Application of Fractal Graphics in Packaging and Decoration (I)

Abstract : This paper studies three kinds of algorithms of fractal images and uses fractal theory to simulate natural scenery and irregular patterns in packaging CAD.

Keywords: packaging CAD, fractal, Mandelbrot set, Julia set, L system

At present, graphics and image processing software commonly used in packaging and decoration design include Photoshop, CorelDraw, AutoCAD and so on. However, these softwares draw geometric patterns with smooth surfaces and regular shapes. Simulating natural scenes often involves interactive methods, or using digital cameras, scanners, and other tools to process pictures after they are entered into a computer. This mechanical method usually does not meet the needs of packaging designers. Based on this consideration, this paper tentatively uses the fractal theory in mathematics to simulate the scenery and irregular abstract graphics in the natural world in order to meet the requirements of packaging designers.

Fractal theory describes irregular geometric shapes. Since its inception, it has been widely used in many fields such as astronomy, computer graphics, meteorology, philosophy and art. Similarly, if we apply it to packaging design, especially packaging and decoration design, we will certainly achieve unexpected results. Fractal graphics can be divided into several types based on the algorithms it implements: self-similar fractals, complex fractals, natural fractals, and random fractals. This article discusses only three of these algorithms.

1. Mandelbrot set

The famous graphic Mandelbrot set developed by Mandelbrot, the founder of fractal theory, is a typical complex fractal. Since there is a one-to-one correspondence between the complex numbers and the points (x, y) on the plane, we can think of the graphic display as a complex plane. The real and imaginary axes of the complex plane correspond to the latitude and longitude of the graph. Each point (x, y) in the graph can be represented as a complex number x + iy. Then, the Mandelbrot set looks like this: The area between -2.0° to +0.5° longitude and -1.25° to +1.25° latitude is a huge sea, it consists of bays, small bays, and several Branches make up.

In fact, Mandelbrot's concentrated oceans and bays are linked by a major cardiogram with a series of disc-shaped protrusions, and each protrusion is surrounded by smaller protrusions. However, this is not all, there are fine "hair-like" branches growing out from the protrusions. These fine hairs have micro-samples similar to the entire Mandelbrot set on each segment. That is, the Mandelbrot set has nested layers and an infinitely fine structure. However, such a graph with a complex structure can be generated with a rather simple complex iterative equation: Zn+1=Zn2+C. According to this formula, taking a number z in the complex plane, multiplying it by itself, adding the initial complex number c, and repeatedly iterating to get the Mandelbrot set. The steps are as follows:

(1) Separate the computer screen into a grid with the center at (0,0), the x-axis representing the real number (from -2.0 to 0.5), and the y-axis representing the imaginary number (from -1.5 to 1.5).

(2) Select a point on the grid to represent a complex number. The real part is represented by the x position and the imaginary part is represented by the y position. This complex number is referred to as c.

(3) Let the complex number Z0=0, the iterative mapping Z=Z2+C, until |z→∞|, or iterate to the number of iterations selected in advance.

(4) If the orbit of Z0 = 0 runs to infinity, draw a point on the screen with a color that is related to the number of iterations required to reach infinity (this point corresponds to the real part of c, the imaginary part value ). If the modulus of Z0 = 0 is still small after it has been iterated for a given number of times, it is assumed that the value of c cannot be iterated to infinity, so the corresponding point is colored blue.

(5) Repeat this step for all points on the grid.

The following figure shows the Mandelbrot set drawn in this way. Since the Mandelbrot set has a layered nested and infinitely fine structure, zooming in on partial areas using zooming techniques can yield completely different pictures. Figures (a) and (b) are the enlarged figures. However, it is not a simple enlargement of the original graphics, but it shows that Mandelbrot sets many details, and if you continue to zoom in, you can have more details. This zooming in process can be carried out without limit, and the resulting graphics are completely different from the original graphics. This feature indicates that fractal graphics are very sensitive to parameters, and small parameter changes can cause completely different graphics. If you use this kind of graphics to make anti-counterfeit packaging, it will be difficult to copy and imitate.

(to be continued)