Application of Iteration Function System for Ceramic Tile Design

2020 IEEE Intl Conf on Parallel & Distributed Processing with Applications, Big Data & Cloud Computing, Sustainable Computing & Communications, Social Computing & Networking (ISPA/BDCloud/SocialCom/SustainCom)

Zhiming Chen Department of Visual Communication School of Art, Xi’an Fanyi University Xi’an, China [email protected]

Abstract—The fractal art pattern is the product of the implement an IFS system that can change parameters to gen- fusion of computer science and art. This pattern expression erate different fractal graphics; this application can choose is based on many traditional aesthetics, and it breaks the different fractal forms, and can set different parameters for standard of conventional aesthetics in a unique way of display. Although it has extremely irregular characteristics, it has the fractal, so as to reach the user desired results. After that, a special aesthetic perspective. To ensure people a unique we can use the creative fractal graphics to design and apply aesthetic sensory experience, the design and application of them to the design of ceramic tiles. fractal patterns on different carriers according to different The rest of this paper is organized as follows. Section II constitutional ways of fractal patterns not only have practical overviews the basic working principle of Iteration Function value, but also have artistic aesthetic value. This paper investi- gates the design of ceramic tiles by Iteration Function System System for generating fractal graphics. In Section III, we (IFS). First, the working principle of IFS is brieﬂy illustrated. concentrate the production of random fractal landscape with Further, a random fractal landscape is generated via IFS codes IFS. Based on the obtained results, this paper discusses and corresponding algorithm. Finally, based on the generated the artistic aesthetic characteristics of fractal graphics in images, we applied them into the design of ceramic tiles. The Section IV. Section V provides a case study of ceramic tile case study demonstrates that our scheme can better decorate the tiles as well as other relevant industrial design. design with fractal objects generated by IFS codes. Then, we discuss the application form of fractal graphics in ceramic Keywords -fractal graphics; ceramic tile; application; IFS; tiles in Section VI. Finally, Section VII concludes this paper.

I. INTRODUCTION II. ITERATION FUNCTION SYSTEM In recent years, a popular nonlinear theory has been Iteration Function System (IFS), one of important branch- developed, which is called fractal. Fractal can be used es in fractal geometry. Due to the powerful ability of IFS to describe regular or irregular objects in nature. Since can describe objects with different shapes, it has been Mandbro started fractal theory, fractal theory has developed widely used to simulate the natural scenes with a small very well, and it has spread in many subjects such as art amount of data [7], [8]. In another words, IFS can be used [1], civil engineering [2], economic management [3], etc. to describe more complex images, thus IFS has a strong At present, there are many kinds of fractals, such as general ability to compress the images data. It is viewed as one fractals and IFS fractals, appearing in front of people. This of the fields with the most vitality and broad application result benefits from the combination of fractals and computer prospects in fractal graphics. The theory of IFS includes graphics. Therefore, fractals attract more and more attention the following aspects: compressed mapping, metric space, [4]. Since the rapid development of fractal graphics, it has existence of invariant compact sets, and measurement theory. become everywhere in our daily life, such as the shape of The corresponding algorithm mainly have two contents: (1) trademarks, packaging style [5], clothing design [6] seen collage rules in the process of acquiring IFS codes; (2) in our daily life; fractal graphics are exquisite, wonderful computer algorithm for displaying geometric objects by IFS and varied, but at present most people only stay in the code, including random iteration algorithm and deterministic stage of appreciating fractal graphics, many people want algorithm. to participate in the dynamic generation process of fractal graphics, and understand how fractal graphics movement A. IFS Iterative Algorithm generation, trying to change the shape of fractal graphics There are two algorithms for generating IFS attractors: by changing the parameters of fractal graphics to create a (1) deterministic iterative algorithm; (2) random iterative more artistic form of graphics process. algorithm. Since deterministic algorithms require large s- In this paper, we first analyze and study the algorithm torage space, thus this type of algorithm is usually not of classical fractal graphics, and compare the classical used for IFS implementations rather than using the random algorithms to study and create new primitives; on the basis iteration algorithms which is characterized by easy computer of classical fractal graphics and creative fractal graphics, we implementation and does not require a large storage space.

