycling on these (hit "+" or "-"). If you haven't been hypnotized by the drawing process, the writhing colors will do it for sure. We have now implemented subliminal messages to exploit the user's vulnerable state; their content varies with your bank balance, politics, gender, accessibility to a Fractint programmer, and so on. A free copy of Microsoft C to the first person who spots them. This type accepts a single parameter, which determines how abruptly the colors change. A value of .5 yields bland clouds, while 50 yields very grainy ones. The default value is 2. Zooming is ignored, as each plasma- cloud screen is generated randomly. The algorithm is based on the Pascal program distributed by Bret Mulvey as PLASMA.ARC. We have ported it to C and integrated it with Fractint's graphics and animation facilities. This implementation does not use floating-point math. Saved plasma-cloud screens are EXCELLENT starting images for fractal "landscapes" created with the "3D" options. Lambdafn (type=lambdafn) Function=[sin|cos|sinh|cosh|exp|log|sqr]) is specified with this type. Prior to version 14, these types were lambdasine, lambdacos, lambdasinh, lambdacos, and lambdaexp. Where we say "lambdasine" or some such below, the good reader knows we mean "lambdafn with function=sin".) These types calculate the Julia set of the formula lambda*fn(Z), for various values of the function "fn", where lambda and Z are both complex. Two values, the real and imaginary parts of lambda, should be given in the "params=" option. For the feathery, nested spirals of LambdaSines and the frost-on-glass patterns of LambdaCosines, make the real part = 1, and try values for the imaginary part ranging from 0.1 to 0.4 (hint: values near 0.4 have the best patterns). In these ranges the Julia set "explodes". For the tongues and blobs of LambdaExponents, try a real part of 0.379 and an imaginary part of 0.479. A co-processor used to be almost mandatory: each LambdaSine/Cosine iteration calculates a hyperbolic sine, hyperbolic cosine, a sine, and a cosine (the LambdaExponent iteration "only" requires an exponent, sine, and cosine operation)! However, Fractint now computes these transcendental functions with fast integer math. In a few cases the fast math is less accurate, so we have kept the old slow floating point code. To use the old code, invoke with the float=yes option, and, if you DON'T have a co-processor, go on a LONG vacation! Mandelfn (type=mandelfn) Function=[sin|cos|sinh|cosh|exp|log|sqr]) is specified with this type. Prior to version 14, these types were mandelsine, mandelcos, mandelsinh, mandelcos, and mandelexp. Same comment about our lapses into the old terminology as above! These are "pseudo-Mandelbrot" mappings for the LambdaFn Julia functions. They map to their corresponding Julia sets via the spacebar command in exactly the same fashion as the original M/J sets. In general, they are interesting mainly because of that property (the function=exp set in particular is rather boring). Generate the appropriate "Mandelfn" set, zoom on a likely spot where the colors are changing rapidly, and hit the spacebar key to plot the Julia set for that particular point. Try "FRACTINT TYPE=MANDELFN CORNERS=4.68/4.76/-.03/.03 FUNCTION=COS" for a graphic demonstration that we're not taking Mandelbrot's name in vain here. We didn't even know these little buggers were here until Mark Peterson found this a few hours before the version incorporating Mandelfns was released. Note: If you created images using the lambda or mandel "fn" types prior to version 14, and you wish to update the fractal information in the "*.fra" file, simply read the files and save again. You can do this in batch mode via a command line like: "fractint oldfile.fra savename=newfile.gif batch=yes" For example, this procedure can convert a version 13 "type=lambdasine" image to a version 14 "type=lambdafn function=sin" GIF89a image. We do not promise to keep this "backward compatibility" past version 14 - if you want to keep the fractal information in your *.fra files accurate, we recommend conversion. Barnsley Mandelbrot/Julia Sets (barnsleym1/.../j3) Michael Barnsley has written a fascinating college-level text, "Fractals Everywhere," on fractal geometry and its graphic applications. (See the bibliography, Appendix. D.) In it, he applies the principle of the M and J sets to more general functions of two complex variables. We have incorporated three of Barnsley's examples in Fractint. Their appearance suggests polarized-light microphotographs of minerals, with patterns that are less organic and more crystalline than those of the M/J sets. Each example has both a "Mandelbrot" and a "Julia" type. Toggle between them using the spacebar. The parameters have the same meaning as they do for the "regular" Mandelbrot and Julia. For types M1, M2, and M3, they are used to "warp" the image by setting the initial value of Z. For the types J1 through J3, they are the values of C in the generating formulas, found in Appendix A. Be sure to try the <O>rbit function while plotting these types. Barnsley IFS Fractals (type=ifs, ifs3d) One of the most remarkable spin-offs of fractal geometry is the ability to "encode" realistic images in very small sets of numbers -- parameters for a set of functions that map a region of two-dimensional space onto itself. In principle (and increasingly in practice), a scene