Algorithmic Modeling Technology for Computer-Aided Fractal Art

Pattern Design

Cui Wang

Shandong Vocational College of Science and Technology, Weifang, Shandong, China

Keywords: Computer-Aided, Fractal Art Pattern Design, Algorithmic Modeling Technology.

Abstract: This paper proposes a solution based on intelligent algorithm to solve the problems of complexity and artistic

expression in the design of fractal art patterns. In the research process, this paper first builds a computer-aided

design system and uses an iterative algorithm to generate fractal patterns. The research is divided into three

steps, including parameter configuration and modeling, model training and optimization, simulation, and

application case analysis and evaluation. The actual results show that the generated fractal pattern complexity

score is 9.8/10 and the visual aesthetic score is 9.5/10 under the optimal parameter configuration. The

comprehensive conclusion shows that the intelligent algorithm can not only effectively improve the

complexity of pattern design, but also significantly enhance its artistic expression, and provide strong data

support for practical applications.

1 INTRODUCTION

In recent years, fractal art design has been widely

used in the field of digital art and scientific

visualization. With the advancement of science and

technology, people have higher requirements for

pattern design, not only requiring it to have high

complexity, but also hoping that the pattern can show

a unique artistic beauty (Dua, Malhotra and Tager,

2024). However, the existing methods face certain

problems when dealing with complex fractal patterns,

such as low efficiency and insufficient artistic

expression (Helm, and Stear, et al. 2023). Some

researchers have proposed that the impact can be

reduced based on an iterative function system, but this

method fails to effectively deal with the high-

complexity pattern design, showing low efficiency

and inability to automate (Hetherington, 2023). At the

same time, some researchers have proposed that the

method of manual adjustment of parameters can be

used to optimize the design of fractal art patterns

(Jing, 2023). Although this method can increase the

complexity of the pattern to some extent, it is too

dependent on the experience of the designer and

cannot be generalized in large-scale production

(Pasca-Tusa, and Solomo, et al. 2023). In addition, it

is also more prone to unpredictable deviations in the

process of generation. In order to effectively solve

these problems, this paper will use computer-aided

methods to design fractal art patterns (Qin, 2024).

This article will be carried out using intelligent

algorithms. The reason for this is that smart methods

can handle complex mathematical calculations,

perform better in highly complex pattern generation

processes, and are much more efficient than

traditional methods (Skaggs, 2023). At the same time,

it has strong adaptability, and can automatically

adjust the parameters according to different design

needs, so as to achieve efficient and accurate pattern

generation (Tauber, 2024). This paper hopes to

achieve the goal of fractal pattern design with high

complexity and high artistic expression through

specific research (Tepavcevic, and Stojakovic, et al.

2023). Based on this, to the field of digital art and

scientific visualization,

Provide more advanced design tools to drive

further development in these areas.

2 RELATED WORKS

2.1 Theory of Fractal Geometry

The fractal geometry theory is the core theoretical

basis of this paper, which was proposed by Benoit

Mandelbrot in the 1970s and is mainly used to

206

Wang, C.

Algorithmic Modeling Technology for Computer-Aided Fractal Art Pattern Design.

DOI: 10.5220/0013538400004664

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 3rd International Conference on Futuristic Technology (INCOFT 2025) - Volume 1, pages 206-212

ISBN: 978-989-758-763-4

describe the mathematical properties of irregular,

self-similar figures (van der Vennet, R and A.

Ciancio, 2024). Fractal geometry emphasizes the

need to generate patterns based on recursive methods

to ensure that their structures are self-similar at all

scales. In artistic pattern design, the theory of fractal

geometry can generate complex and exquisite

patterns. This theory supports pattern generation with

high complexity and detailed expressiveness. In this

paper, the recursive nature of fractal geometry is used

to optimize the generation process based on

intelligent algorithms, and then the complexity of the

pattern is greatly improved.

2.2 Iterative Function System Theory

Iterative function system theory is a key method to

generate fractal patterns, which maps one initial point

to multiple spatial positions under continuous

iteration through a set of linear and nonlinear

transformations to generate complex fractal

structures. In fractal geometry, this theory is widely

used because it can effectively generate adaptive

patterns and has relatively high computational

efficiency. In this study, the iterative function system

theory is applied to the infrastructure design of

intelligent algorithms, and by adjusting its

transformation parameters, the system can

automatically generate fractal patterns with different

styles and complexities. It provides a flexible and

efficient framework that supports extensive

exploration and innovative design of fractal patterns.

2.3 Theory of Adaptive Optimization

Algorithms

The theory of adaptive optimization algorithm is a

theory that needs to be applied in the research of this

paper. It can be used to improve the efficiency and

quality of fractal pattern generation. Adaptive

optimization algorithms, such as genetic algorithm

and particle swarm optimization algorithm, can find

the optimal solution in the multi-dimensional

parameter space, especially for complex nonlinear

problems. These algorithms dynamically adjust the

search path based on the feedback and avoid local

optimal traps to find the global optimal solution. In

this study, the adaptive optimization algorithm is used

to optimize the parameter configuration of the

iterative function system, so that the generated fractal

patterns are complex and visually appealing. Based

on the application of this algorithm, the research can

achieve accurate control of the generation process and

ensure efficient and stable pattern generation results.