978-0-7381-3199-3/20/$31.00 ©2020 IEEE 1475 DOI 10.1109/ISPA-BDCloud-SocialCom-SustainCom51426.2020.00222 This algorithm selects transformswi through a random (2) Set up pixel amplifier; pi( =1) process based on the probability pi assigned by (3) Given the loop variable, from (number of points) each transform wi in the IFS system. Intuitively, the bigger (4) Set up a random interval generator pi is, the more chance of being selected. k=random(i)+1; In the IFS iterative algorithm, we firstly determine the (5) x = d[k, 1] ∗ tempx + d[k, 2] ∗ y + d[k, 5]; initial point (x0,y0). Then, we can randomly select a IFS x = d[k, 3] ∗ tempx + d[k, 4] ∗ y + d[k, 6] function from w1,w2, ··· ,wN for transforming, and further x1 = round(x ∗ j)+Δx generating the new point (x1,y1); the selection of the next y1 = round(y ∗ j)+Δy; point can be conducted according to w1,w2, ··· ,wN , the Remark: tempx indicates the temporary storage selected wi makes the transform for (x1,y1) and obtaining information of x; Δx and Δy denote the pan the new point (x2,y2). By repeating the above process, a distances on the display. series of points will be generated and consequently displayed (6) Scan each pixel to draw graphics. on the monitors. According to the known IFS code, using the above algo- B. Affine Transformation rithm, we plotted the number of points as follows. Therefore, the Bernsley fern leaves with N = 1000, N = 10000, In the IFS system, affine transformation mapping con- N = 100000, N = 1000000 are produced as shown in trols the structure and shape of the set of attractors. Each Figure 1. affine transformation corresponds to a probability, which characterizes the probability that the affine transformation is selected. It has no effect on the attraction set, but it is also an important information for drawing graphics. It can be seen that when the probability pi is selected properly, the basic shape of the fractal graphic is relatively complete regardless of the number of iterations, but if the selection is not appropriate, although the shape of the fractal graphic can be restored when the number of iterations is large, the fractal graphic is unrecognizable when the number (a) N = 1000 (b) N = 10000 is small; and it will also affect the imaging speed of the graphics. For example, if the probability should be too small, it will take a long time before enough points fall into the area. C. Algorithm Design In order to elaborate the procedure of fractal algorithm in IFS, this section provide the algorithm design for a simple fern leaf. (c) N = 100000 (d) N = 1000000 For a simple fern leaf, it is composed of 4 affine transformation. x ab x e Figure 1. Bernsley Fern Leaves Generated by Random Iteration w = + i y cd y f (1) Clearly, as the number of points increasing, the Bernsley The probabilities for affine transformation, i.e, IFS codes fern leaves is becoming more realistic. are listed in Table I. III. RANDOM FRACTAL LANDSCAPE Table I IFS CODES To pave the design for ceramic tiles, this section inves- tigates the random fractal landscape by designing different abcdefp IFS codes of various objects. w1 0 0 0 0.16 0 0 0.01 Many strangely shaped mountains, rivers, continents and w2 0.85 0.04 -0.04 0.85 0 1.6 0.85 islands in nature are products of nature. They cannot be w 3 0.2 -0.26 0.23 0.22 0 1.6 0.07 described with classic Euclidean geometry. They have sta- w4 -0.15 0.28 0.46 0.24 0 0.44 0.07 tistical self-similarity, i.e, they have the fractal features. This is because they do not have the strict self-similarity like The steps for generating the fern leaf are as follows. Mandelbrot set. For example, the blood vessels missing from (1) Define a two-dimensional array and store the IFS the human body, from large blood vessels to small blood codes in the above Table I into the array; vessels to microvessels, the entire system has self-similarity,