of any level of complexity and detail can be stored as a handful of numbers, achieving amazing "compression" ratios... how about a super-VGA image of a forest, more than 300,000 pixels at eight bits apiece, from a 1-KB "seed" file? Again, Michael Barnsley and his co-workers at the Georgia Institute of Technology are to be thanked for pushing the development of these iterated function systems (IFS). Invoking Fractint with "TYPE=ifs" defaults to an IFS "fern". Other IFS images can be created via the *.IFS files included with Fractint (FERN, TRIANGLE, TREE, and more), or from user-written files with the default extension *.IFS. Use the command-line argument "IFS=filename" or the main-menu <I> command to use or manipulate one of these files. Specify whether 2D or 3D (the files supplied with Fractint are almost all 2D). You will then be asked to pick which function to edit. Hitting <Enter> leaves values unchanged. Each line in an IFS code file contains the parameters for one of the generating functions, e.g. in FERN.IFS: a b c d e f p ___________________________________ 0 0 0 .16 0 0 .01 .85 .04 -.04 .85 0 1.6 .85 .2 -.26 .23 .22 0 1.6 .07 -.15 .28 .26 .24 0 .44 .07 The "p" values are the probabilities assigned to each function (how often it is used), which add up to one. Fractint supports up to 32 functions, although usually three or four are enough. Note that some Barnsley IFS values generate images quite a bit smaller than the initial (default) screen. Just bring up the zoom box, center it on the small image, and hit the ENTER key to get a full-screen image. In order to fully appreciate the 3D fern, since your monitor is presumably 2D, we have added rotation, translation, and perspective capabilities. These share values with the same variables used in Fractint's other 3D facilities; for their meaning see "3D Images", farther on in this section. You can enter these values from the command line using: rotation=xrot/yrot/zrot (try 30/30/30) shift=xshift/yshift (shifts BEFORE applying perspective!) perspective=viewerposition (try 200) Alternatively, entering <I> from main screen and then selecting "3D Transform Parameters" will allow you to modify these values. The defaults are the same as for regular 3D, and are not always optimum for the 3D fern. Try rotation=30/30/30. Note that applying shift when using perspective changes the picture -- your "point of view" is moved. A truly wild variation of 3D may be seen by entering "2" for the stereo mode, putting on red/blue "funny glasses", and watching the fern develop with full depth perception right there before your eyes! This feature USED to be dedicated to Bruce Goren, as a bribe to get him to send us MORE knockout stereo slides of 3D ferns, now that we have made it so easy! Bruce, what have you done for us *LATELY* ?? (Just kidding, really!) Sierpinski Gasket (type=sierpinski) Another pre-Mandelbrot classic, this one found by W. Sierpinski around World War I. It is generated by dividing a triangle into four congruent smaller triangles, doing the same to each of them, and so on, yea, even unto infinity. (Notice how hard we try to avoid reiterating "iterating"?) If you think of the interior triangles as "holes", they occupy more and more of the total area, while the "solid" portion becomes as hopelessly fragile as that gasket you HAD to remove without damaging it -- you remember, that Sunday afternoon when all the parts stores were closed? There's a three-dimensional equivalent using nested tetrahedrons instead of triangles, but it generates too much pyramid power to be safely unleashed yet. There are no parameters for this type. We were able to implement it with integer math routines, so it runs fairly quickly even without an FPU. Quartic Mandelbrot/Julia (type=mandel4/julia4) These fractal types are the moral equivalent of the original M and J sets, except that they use the formula Z(n+1) = Z(n)^4 + C, which adds additional pseudo-symmetries to the plots. The "Mandel4" set maps to the "Julia4" set via -- surprise! -- the spacebar toggle. The M4 set is kind of boring at first (the area between the "inside" and the "outside" of the set is pretty thin, and it tends to take a few zooms to get to any interesting sections), but it looks nice once you get there. The Julia sets look nice right from the start. Other powers, like Z(n)^3 or Z(n)^7, work in exactly the same fashion. We used this one only because we're lazy, and Z(n)^4 = (Z(n)^2)^2. Distance Estimator (distest=nnn) This used to be type=demm and type=demj. These types have not died, but are only hiding! They are equivalent to the mandel and julia types with the "distest=" option selected with a predetermined value. The "Distance Estimator Method" (described under "Doodads, Bells and Whistles") can be used to produce higher quality images of M and J sets, especially suitable for printing in black and white. If you have some *.fra files made with the old types demm/demj, you may want to convert them to the new form. See "MANDELFN" section for directions to carry out the conversion. Pickover Mandelbrot/Julia Types (type=manfn+zsqrd/julfn+zsqrd, manzpowr/julzpowr, manzzpwr/julzzpwr, manfn+exp/julfn+exp - formerly included man/julsinzsqrd and man/julsinexp which have now been generalized) These types have been explored by Clifford A. Pickover, of the IBM Thomas J. Watson Research center. As implemented in Fractint, they are