Discontinuous and alternate q-system fractals

This paper presents various patterns and computer art from three extensions to common fractal methods: use of discontinuous functions (such as boolean functions, and the conditional function), plots of root-finding methods other than Halley’s and Newton’s methods (such as Aitken’s and Muller’s methods), and fractals produced using arithmetic in q-systems other than the complex number system. Still fractal images and parametrization "movies" are discussed; these allow several new properties of these algorithms (and their combinations) to be uncovered. Also, the interaction of randomness with fractals is explored. The advent of high-speed computers and graphical display devices has made possible the recent plethora of images produced by chaotic algorithms (see [1-3], and references therein). Two common methods of producing fractal images are Julia sets [4,5] and root-finding method plots (e.g., [6]). These fractals are usually produced by iterating a function employing a small set of mathematical operations (arithmetic, and sometimes trigonometric functions), which are defined within the complex number system. In this paper, I demonstrate two extensions of these methods, explore the interaction of Julia sets with randomness (noise), and show fractals associated with several types of root-finding methods. The Basic Fractals 1. Algorithms: Julia sets are produced by viewing points in the R2 plane as complex numbers and plotting the behavior of a function that is recurrently applied to each number in turn. They have applications in the study of phase transformations [7,8], dynamical systems [9,10], and developmental biology [11]. The basic Julia set algorithm used in this paper will be that defined by [12]. This is an escape-time fractal (where the escape value is given by the constant ε and is typically on the order of 100.0), and the pseudo-code (for a display using 8 colors and a black background) is as follows: for each point (x,y) in some range minxε, break loop. if |real(z)|<ε or |imaginary(z)|<ε, color = g mod 8, otherwise color = black Root-method plots (e.g., [6,12,13]) are produced by considering how fast a particular iterative root-finding method (such as Newton’s method) converges on a root for a given function, when it is started with an initial guess x+yi. The rate of convergence determines the color of each point (x,y) and results in a fractal when a set of points in the R2 plane are taken in turn as the initial guesses. The basic algorithm is as follows: for each point (x,y) in some range minx2.02, y: 0.55 -> 1.204), the movie contains no deformations. Instead, it looks as if one was moving at varying speeds with a fixed-size viewport over an unchanging (static) image. In contrast to this linear translation effect, all other movies show clear deformations in place (structures expand and contract, there is bending and shearing, etc.). By experiment, it is seen that this is a property of the forms z=cos(z+c)-c and z=sin(z+c)-c for small n, but not of z=cosh(z+c)-c or z=sinh(z+c)-c, nor of any other formula found so far. Extension 1: other root-finding methods Other root-finding methods can be found in Grove, 1966. These can be extended to the complex numbers. Of these, Newton’s method is the least computationally-demanding; Halley’s method is intermediately so. Muller’s method and Stephenson’s methods are the most demanding. Figures 15-17 show some other sample root method plots. Each method has certain individual characteristics. For example, Newton and Halley plots are usually quite similar. Stephenson plots contain much empty space and separated islands of color, and occasionally the "bullseye" patterns found in Newton and Halley plots. Muller plots contain many small whirl-like regions. Aitken plots often contain very fuzzy areas (these are regions where points of different color intermix, without any large solidly-colored regions). Movies made with these methods are visually very impressive (especially Muller plots). Extension 2: discontinuous functions Most fractals are drawn using the common functions (rows 8-9 of Table 1). However, other functions (such as trigonometric, and hyperbolic trigonometric) can be defined over the complex number data type (rows 6-7 of Table 1). In addition, it is possible to define discontinuous functions such as the boolean (logic) functions, and some others (rows 1-5 of Table 1) over the complex numbers. In these cases, True and False is defined as being precisely equal to 1.0 and 0.0 respectively. These can then be used to produce fractal images (as Julia sets, or root-method plots) of a much wider range of complex number functions. One last and very powerful addition to this function set