Fractal Art: Fractal Image and Music Generator

Transcription

1 Proceedings o the 7th WSEAS Int. Con. on Signal Processing, Computational Geometry & Artiicial Vision, Athens, Greece, August 4-6, Fractal Art: Fractal Image and Music Generator RAZVAN TANASIE 1 IEEE Member, MIHAI POPESCU, DANA BOGHEANU, GABRIELA CIOCOIU, DORIAN COJOCARU 3 IEEE Member Sotware Engineering Department 1,, Mechatronics Department 3 Faculty o Automation, Computers and Electronics, University o Craiova Bvd. Decebal, Nr. 107 ROMANIA tanasie_razvan@sotware.ucv.ro 1 Abstract: - This paper presents a sotware that creates a ractal screensaver. The application generates both ractal images and music based on the Julia and Mandelbrot ractals. The our coloring algorithms implemented are ounded on: escape-time, distance estimators, escape angle and curvature estimation algorithm. The music is created or dierent instruments on 3 voices. The horizontal scan is an 1/3 notes at a tempo o quarter note with the value 60. The iteration data rom the Julia calculations is mapped to key velocity. The higher the iteration result, the louder the pitch presence. The application is conigurable and permits the user to select the color palette, a region in the ractal and even the instruments to be played. Key-Words: - ractals, Julia, Mandelbrot, ractal music, ractal images, ractal screensaver 1 Introduction In general, a ractal is reerred to as "a rough or ragmented geometric shape that can be subdivided in parts, each o which is (at least approximately) a reduced-size copy o the whole". The term was created by Benoît Mandelbrot in 1975 and is derived rom the Latin ractus meaning "broken" or "ractured" [1,,3]. A ractal as a geometric object generally has the ollowing eatures: It is simple and recursive in its deinition. It has a ine structure at arbitrarily small scales. It is sel-similar (at least approximately or stochastically). It is too irregular to be easily described in traditional Euclidean geometric language. It has a Hausdor dimension that is greater than its topological dimension. Fractals can be considered to be ininitely complex because o the similitude at all their levels. Not all sel similar objects are ractals, because they don t meet all ractals characteristics. Some objects that approximate ractals are: lightings, clouds, mountains. Fundamentals.1 Fractal Types Coloring algorithms do not cover some types o ractals because they are not common and are not used in most o the ractal sotware packages. Basically, the ractal types can be classsiied in six main groups, but the sotware is based on the ollowing ractals: Julia and Mandelbrot [1,4]. These are the most widely used and known ractals and are generated by the iteration o complex polynomials. Due to their ame, they were experimented with dierent coloring algorithms by the ractal artists. Many ractals sets can be considered subgroups rom these types, or example ractal terrains are a three dimensional representation o a plasma ractal.. Fractal Mathematics..1 The Julia Sets The Julia set is composed o all starting points z which have unusual orbits, that is, those points whose long-time behavior under repeated iterations can change drastically under arbitrarily small perturbations.