regular Mandelbrot/Julia set pairs that may be plotted with or without the "biomorph" option Pickover used to create organic-looking beasties (see below). These types are produced with formulas built from the functions z^z, z^n, sin(z), and e^z for complex z. Types with "power" or "pwr" in their name have an exponent value as a third parameter. For example, type=manzpower params=0/0/2 is our old friend the classical Mandelbrot, and type=manzpower params=0/0/4 is the Quartic Mandelbrot. Other values of the exponent give still other fractals. Since these WERE the original "biomorph" types, we should give an example. Try: FRACTINT type=manfn+zsqrd biomorph=0 corners=-8/8/-6/6 function=sin to see a big biomorph digesting little biomorphs! Pickover Popcorn (type=popcorn/popcornjul) Here is another Pickover idea. This one computes and plots the orbits of the dynamic system defined by: x(n+1) = x(n) - h*sin(y(n)+tan(3*y(n)) y(n+1) = y(n) - h*sin(x(n)+tan(3*x(n)) with the initializers x(0) and y(0) equal to ALL the complex values within the "corners" values, and h=.01. ALL these orbits are superimposed, resulting in "popcorn" effect. You may want to use a maxiter value less than normal - Pickover recommends a value of 50. As a bonus, type=popcornjul shows the Julia set generated by these same equations with the usual escape-time coloring. Turn on orbit viewing with the "O" command, and as you watch the orbit pattern you may get some insight as to where the popcorn comes from. Although you can zoom and rotate popcorn, the results may not be what you'd expect, due to the superimposing of orbits and arbitrary use of color. Just for fun we added type popcornjul, which is the plain old Julia set calculated from the same formula. Peterson Variations (xxmarksxx types) (type=marksmandel, marksjulia, cmplxmarksmand, cmplxmarksjul) These fractal types are contributions of Mark Peterson. MarksMandel and MarksJulia are two families of fractal types that are linked in the same manner as the classic Mandelbrot/Julia sets: each MarksMandel set can be considered as a mapping into the MarksJulia sets, and is linked with the spacebar toggle. The basic equation for these sets is: Z(n+1) = ((lambda^n) * Z(n)^2) + lambda where Z(0) = 0.0 and lambda is (x + iy) for MarksMandel. For MarksJulia, Z(0) = (x + iy) and lambda is a constant (taken from the MarksMandel spacebar toggle, if that method is used). The exponent is a positive integer or a complex number. We call these "families" because each value of the exponent yields a different MarksMandel set, which turns out to be a kinda-polygon with (exponent+1) sides. The exponent value is the third parameter, after the "initialization warping" values. Typically one would use null warping values, and specify the exponent with something like "PARAMS=0/0/4", which creates an unwarped, pentagonal MarksMandel set. Unity (type=unity) This Peterson variation began with curiosity about other "Newton-style" approximation processes. A simple one, One = (x * x) + (y * y); y = (2 - One) * x; x = (2 - One) * y; produces the fractal called Unity. One of its interesting features is the "ghost lines." The iteration loop bails out when it reaches the number 1 to within the resolution of a screen pixel. When you zoom a section of the image, the bailout criterion is adjusted, causing some lines to become thinner and others thicker. Only one line in Unity that forms a perfect circle: the one at a radius of 1 from the origin. This line is actually infinitely thin. Zooming on it reveals only a thinner line, up (down?) to the limit of accuracy for the algorithm. The same thing happens with other lines in the fractal, such as those around |x| = |y| = (1/2)^(1/2) = .7071 Try some other tortuous approximations using the TEST stub (see below) and let us know what you come up with! Scott Taylor / Lee Skinner Variations (type=fn(z*z), fn*fn, fn*z+z, fn+fn, sqr(1/fn), sqr(fn), spider, tetrate, manowar) Two of Fractint's faithful users went bonkers when we introduced the "formula" type, and came up with all kinds of variations on escape-time fractals using trig functions. We decided to put them in as regular types, but there were just too many! So we defined the types with variable functions and let you, the, overwhelmed user, specify what the functions should be! Thus Scott Taylor's "z = sin(z) + z^2" formula type is now the "fn+fn" regular type, and EITHER function can be one of sin, cos, sinh, cosh, exp, log, or sqr. Plus we give you 4 parameters to set, the complex coefficients of the two functions! Thus the innocent- looking "fn+fn" type is really at least 36 different types in disguise, not counting the damage done by the parameters! Lee informs us that you should not judge fractals by their "outer" appearance. For example, the images produced by z = sin(z) + z^2 and z = sin(z) - z^2 look very similar, but are different when you zoom in. Kam Torus (type=kamtorus, kamtorus3d) This type is created by superimposing orbits generated by a set of equations, with a variable incremented each time. x(0) = y(0) = orbit/3; x(n+1) = x(n)*cos(a) + (x(n)*x(n)-y(n))*sin(a) y(n+1) = x(n)*sin(a) - (x(n)*x(n)-y(n))*cos(a) After each orbit, 'orbit' is incremented by a step size. The parameters are angle "a", step size for incrementing 'orbit', stop value for 'orbit', and points per orbit. Try this with a stop value of 5 with sound=x for some