1476 Table VI but micro-vessels cannot branch without limit, so they do not IFS CODESOFASUN have strict self-similarity. Not only that, there are almost two things in the creation of nature that are completely identical, abc defp that is, the branches and buds of the same tree also have W1 0.15596 0.98776 -0.98776 0.15596 -0.779 9.124 0.9866 W2 0.04428 0 0 0.04116 0.641 4.829 0.0032 different shapes, which is randomness. W3 0.05566 0 0 0.04527 0.998 4.779 0.0029 W4 0.1154 0 0 0.05094 1.428 4.761 0.0036 Table II W5 0.27142 0 0 0.04923 2.38 4.781 0.0037 IFS CODES OF A MOUNTAIN

abcdefp

W1 0.08 -0.031 0.084 0.0306 5.17 7.97 0.03 W2 0.0801 0.0212 -0.08 0.0212 6.62 9.4 0.025 W3 0.75 0 0 0.53 -0.375 1.106 0.22 W4 0.943 0 0 0.474 -1.980 -0.65 0.245 W5 -0.402 0 0 0.402 15.513 4.588 0.21 W6 0.217 -0.052 0.075 0.15 3 5.741 0.07 W7 0.262 -1.105 0.114 0.241 -0.473 3.045 0.1 W8 0.22 0 0 0.43 14.6 4.286 0.1

Table III IFS CODES OF A GREEN TREE

ab c defp Figure 2. A Random Fractal Landscape W1 0.05 0 0 0.6 0 0 0.1 W2 0.05 0 0 -0.5 0 0.8 0.14 W3 0.4596 -0.3214 0.3857 0.383 0 0.48 0.2 W4 0.4698 -0.1539 0.171 0.4229 0 0.88 0.18 characteristics of artistic beauty of conventional graphics W5 0.433 0.275 -0.25 0.4763 0 0.8 0.18 W6 0.4505 0.2294 -0.3155 0.3277 0 0.4 0.2 such as unity and change, uniformity and symmetry, rhythm and rhythm, proportion and scale. Fractal patterns are simple and complex; ordered and disordered; stable and unstable; Table IV perfect unity of certainty and randomness. IFS CODESOFAPALMTREE • Unity and change, chaos contains expressions of abcde f p order, order and disorder. Unity and change are the W1 0.195 -0.488 0.344 0.433 0.4431 0.2452 0.2 most basic rules followed by fractal patterns. The unity W2 0.462 0.414 -0.252 0.361 0.2511 0.5692 0.2 and change are both opposite and interdependent. The W3 -0.058 -0.07 0.453 -0.11 0.5976 0.0969 0.2 fractal pattern has to go through countless iterations. W4 -0.035 0.07 -0.469 -0.022 0.4884 0.5069 0.2 W5 -0.637 0 0 0.501 0.8562 0.2513 0.2 The iteration is orderly and stable. However, during the process of many iterations, an unpredictable sit- uation has occurred, which has plunged into chaos Table V and germinated a disordered seed in the order. In IFS CODES OF A GRAY TREE order to find order in the chaos, these complicated structures are composed of similar or even local parts. abcdefp All elements in the fractal pattern together form a W1 -0.04 0 0.23 -0.65 -0.08 0.26 0.25 unified and harmonious whole (as shown in Figure 3). W2 0.61 0 0 0.31 0.07 2.5 0.25 The existence of any element is valid and unchangeable, W3 0.65 0.29 -0.3 0.48 0.54 0.39 0.25 W4 0.64 -0.3 0.16 0.56 -0.56 0.4 0.25 otherwise it will destroy the overall quality. Due to the similarity of the elements that make up the image, the With the above IFS codes of various objects, the random pattern structure and color in the image will have an fractal landscape is generated as shown in Figure 2. effect of mutual echo. This combination breaks through the scope of simple splicing and superposition, and is IV. ARTISTIC AESTHETIC CHARACTERISTICS OF a continuous extension of recursion and iteration. From FRACTAL GRAPHICS the visual and connotation of the picture, it is integrated With the intuitive display of fractal geometry in computer into a unified whole with common trends. graphic art design, it directly promotes the development in • Rhythm and rhythm: the expression of unit shape the art field. In addition to the infinite fineness and self- and overall shape. The rhythm and rhythm in the similarity of fractal graphics, fractal art also has the basic pattern are beautiful and harmonious trends and charms

