A Multidimensional Artistic Approach to Enhance Understanding of Julia Sets through Computer Programming Global Journal of Education and Allied Research (GJEAR) Volume.14, Number 3; March-2023; ISSN: 2837-3707 | Impact Factor: 7.80 https://zapjournals.com/Journals/index.php/gjear Published By: Zendo Academic Publishing pg. 1 ART AS A CATALYST: UNLOCKING MATHEMATICAL CREATIVITY WITH JULIA SETS 1Maria Sofia Gonzalez Martinez Article Info Abstract Keywords: Art Integration, Mathematics Education, Computer-Generated Art, Julia Sets, Educational Practices This paper addresses the integration of art and mathematics in education as a means to transcend traditional teaching methods. Historically, both the incorporation of art into mathematics and computer-generated art into the art realm have faced resistance. However, recognizing art's potential for resistance and transformation, this study proposes innovative educational practices at the intersection of art and mathematics. Section 1.1 provides an overview of the integration of mathematical, artistic, playful, and computational dimensions, focusing on Julia sets, computer-generated art, and education. Section 1.2 outlines essential mathematical definitions for the proposed approach. Additionally, section 1.3 outlines a method for creating graphical representations of Julia sets using the CFDG language. Section 2 introduces four examples of computer-generated artworks based on Julia sets, serving as foundational templates for the proposed practices. Sections 3 and 4 detail the applied methodology and present empirical findings. Ultimately, section 5 engages in a comprehensive discussion of the obtained results. 1. INTRODUCTION There is a relatively recent tendency to incorporate art into the teaching of mathematics that criticizes mathematics education in the sense of overcoming the perspective of traditional teaching [1]. However, over the years, there has been rejection of incorporating art into the mathematical world [2], and there has also been rejection of incorporating computer-generated art into the art world [3]. But, as Ferreira and Lessa [1], the author believes in the potential of Art as an element of struggle and resistance, in the sense of breaking with the reified reality, pointing to horizons of transformation. So, as called by [1], this paper proposes some ideas for the dissemination of educational practices that mobilize the Art and Mathematics interface, also considering the discussions provoked. 1 Department of Mathematics, University of El Salvador, El Salvador https://zapjournals.com/Journals/index.php/gjear Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 2 Section 1.1 presents a general background reference on the integration of mathematic, artistic, playful and computational dimensions, particularly on Julia sets, computer-generated art, and education. Then, in section 1.2, the mathematical definitions necessary to develop this proposal are presented. Section 1.3 briefly explains how to generate graphical representations of Julia sets using the CFDG language. Section 2 presents four examples of computer-generated artworks based on Julia sets to use as a starting point for the proposal. Sections 3 and 4 explain the applied methodology and some of the empirical results obtained. Finally, section 5 presents the discussion of the results. 1.1. Background As Bergweiler [4] explains, the study of iteration theory is fundamental in mathematics, and its classic problem is the study of the iterative behaviour of a family of functions that depend on a parameter. In this sense appears the study of what we now know as Julia Sets in the early twentieth century. Of course, at the time there were no computers and the study of these sets was very difficult. However, Julia sets play a critical role in the understanding of the dynamics of families of mappings [5]. In recent decades, as Hitt [6, p. 214] points out: “Technological advancement has significantly influenced the development of theoretical notions that were previously taken into account but were not considered crucial in terms of explaining the learning of mathematical concepts. These theoretical aspects are the basis for understanding the study of the different representations of mathematical objects and their role in the construction of concepts. Now, with technology, it is important to study the different representations of mathematical objects in environments very different from those that were followed in the past”. The study of many areas of mathematics and mathematics education have been modified with the popularization