weird fractal music (ok, ok, fractal noise)! You will also see the KAM Torus head into some chaotic territory that Scott Taylor wanted to hide from you by setting the defaults the way he did, but now we have revealed all! The 3D variant is created by treating 'orbit' as the z coordinate. Bifurcation (type=bifxxx) The wonder of fractal geometry is that such complex forms can arise from such simple generating processes. A parallel surprise has emerged in the study of dynamical systems: that simple, deterministic equations can yield chaotic behavior, in which the system never settles down to a steady state or even a periodic loop. Often such systems behave normally up to a certain level of some controlling parameter, then go through a transition in which there are two possible solutions, then four, and finally a chaotic array of possibilities. This emerged many years ago in biological models of population growth. Consider a (highly over-simplified) model in which the rate of growth is partly a function of the size of the current population: New Population = Growth Rate * Old Population * (1 - Old Population) where population is normalized to be between 0 and 1. At growth rates less than 200 percent, this model is stable: for any starting value, after several generations the population settles down to a stable level. But for rates over 200 percent, the equation's curve splits or "bifurcates" into two discrete solutions, then four, and soon becomes chaotic. Type=bifurcation illustrates this model. (Although it's now considered a poor one for real populations, it helped get people thinking about chaotic systems.) The horizontal axis represents growth rates, from 190 percent (far left) to 400 percent; the vertical axis normalized population values, from 0 to 4/3. Notice that within the chaotic region, there are narrow bands where there is a small, odd number of stable values. It turns out that the geometry of this branching is fractal; zoom in where changing pixel colors look suspicious, and see for yourself. The default display has been filtered by allowing the population to settle from its initial value for 5000 cycles before plotting maxiter population values. To override this filter value, specify a new (smaller) one as the first "PARAMS=" value. "PARAMS=1" produces an unfiltered map. Many formulae can be used to produce bifurcations. Mitchel Feigenbaum studied lots of bifurcations in the mid-70's, using a HP-65 calculator (IBM PCs, Fractals, and Fractint, were all Sci-Fi then !). He studied where bifurcations occurred, for the formula r*p*(1-p), the one described above. He found that the ratios of lengths of adjacent areas of bifurcation were four and a bit. These ratios vary, but, as the growth rate increases, they tend to a limit of 4.669+. This helped him guess where bifurcation points would be, and saved lots of time. When he studied bifurcations of r*sin(PI*p) he found a similar pattern, which is not surprising in itself. However, 4.669+ popped out, again. Different formulae, same number ? Now, THAT's surprising ! He tried many other formulae and ALWAYS got 4.669+ - Hot Damn !!! So hot, in fact, that he phoned home and told his Mom it would make him Famous ! He also went on to tell other scientists. The rest is History... (It has been conjectured that if Feigenbaum had a copy of Fractint, and used it to study bifurcations, he may never have found his Number, as it only became obvious from long perusal of hand-written lists of values, without the distraction of wild color-cycling effects !). We now know that this number is as universal as PI or E. It appears in situations ranging from fluid-flow turbulence, electronic oscillators, chemical reactions, and even the Mandelbrot Set - yup, fraid so: "budding" of the Mandelbrot Set along the negative real axis occurs at intervals determined by Feigenbaum's Number, 4.669201660910..... Fractint does not make direct use of the Feigenbaum Number (YET !). However, it does now reflect the fact that there is a whole sub-species of Bifurcation-type fractals. Those implemented to date, and the related formulae, (writing P for pop[n+1] and p for pop[n]) are : bifurcation P = p + r*p*(1-p) Verhulst Bifurcations. biflambda P = r*p*(1-p) Real equivalent of Lambda Sets. bif+sinpi P = p + r*sin(PI*p) Population scenario based on... bif=sinpi P = r*sin(PI*p) ...Feigenbaum's second formula. It took a while for bifurcations to appear here, despite them being over a century old, and intimately related to chaotic systems. However, they are now truly alive and well in Fractint! Lorenz Attractors (type=lorenz/lorenz3d) The "Lorenz Attractor" is a "simple" set of three deterministic equations developed by Edward Lorenz while studying the non- repeatability of weather patterns. The weather forecaster's basic problem is that even very tiny changes in initial patterns ("the beating of a butterfly's wings" - the official term is "sensitive dependence on initial conditions") eventually reduces the best weather forecast to rubble. The lorenz attractor is the plot of the orbit of a dynamic system consisting of three first order non-linear differential equations. The solution to the differential equation is vector-valued function of one variable. If you think of the variable as time, the solution traces an orbit. The orbit is made up of two spirals at an angle to each other in three dimensions. We