1477 the beauty of mathematics. St. Augustine, a famous mathematician in the Middle Ages, believes that ”number is the foundation of all beauty”. For example, the famous Swedish mathematician Koch designed a Koch curve similar to snowflakes or island edges in 1904. No matter how many curves and how long this curve is, the area enclosed by it is always smaller than the area of the circumscribed circle of the original triangle; Mandelbrot set adjusts the plane rotation coefficient and the number of iterations, and can use a combination of different transformation functions to get a pattern model. The initial coordinates, the number of iterations, the Figure 3. Self Similarity of Fractal Graphic plane rotation coefficient, and the weight coefficients between layers are arranged and combined within the range of values, and thousands of new flower patterns with similar shapes and different characteristics are formed on the basis of repetition, and are derived from expanded to achieve flower pattern batches. Generated the regular changes in nature, such as the rolling hills, functions. The creation of fractal patterns is inseparable the ebb and flow of the sea, and the beat of the human from numbers. Using the relationship between propor- heartbeat . Any structure in the whole of the fractal tion and numbers, from a scientific and mathematical pattern of Julia set is a branch point, composed of point of view, according to the intention of the creation, countless details, it is the whole of itself, and it is also through the size, height, angle, etc. of the form in the a part of a larger whole, gradually repeating to infinity. picture, a pattern with mathematical beauty is created. In the range of infinity to infinity, the kaleidoscope is such an analogue. The complex structure of the gradual rhythm is seamless and beautiful, forming a gradual V. D ESIGN OF CERAMIC TILE rhythm and rhythmic beauty. The generation principle On the decoration design of fractal technology, it has been of the Sierpinski carpet is increasing with the number of used for a long time both in China and abroad. Similar to iterations. It has been repeated to divide the remaining the roof, appearance and interior of some Gothic buildings in squares into nine and remove the repeated operation of Europe, it can reflect the application of fractal technology the middle square. The generated fractal pattern has a because they all have Obvious self-similarity. In addition, repeated sense of rhythm; the Koch curve passes node my country also used a decoration method with fractal art a changes generate a rhythmic linear pattern. long time ago. • Symmetry and balance: local and overall, dynam- In general, more and more exquisite fractal graphics are ic and static performance. Symmetry refers to the produced. For decoration design, whether it is the overall relationship that the pattern shapes in the picture com- structure or simple interior patterns, there will be more and pletely overlap in size, shape, and arrangement; balance more attempts and breakthroughs. This paper focuses on refers to a relationship in which the pattern shapes are generation of the fractal graphic with IFS system, and then different, but through the size, number, color, position, innovatively implement the design of ceramic tile. etc. of the shape, the visual balance is achieved through Example 1: In order design a wonderful ceramic tile, the shape change. The Sierpinski triangle mat, except we firstly generate the fractal graphic elements. Given two that the triangle itself belongs to the symmetry of the initial points (x1,y1) and (x2,y2), then we can calculate central axis; the Julia set (Julia set) is based on the the middle points (x3,y3), (x4,y4), (x5,y5), (x6,y6) and theory of z = x + iy, the point on the x axis of the (x7,y7) between them and connect these points each other plane corresponds to the real number, and the point on with lines. Obviously, this iteration process can be resolved the y axis can correspond to Because of pure imaginary with the following reclusive algorithm. numbers, the plane of the complex number z can be The process of Algorithm 1 is working as follows. At the called the complex plane, and for each different value beginning, we take two points (x1,y1) and (x2,y2) as the of c, f can generate various mysterious fractal figures, initial input of the algorithm; the output is the generated which have a balanced beauty in themselves and are fractal graphic; Lines 1-10 are in charge of calculating generated in a symmetrical manner during the iteration the coordinates of middle points x3, x4, x5, x6, and x7 process Arranged to show the beauty of advancing layer according to the input points. Then, it goes to the recursion by layer, exuding indescribable charm. section and invoke the function Tile(Lines 11-18). Then the • Proportion and scale, rational and emotional fusion neighboring points are connected with a line (Lines 20-26). performance. Both proportions and numbers represent