of different technologies such as personal computers, including Fractal Geometry. So, building multiple computer-generated images to form a richer mental image (“mental image” in the Vinner’s [7] sense) of fractals like Julia Sets is quite affordable for the students of our day. One way to implement this construction to achieve a better understanding of Julia Sets (and other types of fractals) is by artistic means, motivating the learner in a purely playful process to build (in the sense of creating) images not only aesthetically pleasing but also endowed with some meaning through computer programs. This pathway is not widely used in our time in mathematics in general, because although mathematics and art have been very close since the first manifestations of rationality of the human species, unfortunately, we have seen that these two areas of knowledge have distanced in school programs [8]. In his book, D’Amore [9] makes a brilliant exposition of the presence of mathematics at the dawn of humanity. The playful dimension is more common. Bishop [10, p. 21] discusses the role of games in mathematics education and notes: “Educators in mathematics have discovered through their experience, and they have supported with theoretical research, that playing can be an integral part of learning. This has made the act of playing and the idea of gaming a much more widespread teaching and learning activity than it had been before”. For further reading, in the literature review made by González Peralta, et al. [11], one can find possible research lines about gaming in mathematics education. There is also the computational dimension, which is becoming more common. For example, Hoffmann [12] presents a sixth-grade primary experience using a Monte Carlo simulation for introducing the concept of area of a unit circle (which is the approximation of the number π). There is also the Experience of DeJarnette [13] in which students use the Scratch programming environment to help themselves understanding of how distances travelled by certain objects are functions of time. Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 3 So, we have the seemingly strange conjunction of the four dimensions: mathematic, artistic, playful and computational. This conjunction is not directly welcomed by mathematicians, artists and art critics, or traditional computer scientists. In fact, pioneers in Computational Art faced rejection by the mathematical community, as Mumford, et al. [2, p. viii] emphasizes: “What to do with the pictures? [...] they were unpublishable in the standard way. There were no theorems, only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were the coin of the realm and the conventions of that day dictated that journals only publish theorems”. A similar marginalization (and/or misunderstanding) occurs to Rani and Kumar [14] whose article was published under Series D journal, “Research in Mathematical Education” and not under Series B journal, “The Pure and Applied Mathematics” of the same Organization (http://www.ksme.info/eng/), despite their article is not about education at all, but about Superior Mandelbrot Set. This paper is clearly about Pure Mathematics even though there are no new theo- rems. It has “only new pictures and conjectures”. There are also problems with artistic acceptance according to Franke [3, p. 186]: “[Images] were considered [only] drawings from the plotter, the main problem was the uncertainty of the experts, the art historians and the critics, and above all the gallery owners. The problem was that the computer can produce an arbitrary number of equally good ‘originals’, which can be a detriment in the business world of art”. On the apparent incompatibility between science and art, and referencing the Peitgen and Richter’s book [15], Eilenberg [16, p. 175] offers a conciliation: “It is rather unusual for natural [physical and mathematical] scientists to endeavour with such tenacity to bring their results and insights to the general public, [...] Instead of giving an abstract presentation in so many dry words, they have chosen pictures with a direct, universal appeal – a combination of mathematics and art!”. That is, scientific art (whether mathematical art or computational art) can be used primarily to publicize results to the general public. On the confusion between Computational Art and Standard Art, Franke [3, p. 187] proposes the following reflection: “The art of every age has used the means of its time to give form to artistic innovation. [...] Why shouldn't the computer, that universal medium of information and communication which has even invaded our private homes [and our lives], be used as a medium and instrument of art?”