change the orbit color as time goes on to add a little dazzle to the image. The equations are: dx/dt = -a*x + a*y dy/dt = b*x - y -z*x dz/dt = -c*z + x*y We solve these differential equations approximately using a method known as the first order taylor series. Calculus teachers everywhere will kill us for saying this, but you treat the notation for the derivative dx/dt as though it really is a fraction, with "dx" the small change in x that happens when the time changes "dt". So multiply through the above equations by dt, and you will have the change in the orbit for a small time step. We add these changes to the old vector to get the new vector after one step. This gives us: xnew = x + (-a*x*dt) + (a*y*dt) ynew = y + (b*x*dt) - (y*dt) - (z*x*dt) znew = z + (-c*z*dt) + (x*y*dt) (default values: dt = .02, a = 5, b = 15, c = 1) We connect the successive points with a line, project the resulting 3D orbit onto the screen, and voila! The Lorenz Attractor! We have added two versions of the Lorenz Attractor. "Type=lorenz" is the Lorenz attractor as seen in everyday 2D. "Type=lorenz3d" is the same set of equations with the added twist that the results are run through our perspective 3D routines, so that you get to view it from different angles (you can modify your perspective "on the fly" by selecting the "3D Transform Parameters" option of the <I> command.) If you set the "stereo" option to "2", and have red/blue funny glasses on, you will see the attractor orbit with depth perception. Hint: the default perspective values (x = 60, y = 30, z = 0) aren't the best ones to use for fun Lorenz Attractor viewing. Experiment a bit - start with rotation values of 0/0/0 and then change to 20/0/0 and 40/0/0 to see the attractor from different angles.- and while you're at it, use a non-zero perspective point Try 100 and see what happens when you get *inside* the Lorenz orbits. Here comes one - Duck! While you are at it, turn on the sound with the "X". This way you'll at least hear it coming! Different Lorenz attractors can be created using different parameters. Four parameters are used. The first is the time-step (dt). The default value is .02. A smaller value makes the plotting go slower; a larger value is faster but rougher. A line is drawn to connect successive orbit values. The 2nd, third, and fourth parameters are coefficients used in the differential equation (a, b, and c). The default values are 5, 15, and 1. Try changing these a little at a time to see the result. Rossler Attractors (type=rossler3D) This fractal is named after the German Otto Rossler, a non-practicing medical doctor who approached chaos with a bemusedly philosophical attitude. He would see strange attractors as philosophical objects. His fractal namesake looks like a band of ribbon with a fold in it. All we can say is we used the same calculus-teacher-defeating trick of multiplying the equations by "dt" to solve the differential equation and generate the orbit. This time we will skip straight to the orbit generator - if you followed what we did above with Lorenz you can easily reverse engineer the differential equations. xnew = x - y*dt - z*dt ynew = y + x*dt + a*y*dt znew = z + b*dt + x*z*dt - c*z*dt Default parameters are dt = .04, a = .2, b = .2, c = 5.7 Henon Attractors (type=henon) Michel Henon was an astronomer at Nice observatory in southern France. He came to the subject of fractals via investigations of the orbits of astronomical objects. The strange attractor most often linked with Henon's name comes not from a differential equation, but from the world of discrete mathematics - difference equations. The Henon map is an example of a very simple dynamic system that exhibits strange behavior. The orbit traces out a characteristic banana shape, but on close inspection, the shape is made up of thicker and thinner parts. Upon magnification, the thicker bands resolve to still other thick and thin components. And so it goes forever! The equations that generate this strange pattern perform the mathematical equivalent of repeated stretching and folding, over and over again. xnew = 1 + y - a*x*x ynew = b*x The default parameters are a=1.4 and b=.3. Pickover Attractors (type=pickover) Clifford A. Pickover of the IBM Thomas J. Watson Research center is such a creative source for fractals that we attach his name to this one only with great trepidation. Probably tomorrow he'll come up with another one and we'll be back to square one trying to figure out a name! This one is the three dimensional orbit defined by: xnew = sin(a*y) - z*cos(b*x) ynew = z*sin(c*x) - cos(d*y) znew = sin(x) Default parameters are: a = 2.24, b = .43, c = -.65, d = -2.43 Gingerbreadman (type=gingerbreadman) This simple fractal is a charming example stolen from "Science of Fractal Images", p. 149. Look closely, and you will see that the gingerbreadman contains infinitely many copies of himself at all different scales. xnew = 1 - y + |x| ynew = x There are no parameters. Test (type=test) This is a stub that we (and you!) use for trying out new fractal types. "Type=test" fractals make use of Fractint's structure and features for whatever code is in the routine 'testpt()' (located in the small source file TESTPT.C) to determine the color of a particular pixel. If you have a favorite fractal type that you believe would fit nicely into Fractint, just rewrite the C function in TESTPT.C (or use the