2 Proceedings o the 7th WSEAS Int. Con. on Signal Processing, Computational Geometry & Artiicial Vision, Athens, Greece, August 4-6, Deinition Orbit o z: Let be a unction. Then the list o successive iterations o a point or number is called the orbit o that point. For example consider = z : I starting with z=1, then the orbit is (1, 1, 1,...), so the orbit converges to the number 1, so 1 is an attracting point. Starting with z=0.5, then the orbit is (0.5, 0.5, 0.065, ,...), which converges to 0, so 0 is an attracting point, too. Stating with z=, the orbit is (, 4, 16, 56, 65536,...), which converges (diverges) to ininity, so ininity is an attracting point, too. In this case it can be seen that the orbits o all initialization values lead to dierent attracting point [8]. The explanations are limited mainly to quadratic systems, i.e. systems where (z) is a polynomial o the second degree: ( z) = a z + b z c + Some things will be deined as: (1) Deinition Critical Point o unction : Let be a unction. Then every point z with '(z)=0 (irst derivation o at point z is null) is called Critical point o. Deinition ixed point o : Let be any unction. Then every point z with z=(z) ( lets z invariable) is called ixed point. Additionally, i ininity is invariable, i.e. ( ) =, then ininity is a ixed point, too. The unction ( z) = a z + b z + c can be simpliied by applying a transormation which gives special values to some points. Two orms are widely used: ( z) z z c = + () and ( z) m z + ( m) z z = 1 (3) In the irst case the critical point o is at 0, and in some situations this can be very useul or theoretical and practical purposes. The ixed points o are: c and c. In the second case the ixed points o are at 0 and 1. Most o the time ractal programs use standard ormula (). The attracting ixed points o a Julia set determine its property. Consider p is a ixed point o, i.e. (p)=p Zooming in at p very deep and taking z 0 close to p, yields z 1 =(z 0 ), similarly with: z 1 -p=m*(z 0 -p), (4) with m='(p) ' Writing it in another way will result: ( p) = lim z p ( z) ( p) z p, (5) the irst derivation o at point p. Choosing z 0 close to p, then m='(p) is approximated: ( z ) ( p) ( z ) ( p) 0 1 m = = (6) z p z p 0 The so called eigen value m o a ixed point p o is m='(p). As seen earlier, i m is less than 1, then z 1 is closer to p than z 0. Thus an attracting ixed point is obtained. I m is greater than 1, then z 1 is more ar away than z 0, so a repulsing ixed point is computed. I m =1, then the ixed point is neutral or indierent. Then numerous interesting eects can occur (at least theoretically interesting) [6]... The Mandelbrot Set Consider J() any Julia set using unction. I a complex number c is taken and the Julia set J() is calculated using that number, i.e. J(z +c), then what it must become known i the Julia set J(z +c) is connected or like dust [7]. () c 0 = 1 i i J G J 1 ( z + c) ( z + c) is is connected dust like (7) The type o the Julia set is shown by every point o the Mandelbrot set. An arbitrary point c can be taken and lied inside the Mandelbrot set (i.e. normally the black region), thereore the Julia set J(z +c) is connected. Taking another point d rom outside the Mandelbrot set, thereore the Julia set J(z +d) is dust like.

3 Proceedings o the 7th WSEAS Int. Con. on Signal Processing, Computational Geometry & Artiicial Vision, Athens, Greece, August 4-6, It is not necessary to calculate the whole Julia set, however the orbits o speciic points must be examined. The speciic points mentioned above are the critical points, i.e. all points where the irst derivation vanishes ('(z)=0). The type o the Julia set is deined by the orbits o the critical points [8]. When all orbits remain limited, the Julia set is connected. (I at least one orbit tends to converge to ininity, then the Julia set is dust like). Consider the ollowing example: In the case o examining z +c there is only one critical point: '(z)=*z=0, i.e. z=0 is the only critical point. Thus in this case only the orbit o 0 must be examined. The most interesting Julia sets related to the Mandelbrot set are those that lie near its border. In order to calculate an interesting Julia set the parameter c must be set to a value lying near the border o the Mandelbrot set [9]. The Mandelbrot set appears as a ringed horizontal eight that is symmetrical on the real axis, as in the ollowing igure: Note: The dierence between Mandelbrot and Julia sets consists o the act that the complex coordinates o the point are substituted to c and not to z (established at irst with a 0 value). The complex number c should not be a constant deined when starting the elaboration. An example o a Julia set obtained rom the application is shown in Fig...3 The sotware The sotware was developed in order to ollow two major lines: visual and acoustic, merged into a single inal product - a screen saver. The application is based upon the previous described sets, making a visual link between them, generating the Julia set by using a constant selected rom the Mandelbrot set. It can be observed that i the constant is outside the Mandelbrot set, the Julia set is represented as a ractal image with little complexity. Also, i the constant is inside the Mandelbrot set, the image is not interesting. Ininite complexity can be achieved by choosing constants merely on the coast line o the Mandelbrot set [10]. The music can be generated upon the ractal image created and is strongly depending on the image. For example, a zoom could be made in or out on the same spot in the complex plane and the result is that the music changes accordingly [11]. Fig. 1 A Mandelbrot set 3 Fractal Generation Algorithms The new ractal techniques tend to lean rom ractal ormulas towards algorithms. The mathematical equations are substituted by coloring algorithms, and, more interesting, music generation algorithms [1]. Fig. A Julia set 3.1 Fractals and Dynamical Systems Dynamical systems are a well-known branch o mathematics, but until the arrival o computers the sheer number o calculations involved made them impractical or real use. Benoit Mandelbrot was the irst one to use computer s ability to perorm rapid calculations to produce graphical representations o dynamical systems in the complex plane. This representation was based on the quadratic ormulas described by the French mathematician Gaston Julia at the beginning o the 0th century. The 1980s was the period when ractals started to be explored not only or their mathematical signiicance, but also or their artistic nature. The