. Moreover, Zaleski Filho [8] tells us: “Thus, true art, which has no end in any of its external realizations, has as its identification a spiritual principle that enlivens and surpasses all of them”. So, after all, computational art, mathematical art and scientific art in general, as well as all other types of art, need no more justification than their ability to encourage human beings to rejoice in the artworks themselves. This is how various collections and producers of Computer Generated Mathematical Art have proliferated, such as the Peitgen and Richter’s [15] collection which includes many fractal graphics of complex dynamic systems, such as The Bridges Organization [17] which annually holds an international exhibition and competition for mathematical art (not just computergenerated), such as Aslaksen's [18] collection of university courses about art and mathematics, such as the computer-generated mathematical art collection presented by Navas-López [19] which includes various types of fractals and other types of computer graphing techniques, etc. (explained in their catalogue [20]). There are also other less formal but not less impressive collections, such as Nylander's [21], The Context Free Art [22] community gallery, and Math Munch's [23] blog which included not only mathematical art but many interesting things. There are some collections that are very specific like Ross' [24] mathematical analyses from the Sacks number spiral. Speaking now of mathematical art in the classroom and in the curriculum in general, the author supports the reasoning of Figueiras, et al. [25, p. 46]: “In the Mathematics that are taught, those that in Obligatory Education Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 4 it is said that they will serve to acquire what is needed in life, who can deny a place to beauty? Do we intend to let ourselves be carried away only by the dubious pragmatism of a mathematics cut short both in the time available for its teaching and in the potentiality of its values?”. Moreover, “The study of fractals is a motivating element in students, due to the implicit aesthetics in their constructions and the suggestive that their designs can be” [25, p. 47]. For Bosque, et al. [26, p. 20]: “[...] Aesthetics transcends the Philosophy of Art finding a place in the Philosophy of Science. In this way, if aesthetics is a component of knowledge and also has a role in scientific and mathematical activity, it makes sense to study its influence on the teaching and learning of mathematics”. The proposed Fractal Geometry Activities in the Redondo and Haro [27, 28] High School Classroom are an excellent source of ideas for planning outreach activities to different types of fractals, including Julia Sets (see [28, p.17]). However, in this vast and general overview, the artistic approach and colours are scarce. The presentation of the Mandelbrot and Julia Sets by Varona [29] explains technical details about how to graph them in Mathematica(R), which is proprietary software, and proposes the inclusion of colour palettes to enhance the images although it can only display them in grey-scale due to the type of publication. This is an example that although there is acceptance of the subject, not all journals, editors or publishers are interested or prepared to accept “mathematics with colours”. So, this paper then aims to expand the part of Redondo and Haro's [27, 28] proposal on the Julia Sets topic, using computer programming as in the Varona’s [29] paper, but with the following differences: (a) emphasizing the artistic approach and not the mathematical one, (b) not reducing to the aesthetic dimension but incorporating the communicative and didactic dimensions, (c) adding colour, and (d) using free software, in this case the ContextFree software (www.contextfreeart.org). 1.2. Julia Sets We can take the definition of Julia Set from [4, p. 153]: Let be a meromorphic function, where is de complex plane and . […] we shall always assume that is neither constant nor a linear transformation. Denote by the th iterate of , that is, and for . The basic objects studied in iteration theory are the Fatou set and the Julia set of a meromorphic function . Roughly speaking, the Fatou set is the set where the iterative behaviour is relatively tame in the sense that points close to each other behave similarly, while the Julia set is the set where chaotic phenomena take place. The formal definitions are: and is defined and normal in some neighbourhood of However, here will be chosen the most simplified version of [30, p. 263]: “Julia set of a function with seed , denoted by , is the set composed by all , such that the sequence is bounded, where and [for ]. Where , in its simplest . Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 5 form, uses ”. Nevertheless, many other different functions can be used that provide interesting results such as those presented by Entwistle [31], Garijo, et al. [32], Liu, et al. [33], Peitgen and Richter [15], Pickover and Khorasani [34], Rochon [35] and Rani and Kumar [14]. The criterion used to determine whether the sequences diverge is if , for some , has a modulus greater than 2, that is . Since it cannot be evaluated to infinity, a bound is used: . If the sequence does not “diverge” before reaching the th term (when ), it is considered not divergent, that is, it is bounded. The larger , the greater the precision of the set [30, p. 263]. 1.3. Implementation of Julia sets pictures on CFDG language The CFDG language, version 3, of Context Free (https://www.contextfreeart.org/) software and adapted computer graphics techniques from [36] and some from [37] (from their respective chapters on fractal graphing) are used for implementation here. No extra libraries are needed. The CFDG language is not a programming language properly. It is actually a language in which can be defined a particular type of context-free grammars whose terminal symbols are primitive figures: squares, circles and triangles. Different related transformations (displacement, scaling, rotation, etc.) can be applied to these figures. However, CFDG language supports Functional Programming when it is needed, especially for numerical algorithms. For further reading, visit the Context Free Art documentation page [38]. The basic geometric object to use in the examples is the square, that will be each pixel of images. This is constructed using the following primitive in CFDG language: SQUARE [ x y size # Can be abbreviated as s hue # Can be abbreviated as h saturation # Can be abbreviated as sat brightness # Can be abbreviated as b ] Where and indicate the displacement from the origin, and determine the size of the figure, is an angle between 0° and 360° indicating the colour of the figure according to the HSV colour model, and and indicate the corresponding. For more information about HSV colour model, see [39]. The function required to determine the convergence of a point (z_r, z_i) in the complex plane in CFDG language is in listing 1 (lines 3–8). Where MAXSTEPS is the value of in the criterion described in section 1.2. The initial call must be in the form steps(0, z_r, z_i, c_r, c_i), where (c_r, c_i) is the seed value. This call returns the number of iterations executed from which the sequence diverges, or returns MAXSTEPS if the sequence has not yet diverged at the Nth iteration. Listing 1: Basic source code for Julia set in CFDG language 1. 2. 3. 4. 5. 6. 7. 8. 9. https://www.contextfreeart.org/ Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 6 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. Then a Julia set figure must be implemented, determined by the seed, with the definitions in lines 10–38 from listing 1. The startshape directive is used to indicate which is the generating/starting shape (see listing 1, line 1). Result of execution of listing 1 source code with MAXSTEPS values equal to 40, 60, 80 and 100 is presented in figure 1. startshape julia(-0.381966, 0.618034) MAXSTEPS = 40 steps(numSteps, z_r,z_i, c_r,c_i) = if((numSteps < MAXSTEPS) && (z_r*z_r+z_i*z_i<4), steps(numSteps+1, z_r*z_r - z_i*z_i + c_r, 2*z_r*z_i + c_i, c_r, c_i), numSteps) LIMIT = 1000 # Image resolution # Borders of the complex plane to show: LIMLEFT = -1.4 LIMRIGHT = 1.4 LIMTOP = 1.4 LIMBOT = -1.4 # Width and height of the squares that will discretize the image: SIZEX = (LIMRIGHT-LIMLEFT)/(LIMIT-1) SIZEY = (LIMTOP-LIMBOT)/(LIMIT-1) shape julia(c_r,c_i) { loop i = (LIMIT) [] { z_i = (LIMTOP- LIMBOT)*i/(LIMIT-1) + LIMBOT # y loop j = LIMIT [] { z_r = (LIMRIGHT-LIMLEFT)*j/(LIMIT-1) + LIMLEFT # x numSteps = steps(0, z_r, z_i, c_r, c_i) if (numSteps==MAXSTEPS){ # Black SQUARE[x z_r y z_i size SIZEX SIZEY b 0] } else { # Gray SQUARE[x z_r y z_i size SIZEX SIZEY b 0.9] } } } } Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 7 2. ARTWORKS RAISED AS EXAMPLES It is a recommended collection of examples as a proposal for experiment with the parameters like seed number, viewport (drawn interval), colours, bright, saturation and hue formulae, etc. 2.1. Frozen Fjords This artwork (Figure 2) shows an aerial view of snow-capped fjords, its thin dark sand shores and the deeply blue sea. The motivation is that fjords have a natural fractal shape. Figure 1. Basic Julia set image generated by CFDG code, with values of =40 , 60, 80 and 100. Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 8 Figure 2. Frozen Fjords From a technical point of view, it is a view of a Julia set in the range of [0.01, 0.09] [0.02i, 0.1i], with see −1.384286+0.004286i. The colouration of this artwork has a constant hue as well as saturation, but the brightness is variable depending on the number of steps in which it is determined that the point belongs or does not belong to the set. See Listing 2. Listing 2: CFDG source code for Frozen Fjords 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. # This file is Free Software released under the GNU GPLv3 license or # its latest version: # http://www.gnu.org/licenses/gpl.html # To generate the image run the following line: # $ cfdg -b 0 -s 1000 fjords.cfdg fjords.png startshape fjords(-1.384286,0.004286) LIMIT = 1000 # Image resolution MAXSTEPS = 300 # Borders of the complex plane to show: LIMLEFT = 0.01 LIMRIGHT = 0.09 LIMTOP = 0.10 LIMBOT = 0.02 # Width and height of the boxes that will discretize the image: SIZEX = (LIMRIGHT-LIMLEFT)/(LIMIT-1) Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 9 38. 39. 40. 41. 42. 43. 44. 45. 46. 2.2. The Wail of the Pripyat Forest This artwork (Figure 3) shows a sick forest around the city of Pripyat. This ghost town is known for being affected by the worst accident in nuclear power history on April 26, 1986, when the Chernobyl Nuclear Power Plant reactor number 4 was overheated and exploded during a shutdown test. The motivation came after seeing a detailed documentary about the nuclear disaster at the Chernobyl nuclear power plant. Figure 3. The Wail of the Pripyat Forest From a technical point of view, it is a view of a Julia set in the range [−0.052857, 0.188571] SIZEY = (LIMTOP-LIMBOT)/(LIMIT-1) steps(numSteps,z_r,z_i,c_r,c_i) = if((numSteps < MAXSTEPS) && (z_r*z_r+z_i*z_i<4), steps(numSteps+1, z_r*z_r - z_i*z_i + c_r, 2*z_r*z_i + c_i, c_r, c_i), numSteps) shape fjords(c_r,c_i) { FILL[h 214 sat 0.89 b 0.95] # Blue ocean loop i = (LIMIT) [] { z_i = (LIMTOP- LIMBOT)*i/(LIMIT-1) + LIMBOT # y loop j = LIMIT [] { z_r = (LIMRIGHT- LIMLEFT)*j/(LIMIT-1) + LIMLEFT # x numSteps = steps(0, z_r, z_i, c_r, c_i) if(numStepsPROPORTION*MAXSTEPS) { SQUARE[x z_r y z_i size SIZEX SIZEY b 1 sat ((numSteps-1)/(MAXSTEPS-1))] } } } } Global Research Journal of Management and Social Sciences (GRJMSS) Vol. 14 (3) pg. 20 47. 48. 49. 50. 5. DISCUSSION After the course, the students expressed being a little surprised by this strange mix-of-maths-andart sessions, where they were free to experiment and play with the parameters. Moreover, students noted that the fine structures of these images are manifestations of the fact that the smallest variations (mainly the value of the seed) at the beginning of a procedure can result in huge differences later (the different Julia sets are very different from each other), and as [16] tells us, the research of dynamic systems indicates that this is typical of natural processes. As González Peralta et al. [11] say, there is certainly potential in the inclusion of playful activities in teaching but precautions must be taken to make the sessions useful for the purposes of the curriculum. So, this type of activities should be done mainly in extracurricular spaces, since students have different levels of aptitude and artistic sensitivity. Artistic activities in general are very enriched thanks to the computer offering the possibility of experimentation, since one can check the influence of parameters on the results, one can check the result of the transformations, the limiting values of interactively applied calculations, etc. [3]. “Modern art studies have shown, however, that meeting the classical definition of beauty is not in itself sufficient to create a work of art. In addition, there must be something to stimulate interest, demand involvement, and motivate further thoughts [3, p. 184]”. So, it is not enough to “create” complicated fractal images that are aesthetically beautiful, but they must have a more transcendent meaning. So, as Sethi and Subramoniam [40] claim, this proposal is meant to accomplish a type of a holistic understanding of Julia Sets and colour variation models, for students to discover meaningful relationships, and develop new knowledge that was difficult to do in the past. 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