prototype function there, which is a simple M-set implementation) with an algorithm that computes a color based on a point in the complex plane. After you get it working, send your code to one of the authors and we might just add it to the next release of Fractint, with full credit to you. Our criteria are: 1) an interesting image and 2) a formula significantly different from types already supported. (Bribery may also work. THIS author is completely honest, but I don't trust those other guys.) Be sure to include an explanation of your algorithm and the parameters supported, preferably formatted as you see here to simplify folding it into the documentation. Formula (type=formula) Fractint now has a new 'type=formula' fractal type. This is a "roll- your-own" fractal interpreter - you don't even need a compiler! To run a "type=formula" fractal, you first need a text file containing formulas (there's a sample file - FRACTINT.FRM - included with this distribution). When you select the "formula" fractal type, you are prompted first for the name of your file. Then, after the program locates the file and scans it for formulas, you are prompted for the formula name you wish to run. After prompting for any parameters, the formula is parsed for syntax errors and then the fractal is generated. There are also two command-line options that work with type=formula ("formulafile=" and "formulaname=") if you are using this new fractal type in batch mode. The following documentation is supplied by Mark Peterson, who wrote the formula interpreter: Formula fractals allow you to create your own fractal formulas. The general format is: Mandelbrot(XAXIS) { z = Pixel: z = sqr(z) + pixel, |z| <= 4 } | | | | | Name Symmetry Initial Iteration Bailout Condition Criteria Initial conditions are set, then the iterations performed until the bailout criteria is true or 'z' turns into a periodic loop. All variables are created automatically by their usage and treated as complex. If you declare 'v = 2' then the variable 'v' is treated as a complex with an imaginary value of zero. Predefined Variables (x, y) -------------------------------------------- z used for periodicity checking p1 parameters 1 and 2 p2 parameters 3 and 4 pixel screen coordinates Precedence -------------------------------------------- 1 sin(), cos(), sinh(), cosh(), sqr(), log(), exp(), abs(), conj(), real(), imag() 2 - (negation), ^ (power) 3 * (multiplication), / (division) 4 + (addition), - (subtraction) 5 = (assignment) 6 < (less than), <= (less than or equal to) Precedence may be overridden by use of parenthesis. Note the modulus squared operator |z| is also parenthetic and always sets the imaginary component to zero. This means 'c * |z - 4|' first subtracts 4 from z, calculates the modulus squared then multiplies times 'c'. Nested modulus squared operators require overriding parenthesis: c * |z + (|pixel|)| The formulas are performed using either integer or floating point mathematics depending on the type of math you have setup. If you do not have an FPU then type MPC math is performed in lieu of traditional floating point. Remember that when using integer math there is a limited dynamic range, so what you think may be a fractal could really be just a limitation of the integer math range. God may work with integers, but His dynamic range is many orders of magnitude greater than our puny 32 bit mathematics! Always verify with the floating point. Julibrots (type=julibrot) The following documentation is supplied by Mark Peterson, who "invented" the Julibrot algorithm. There is a very close relationship between the Mandelbrot set and Julia sets of the same equation. To draw a Julia set you take the basic equation and vary the initial value according to the two dimensions of screen leaving the constant untouched. This method diagrams two dimensions of the equation, 'x' and 'iy', which I refer to as the Julia x and y. z(0) = screen coordinate (x + iy) z(1) = (z(0) * z(0)) + c, where c = (a + ib) z(2) = (z(1) * z(0)) + c z(3) = . . . . The Mandelbrot set is a composite of all the Julia sets. If you take the center pixel of each Julia set and plot it on the screen coordinate corresponding to the value of c, a + ib, then you have the Mandelbrot set. z(0) = 0 z(1) = (z(0) * z(0)) + c, where c = screen coordinate (a + ib) z(2) = (z(1) * z(1)) + c z(3) = . . . . I refer to the 'a' and 'ib' components of 'c' as the Mandelbrot 'x' and 'y'. All the 2 dimensional Julia sets correspond to a single point on the 2 dimensional Mandelbrot set, making a total of 4 dimensions associated with our equation. Visualizing 4 dimensional objects is not as difficult as it may sound at first if you consider we live in a 4 dimensional world. The room around you is three dimensions and as you read this text you are moving through the fourth dimension of time. You and everything around your are 4 dimensional objects - which is to say 3 dimensional objects moving through time. We can think of the 4 dimensions of our equation in the same manner, this is as a 3 dimensional object evolving over time - sort of a 3 dimensional fractal movie. The fun part of it is you get to pick the dimension representing time! To construct the 4 dimensional object into something you can view on the computer screen you start with the simple 2 dimensions of the Julia set. I'll treat the two Julia dimensions as the spatial dimensions