4 Proceedings o the 7th WSEAS Int. Con. on Signal Processing, Computational Geometry & Artiicial Vision, Athens, Greece, August 4-6, mathematical apparatus is still the base, but the purpose changed to art representation. Fractal artists tested hundreds o dierent ractal equations which lead to the discovery o many new dierent ractal types. The choosing o the parameters was very important to reine the ractal orm, aspect and even sound. Ater 1995, ew new major ractal types have been introduced. This is because the newest innovations in ractal art do not come rom changing the ractal equation, but rom new ways o coloring the results o those equations. The increasing complexity o the coloring algorithms made it possible to use simpler equations. This lexibility made more space or personal artistic expression in the crisp ractal world. 3. Coloring Algorithms Every dynamical system produces a sequence o values z 0, z 1, z z n (the orbit o z start point). Fractal images are created by producing one o these sequences or each pixel in the image and the coloring algorithm is what interprets this sequence to produce a inal color. An example is presented in Fig. 3. Fig. 3 Example o an orbit Typically, the coloring algorithm produces a single value or each pixel. Since color is a threedimensional space, this one-dimensional value must be expanded to produce a color image. A spread method is to create a linear color palette as a sequence o 3D color values. The value computed by the coloring algorithm is used as a gradient along this line. I the last palette color is connected to the irst, a closed, segmented loop is ormed and any real value rom the coloring algorithm can be mapped to a deined color in the gradient. Generally, gradients are linearly interpolated in RGB space, but they can be interpolated in other spaces like HSL and interpolated with spline curves instead o straight line segments. The gradient selection is essential in creating a ractal image. This value determines which parts o the image are stressed out. In extreme cases, two images with the same ractal parameters, but dierent color schemes may be totally dierent The Escape-Time Algorithm The escape-time algorithm is one o the earliest coloring algorithms, and in many programs it is still the only option available. Due to its simplicity, it is recommended or those that are rookies in ractal sotware. Unortunately it is not very interesting rom the artistic point o view because it produces discrete values. The algorithm is based on the number o iterations used to compute i the orbit sequence converges to ininity or not. It can be demonstrated that when the orbit o any value o z 0, z 1, z z n exceeds a border region R, it always diverges towards ininity. For each ractal type a dierent shape R, with a minimum size is deined. I the orbit is stopped when z n is outside o R, the escapetime algorithm uses as coloring value the length n. In sotware implementations it is necessary to set a limit to the number o iterations to prevent ininite loops. Generally, R is set as a circle, centered in the origin, with radius. The reason is that or the Mandelbrot set, it can be proven that as soon as z >, the orbit will diverge too. Dierent types o image were created by using other shapes and positions or R: stars, triangles, ellipses and others. Although these shapes must, mathematically speaking, include the circle o radius in order to correctly compute the divergence, some tests were made on other shapes (smaller radius). Other complex coloring algorithms are implemented but this article is ocused on how a simple ormula and simple algorithms can generate interesting images and music. 3.3 Fractal Music The data can be mapped into a single tonality which could help illustrating the pitch data in relation to itsel, but no tonal motion is allowed in