1478 Input: x1,y1,x2,y2 Output: Fractal Graphic 1 x3 = x1 + pow(3, 0.5) ∗ (x2 − x1)/(2.0 ∗ (pow(3, 0.5) + 1)) + (y2 − y1)/(4.0 ∗ (pow(5, 0.5) + 3)); 2 y3 = y1 + pow(3, 0.5) ∗ (y2 − y1)/(2.0 ∗ (pow(3, 0.5) + 1)) − (x2 − x1)/(4.0 ∗ (pow(5, 0.5) + 3)); 3 x4 = x3 − (y2 − y1)/(pow(3, 0.5) + 1); 4 y4 = y3 +(x2 − x1)/(pow(3, 0.5) + 1); 5 x5 = x4 +(x2 − x1)/(2.0 ∗ (pow(3, 0.5)+1))+(y2 − y1) ∗ pow(3, 0.5)/(4.0 ∗ (pow(5, 0.5) + 3)); 6 y5 = y4 +(y2 − y1)/(2.0 ∗ (pow(3, 0.5) + 1)) − (x2 − x1) ∗ pow(3, 0.5)/(4.0 ∗ (pow(5, 0.5) + 3)); 7 x6 = x4 +(x2 − x1)/(pow(3, 0.5) + 1); y = y +(y − y )/(pow(3, 0.5) + 1) 8 6 4 2 1 ; Figure 5. Our Designed Ceramic Tile 9 x7 = x3 +(x2 − x1)/(pow(3, 0.5) + 1); 10 y7 = y3 +(y2 − y1)/(pow(3, 0.5) + 1); 11 if (n>1) then wall of the living room and bed room. Figure 6 shows our 12 Tile(x1,y1,x3,y3,n− 1); applications. Obviously, our design can be penetrated into 13 Tile(x3,y3,x4,y4,n− 1); decoration and other relevant industrial design ﬁelds. 14 Tile(x4,y4,x5,y5,n− 1); 15 Tile(x5,y5,x6,y6,n− 1); 16 Tile(x6,y6,x7,y7,n− 1); 17 Tile(x7,y7,x2,y2,n− 1); 18 end 19 else 20 line(x1,y1,x3,y3); 21 line(x3,y3,x4,y4); 22 line(x4,y4,x5,y5); 23 line(x5,y5,x6,y6); 24 line(x6,y6,x7,y7); (a) Living Room 25 line(x7,y7,x2,y2); 26 end Algorithm 1: Algorithm on Generating the Fractal Ele- ments for Ceramic Tiles

With the above algorithm, the fractal graphic element of ceramic tiles is obtained as shown in Figure 4.

(b) Bed Room

Figure 6. Decoration of the Background Wall of the Living Room and Bed Room

Figure 4. Fundamental Fractal Graphic of Ceramic Tiles VI. DISCUSSIONS In this section, we mainly discuss the application form of Based on the above fractal graphic element, we make the fractal graphics in ceramic tiles. Tiles are an indispensable necessary art processing on the ceramic. Figure 5 shows our decorative element in modern environmental decoration de- designed ceramic tile. sign. In addition to special artistic tiles, they can be generally Example 2: Based on our designed random fractal land- divided into rectangles and squares according to the shape scape, we applied it into the decoration of the background characteristics. According to the collage process, two-sided