of height and width, and the Mandelbrot 'y' dimension as the third spatial dimension of depth. This leaves the Mandelbrot 'x' dimension as time. Draw the Julia set associated with the Mandelbrot coordinate (-.83, - .25), but instead of setting the color according to the iteration level it bailed out on, make it a two color drawing where the pixels are black for iteration levels less than 30, and another color for iteration levels greater than or equal to 30. Now increment the Mandelbrot 'y' coordinate by a little bit, say (-.83, -.2485), and draw another Julia set in the same manner using a different color for bailout values of 30 or greater. Continue doing this until you reach (-.83, .25). You now have a three dimensional representation of the equation at time -.83. If you make the same drawings for points in time before and after -.83 you can construct a 3 dimensional movie of the equation which essentially is a full 4 dimensional representation. In the Julibrot fractal available with this release of Fractint the spatial dimensions of height and width are always the Julia dimensions. The dimension of depth is determined by the Mandelbrot coordinates. The program will consider the dimension of depth as the line between the two Mandelbrot points. To draw the image in our previous example you would set the 'From Mandelbrot' to (-.83, .25) and the 'To Mandelbrot' as (- .83, -.25). If you set the number of 'z' pixels to 128 then the program will draw the 128 Julia sets found between Mandelbrot points (-.83, .25) and (-.83, -.25). To speed things up the program doesn't actually calculate ALL the coordinates of the Julia sets. It starts with the a pixel a the Julia set closest to the observer and moves into the screen until it either reaches the required bailout or the limit to the range of depth. Zooming can be done in the same manner as with other fractals. The visual effect (with other values unchanged) is similar to putting the boxed section under a pair of magnifying glasses. The variable associated with penetration level is the level of bailout there you decide to make the fractal solid. In other words all bailout levels less than the penetration level are considered to be transparent, and those equal or greater to be opaque. The farther away the apparent pixel is the dimmer the color. The remainder of the parameters are needed to construct the red/blue picture so that the fractal appears with the desired depth and proper 'z' location. With the origin set to 8 inches beyond the screen plane and the depth of the fractal at 8 inches the default fractal will appear to start at 4 inches beyond the screen and extend to 12 inches if your eyeballs are 2.5 inches apart and located at a distance of 24 inches from the screen. The screen dimensions provide the reference frame. To the human eye blue appears brighter than red. The Blue:Red ratio is used to compensate for this fact. If the image appears reddish through the glasses raise this value until the image appears to be in shades of gray. If it appears bluish lower the ratio. Julibrots can only be shown in 256 red/blue colors for viewing in either stereo-graphic (red/blue funny glasses) or gray-scaled. Fractint automatically loads either GLASSES1.MAP or ALTERN.MAP as appropriate. Diffusion Limited Aggregation (type=diffusion) This type begins with a single point in the center of the screen. Subsequent points move around randomly until coming into contact with the first point, at which time their locations are fixed and they are colored randomly. This process repeats until the fractals reaches the edge of the screen. Use the show orbits function to see the points' random motion. One unfortunate problem is that on a large screen, this process will tend to take eons. To speed things up, the points are restricted to a box around the initial point. The first and only parameter to diffusion contains the size of the border between the fractal and the edge of the box. If you make this number small, the fractal will look more solid and will be generated more quickly. Diffusion was inspired by a Scientific American article a couple of years back which includes actual pictures of real physical phenomena that behave like this. Thanks to Adrian Mariano for providing the diffusion code and documentation. Magnetic Fractals (type=magnet1m/.../magnet2j) These fractals use formulae derived from the study of hierarchical lattices, in the context of magnetic renormalisation transformations. This kinda stuff is useful in an area of theoretical physics that deals with magnetic phase-transitions (predicting at which temperatures a given substance will be magnetic, or non-magnetic). In an attempt to clarify the results obtained for Real temperatures (the kind that you and I can feel), the study moved into the realm of Complex Numbers, aiming to spot Real phase-transitions by finding the intersections of lines representing Complex phase-transitions with the Real Axis. The first people to try this were two physicists called Yang and Lee, who found the situation a bit more complex than first expected, as the phase boundaries for Complex temperatures are (surprise!) fractals. And that's all the technical (?) background you're getting here! For more details (are you SERIOUS ?!) read "The Beauty of Fractals". When you understand it all, you might like to re-write this section, before you start