5 Proceedings o the 7th WSEAS Int. Con. on Signal Processing, Computational Geometry & Artiicial Vision, Athens, Greece, August 4-6, the music. This does not represent a musically creative decision, but mono-tonality allows a clearer illustration o the set data Music Algorithm For each image with the borders: 31 pixels wide by 56 pixels vertical, the distance between each pixel is computed according to the Julia world window. By scanning 8 times rom let to the right the image to cover the 56 pixel vertical height the 3 voices are ormed. The pitch remains the same or each voice throughout, voice 1 the highest pitch and voice 3 the lowest pitch. The horizontal scan is an 1/3 notes at a tempo o quarter note with the value 60. The iteration data rom the Julia calculations is mapped to key velocity. The higher the iteration result, the louder the pitch presence. 31x56 Fig. 4 Fractal image generation sample The velocity data is stratiied to 6 values (dwell bands), the lowest being suppressed (i.e. silent background). Though the Julia set can be mapped to sound in many ways, the purpose here is two-old: 1. Demonstrating the easibility o soniying (as opposed to visualizing) scientiic data, and. Demonstrating the possibility o inding musical structures in the Julia set. When composing music, creative manipulation must be involved. The "composition" part was kept to a minimum in the algorithm; it might lessen the musical qualities but having as a purpose to preserve the objectivity toward the scientiic data. For any dwell band, the initial attack is the only one sounded in any singular voice and the reiteration o attacks on a repeated note in a given voice is suppressed. A note is sounded when or a given voice there is a change o value. Thereore, the ongoing background noise is cleaned up in order to clearly hear the leading edge o the shape. 4 Implementation Every ractal rendering sotware requires massive computational eorts and any optimization o the code is more than welcomed. First, one could observe that the bail out condition could be optimized or speed. z > is equivalent with z > 4, but the latter is much aster because we don t compute the square root. complex P, Z; or each point P on the complex plane { Z = 0; or i = 0 to ITER { i abs(z)^ > 4.0 set point color to palette[i]; break the inner loop and go to the next point; Z = Z * Z + P; } set point color to black; } Another optimization is the implementation o SSE (Streaming SIMD Extensions) Julia computation routines on Intel Pentium 4 Processors. These instructions perorm 4 pixel color computation at the same time using a more precise ormat. The code is written in C++ and uses Win3 API which is optimized also or speed o execution. The application plays all music data on a MIDI device, using a selected instrument rom a list o 17 possible MIDI instruments. One major problem o the application was the music as it was hard to play all data rom the ractal image because o the sel-similarity property was relected in the music too. This issue was solved by playing only the edges o the ractal and the rest o the music data skipped. One can also play a ractal music sample that is the music played during the screen saver period but is 8 times compressed in time and space.

6 Proceedings o the 7th WSEAS Int. Con. on Signal Processing, Computational Geometry & Artiicial Vision, Athens, Greece, August 4-6, Fig. 5 Fractal Art Screen Saver Setup All these inormation are saved into the system registry and used when displaying and animating the ractal. The animation is perormed by cycling the colors rom the palette and simply updating the display. 5 Conclusions This paper presented a sotware that creates a ractal screensaver. The application generates both ractal images and music based on the Julia and Mandelbrot ractals. The our coloring algorithms implemented are ounded on: escape-time, distance estimators, escape angle and curvature estimation algorithm. In this article it was emphasized only the escape-time coloring algorithm and the music generation. The music is created or dierent instruments on 3 voices. The horizontal scan is an 1/3 notes at a tempo o quarter note with the value 60. The iteration data rom the Julia calculations is mapped to key velocity. The higher the iteration result, the louder the pitch presence. The application creates a linear color palette as a sequence o 3D color values. The value computed by the coloring algorithm is used as a gradient along this line. The application is conigurable and permits the user to select the color palette, a region in the ractal and even the instruments to be played. The result is a screen saver application that manages to combine beautiul ractal images and strange ractal music. Reerences: [1] Benoit B. Mandelbrot, Fractals and Chaos: The Mandelbrot Set and Beyond, Springer, 004 [] Benoit B. Mandelbrot, The Fractal Geometry o Nature, W. H. Freeman, 198 [3] Benoit Mandelbrot, Gaussian Sel-Ainity and Fractals, Springer, 001 [4] Benoit B. Mandelbrot, Michael Frame, Fractals, Graphics, and Mathematics Education, The Mathematical Association o America, 00 [5] Charles Madden, Fractals in Music: Introductory Mathematics or Musical Analysis, High Art Press, 1999 [6] Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, Chaos and Fractals, Springer, 004 [7] Kenneth Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, 003 [8] Masahiro Nakagawa, Chaos and Fractals in Engineering, World Scientiic Publishing Company, 1999 [9] Nigel Lesmoir-Gordon, Introducing Fractal Geometry, Totem Books, 006 [10] Nigel Lesmoir-Gordon, The Colors o Ininity: The Beauty, The Power and the Sense o Fractals, Clear Books, 004 [11] Roger T. Stevens, Creating Fractals, Charles River Media, 005 [1] Roger T. Stevens, Fractal Programming in C, M & T Books, 1989