1479 continuous, four-sided continuous, and corner-related fractal graphic changes the single pattern style of the traditional can be designed. The two-sided continuous fractal graphic pattern, and the fractal pattern tile designed by combining refers to a way to tile tiles in a horizontal or vertical direction the material of the tile and the special characteristics of the by one or two or three patterns combined by fractal design, process not only helps the wide application in the field of which can be in symmetrical, scattered, continuous, balanced decoration, but also helps to beautify the environment and and other styles arrangement, usually the lace-like tile style enhance users’ aesthetic sentiment. is often used as a decorative “waist line” on the wall or a VII. CONCLUSIONS decorative line on the ground, using repeated arrangement Aiming to bridge the fractal art and the ceramic art design, of fractal patterns to obtain the beauty of rhythm and rhythm this paper have studied the feasibility of fractal art in the (as shown in Figure 7). particular field, i.e, ceramic tile design. Specifically, the IFS system is firstly over-viewed including the explanation of working principle of IFS based fractal graphics generation. Considering the virtual landscape, we provided the IFS based random fractal landscape production approach which paves the way for further creation of ceramic tiles. By our case study on ceramic tile design, we validated the feasibility and usability of our approach. It is believed that our approach can be applied into other relevant realm of industrial design. ACKNOWLEDGMENT (a) Symmetrical Continuous Two- (b) Scattered Continuous Two- sided Tiles sided Tiles This work is supported by the teaching project of “Deco- ration and pattern” course of Xi’an Fanyi University (Grant Figure 7. Two-sided Continuous Fractal Graphics No.T1908). REFERENCES The four-sided continuous fractal pattern refers to a [1] X. Lin, W. Liu: The Application of Fractal Art in Ceramic method of laying tiles according to a certain skeleton for Product Design[C]//IOP Conference Series: Materials Science each unit-shaped fractal tile. The four-square continuous and Engineering. IOP Publishing, 2019, 573(1):012003. skeleton style has a pattern of tile collage, dot pattern, shad- [2] Y. Xiao, H. Liu, P. Xiao, et al: Fractal crushing of carbonate ing pattern and floating pattern. The four-sided continuous sands under impact loading[J]. Gotechnique Letters, 2016, fractal pattern seeks unity in change, changes in the same, 6(3):199-204. and obtains an orderly beauty in application (as shown in Figure 8). [3] Q. Cheng, J. Jiao: Fractal Features of Fractional Brown- ian Motion and Their Application in Economics[J]. 2019, 37(3):863-868. [4] F. Hao, D.S. Park, Y.S. Jeong, et al. hFractal: A Cloud- Assisted Simulator of Virtual Plants for Digital Agricul- ture[M]//Advanced Multimedia and Ubiquitous Engineering. Springer, Berlin, Heidelberg, 2016: 233-240. [5] J. Wang: Application of Fractal Theory in Packaging Graphic Design[C]//2020 12th International Conference on Measur- ing Technology and Mechatronics Automation (ICMTMA). IEEE, 2020:186-190.

(a) Scattered Four-sided Continu- (b) Ladder Contiguous Four-sided [6] X. Long, W. Li, W. Luo: Design and application of fractal ous Continuous pattern art in the fashion design[C]//2009 International Work- shop on Chaos-Fractals Theories and Applications. IEEE, 2009:391-394. Figure 8. Four-sided Continuous Fractal Graphics [7] P.F.L. Tobing, S. Feranie, F.D.E. Latief: Preliminary Study of The corner fractal graphic refers to the fractal pattern tile 2D Fracture Upscaling of Geothermal Rock Using IFS Fractal style at the turning point of the decorative wall or ground tile Model[J]. JPhCS, 2016, 739(1):012095. shape. The corner fractal graphics are generally arranged and [8] T. Martyn: Realistic rendering 3D IFS fractals in real-time combined according to the balanced and symmetrical styles. with graphics accelerators[J]. Computers and Graphics, 2010, In the application of this combination, the edge is suitable 34(2):167-175. for the application of diagonal or four corners. The fractal

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