your new job as a professor of theoretical physics... In Fractint terms, the important bits of the above are "Fractals", "Complex Numbers", "Formulae", and "The Beauty of Fractals". Lifting the Formulae straight out of the Book and iterating them over the Complex plane (just like the Mandelbrot set) produces Fractals. The formulae are a bit more complicated than the Z^2+C used for the Mandelbrot Set, that's all. They are : / Z^2 + (C-1) \ MAGNET1 : | ------------- | ^ 2 \ 2*Z + (C-2) / / Z^3 + 3*(C-1)*Z + (C-1)*(C-2) \ MAGNET2 : | --------------------------------------- | ^ 2 \ 3*(Z^2) + 3*(C-2)*Z + (C-1)*(C-2) - 1 / These aren't quite as horrific as they look (oh yeah ?!) as they only involve two variables (Z and C), but cubing things, doing division, and eventually squaring the result (all in Complex Numbers) don't exactly spell S-p-e-e-d ! These are NOT the fastest fractals in Fractint ! As you might expect, for both formulae there is a single related Mandelbrot-type set (magnet1m, magnet2m) and an infinite number of related Julia-type sets (magnet1j, magnet2j), with the usual toggle between the corresponding Ms and Js via the spacebar. If you fancy delving into the Julia-types by hand, you will be prompted for the Real and Imaginary parts of the parameter denoted by C. The result is symmetrical about the Real axis (and therefore the initial image gets drawn in half the usual time) if you specify a value of Zero for the Imaginary part of C. Fractint Historical Note: Another complication (besides the formulae) in implementing these fractal types was that they all have a finite attractor (1.0 + 0.0i), as well as the usual one (Infinity). This fact spurred the development of Finite Attractor logic in Fractint. Without this code you can still generate these fractals, but you usually end up with a pretty boring image that is mostly deep blue "lake", courtesy of Fractint's standard Periodicity Checking. See the Finite Attractors section for more information on this aspect of Fractint internals. (Thanks to Kevin Allen for Magnetic type documentation above). L-Systems (type=lsystem) These fractals are constructed from line segments using rules specified in drawing commands. Starting with an initial string, the axiom, transformation rules are applied a specified number of times, to produce the final command string which is used to draw the image. Like the type=formula fractals, this type requires a separate data file. A sample file, FRACTINT.L, is included with this distribution. When you select type lsystem, you are asked for the filename. Next you are asked for the lsystem name you wish to run. Last, you are asked for one parameter - the "order", or number of times to execute all the transformation rules. It is wise to start with small orders, because the size of the substituted command string grows exponentially and it is very easy to exceed your resolution. (Higher orders take longer to generate too.) The command line options "lname=" and "lfile=" can be used to over-ride the default file name and lsystem name. Each L-System entry in the file contains a specification of the angle, the axiom, and the transformation rules. Each item must appear on its own line and each line must be less than 160 characters long. The statement "angle n" sets the angle to 360/n degrees; n must be an integer greater than two and less than fifty. "Axiom string" defines the axiom. Transformation rules are specified as "a=string" and convert the single character 'a' into "string." If more than one rule is specified for a single character all of the strings will be added together. This allows specifying transformations longer than the 160 character limit. Transformation rules may operate on any characters except space, tab or '}'. A line which starts with ;; is a comment line. Here is a sample lsystem: | This comment section would NOT be legal. | Comments inside an L-System start with ;; at | the start of the line. Dragon { | Name of lsystem, { indicates start ;; Comment! | This is a comment Angle 8 | Specify the angle increment to 45 degrees Axiom FX | Starting character string F= | First rule: Delete 'F' y=+FX--FY+ | Change 'y' into "+fx--fy+" x=-FX++FY- | Similar transformation on 'x' } | final } indicates end The standard drawing commands are: F Draw forward G Move forward (without drawing) + Increase angle - Decrease angle | Try to turn 180 degrees. (If angle is odd, the turn will be the largest possible turn less than 180 degrees.) These commands increment angle by the user specified angle value. They should be used when possible because they are fast. If greater flexibility is needed, use the following commands which keep a completely separate angle pointer which is specified in degrees. D Draw forward M Move forward \nn Increase angle nn degrees /nn Decrease angle nn degrees Color control: Cnn Select color nn <nn Increment color by nn >nn decrement color by nn Advanced commands: ! Reverse directions (Switch meanings of +, - and \, /) @nnn Multiply line segment size by nnn nnn may be a plain number, or may be preceded by I for inverse, or Q for square root. (e.g. @IQ2 divides size by the square root of 2) [ Push. Stores current angle and position on a stack ] Pop. Return to location of last push Other characters are perfectly legal in command strings. They are ignored for drawing purposes, but can be used to achieve complex translations.