involves the "cond" function. This is defined as the standard LISP (cond a b c) function form, which returns "b" or "c" depending on whether the parameter "a" is True or False. This function allows one to section the plane. For example (in LISP notation), "z=(cond (r< z0 3) (sin z) (pow z 3.0))", applies sin(z) to the x<3.0 part of the complex plane, and z to the rest of the plane. Note that the "cond" function violates assumptions of analyticity used in the theory of Julia sets. This makes analysis of functions using it more difficult; however, it allows one to simulate systems which include a choice component at each iteration (e.g., [11]). When using functions which perform a choice (such as "cond", "min", "max", etc.), it is useful to have them keep track of and display the ratio in which the two choices were given, during a particular image. This is because sometimes such a function always returns the same choice, and can thus be simplified, reducing compute time (for example, the function "(min z 0)" will always return "0" if z stays positive). Figures 18-20 show some examples of Julia sets and root-method plots produced with discontinuous functions. Extension 3: i-systems other than i2 = -1 The last extension to the common fractal methods involves alternate q-systems [20]. Number systems involving numbers of the form a+bi can be defined by addition and multiplication rules which operate on these numbers and satisfy certain constraints (for example, that addition and multiplication must be commutative and associative, and that multiplication must be distributive over addition). These rules, however, do not constrain the value of i2 to any particular number. In the ordinary complex number system, i2 is made to equal -1. However, it is possible to define consistent number systems with i2 being equal to arbitrary p+qi. This defines the character of the system and determines the properties of operations within it. It turns out that there is not an infinite number of such systems, but that each system can be reduced to one of the following cases: • numbers a+bi with i2 = -1 (the complex numbers) • numbers a+bi with i2 = 1 (the "double" numbers) • numbers a+bi with i2 = 0 (the "dual" numbers) Nevertheless, it is possible to define all common complex number functions (such as arithmetic, trigonometric functions, powers, etc.) over any arbitrary q-number system. The definitions are performed as follows: For a number system with arbitrary A and B such that q2 = A+q(2B) (the factor 2 is there just to simplify the results), the numbers have two individual terms to which expansion formulas can be applied (for example, sin(a+bi) can be simplified to functions of a and bi using the regular sine expansion formula). Defining addition and subtraction is easy since one only has to add and subtract like terms, and neither A nor B is involved since q is never squared. Thus, (a+qb) + (c+qd) = (a+c) + q(b+d) (a+qb) (c+qd) = (a-c) + q(b-d) Multiplication isn’t much harder since the distributive law can be used to give: (a+qb) · (c+qd) = a·c + b·c·q + d·a·q + b·d·q2 collect like terms: a·c + (b·c + d·a)·q + b·d·q2 use the definition of q2: a·c + (b·c + d·a)·q + b·d·(A+2Bq) collect like terms: (a·c+b·d·A) + (b·c+d·a+b·d·2B)·q Division becomes: (x+qy) ÷ (u+qv) = (ux+2Bvx-Avy)/(u2+2Buv-Av2) + (uy-vx)/(u2+2Buv-Av2)q Sines, cosines, and hyperbolic sines and cosines are slightly more tricky. All qsystems can be classified on the basis of the A and B values used in the definition of q2: systems with B2 + A < 0 are isomorphic with the i-system where i2 = -1 systems with B2 + A = 0 are isomorphic with the l-system where l2 = 0 systems with B2 + A > 0 are isomorphic with the h-system where h2 = 1 The i, l, and h-systems are the classical systems. To find the expression for f(a+bi) in an arbitrary q-system, one must convert that problem to a problem involving one of the classical systems. This is because all the common functions have easy definitions in the classical systems. For each q, there is a classical q^ which behaves just like q. Thus, one must find an expression for q in terms of the classical q^, with which it is replaced: q = B + q^·D where D = {sqrt(-(B2+a)) or 1 or sqrt(B2+A)} for the i, l, and h systems respectively. the transformation is later reversed (express the q^ in terms of q) by: q^ = -B/D + q·1/D For example, to find sin(x+qy): a) replace q with the classical q^ getting: sin(x+(B+q^·D)·y) b) collect like terms inside the sine function: sin(x+B·y + q^·D·y) c) use the sine addition formula, getting: sin(x+B·y)·cos(q^·D·y) + cos(x+B·y)·sin(q^·D·y) now, using the power series for sines and cosines, we find sin() and cos() for imaginary values in classical systems: "! ( #^·$ ·% ) = # ^· "!'& ($ ·% )