3 ANALYSIS OF THE FORM OF

ARTISTIC PATTERNS

3.1 Introduction to Algorithm

Modeling Technology

The architecture of the computer-aided fractal art

design system contains a number of functional

modules, which can work in harmony and ensure the

overall performance and stability of the system. First,

the user interface module provides an intuitive

interface and allows users to easily enter parameters

to adjust the design in real time. Secondly, the

modeling module can be used to define and manage

the fractal module, support the selection of multiple

fractal types, and support parameter optimization.

The core part of the system, the algorithm processing

module, is used to generate fractal patterns and

optimize them, which can be used to increase the

processing speed of the system based on parallel

design. The main function of the data storage module

is to manage user data and generated patterns, and to

provide historical record saving, version control, and

data backup functions. The main purpose of the

Rendering & Display module is to ensure that the

graphics are presented with a high sense of quality,

and support real-time preview and multiple output

options. The main task of the system management

module is to manage user rights and configure the

system, provide logging, and ensure the high security

and maintainability of the system. Based on the

efficient collaboration between the modules, the

system can efficiently generate high-level fractal art

troupes with complexity and aesthetics to meet the

needs of art design.

3.2 Algorithm Modeling Technology

Design

During the modeling phase, the basic structure of the

fractal pattern is defined to lay the foundation for the

design. The focus of this phase is to identify the

applicable iterative function system, as described in

Eq. (1).

ii i

i=1

(x)= w f (x)+ b×



)

In Eq. (1),

is the initial point and parameter of

the input represent one of the starting positions in the

fractal pattern.

w is the weight of each iteration

function, which can be used to adjust the influence of

Algorithmic Modeling Technology for Computer-Aided Fractal Art Pattern Design

207

each function in generating the fractal pattern. For

example, if a fractal part needs more complex details,

you can increase the weight of the function

accordingly.

(x)is a specific fractal function that

determines how the current point is converted in each

iteration. Usually these functions can be simple linear

transformations, more complex nonlinear

transformations.

b is an offset value that is used to

pan and adjust the output in each iteration to ensure

that the fractal art pattern is evenly distributed across

the plane.

For the formula for calculating the fractal

dimension, see Eq. (2).

log(N)

log(1 / r)

(2)

In Eq. (2),

is the fractal dimension, which is

mainly used to quantify the fractal complexity. It

represents the degree of filling of the fractal pattern in

space, and the higher the dimension, the more

complex the pattern.

is the number of repeating

elements in the pattern, that is, the number of self-

similarities of the fractal structure. More repeating

units indicates a higher fractal dimension.

is the

scale scale, which indicates how much the pattern is

reduced after each iteration. Smaller scales produce

more complex patterns. The formula for fractal

generation is detailed in (3).

()

x f x

(3)

In Eq. (3),

x is the point of the current iteration,

indicating the position of the fractal pattern during the

iteration.

()

is the result of an iterative function

that generates the next position based on the applied

fractal function. This step is the core of the generation

of fractal art patterns, and complex fractal shapes will

be generated under multiple iterations.

3.3 Algorithm Modeling Technology

Training

The training phase is the process of tuning the initial

model to the desired state. The purpose of this phase

is to gradually refine the parameters of the model so

that the resulting fractal art pattern can be more in line

with expectations. The steps to initialize the

parameters are necessary. Here, we first set the

weights and biases in the iterative function system. In

general, these parameters can be set by random

initialization and prior knowledge. For example, a

specific initial value can be selected according to the

needs of the fractal pattern design, and the pattern has

certain predetermined characteristics from the

beginning. Subsequently, iterative optimization is

required, and convergence checks and optimizations

are performed to better improve the training effect.

When the optimum is reached, the iteration is

stopped.

In the process of optimizing the model, it is

important to note that this is a fine-tuning step for the

resulting fractal art pattern to ensure that it meets the

mathematical requirements and has high artistic

requirements. For multi-objective optimization, see

Eq. (4).

()

i=1

minimize α ×Objective (x)



)

In Eq. (4),

α is the weight factor of different

objective functions, which is mainly used to balance

multiple design goals, such as the symmetry goal and

complexity goal in fractal art design, and the visual

beauty of the wood white gull. By adjusting these

weights, you can achieve the purpose of optimizing

some specific artistic effects.

(x)Objective are

different objective functions that are mainly used to

evaluate the performance of fractal art patterns in

certain aspects. For example, the symmetry objective

function can be used to ensure good symmetry of the

fractal pattern, and the complexity objective can

ensure that the pattern can meet the complexity

requirements to show the unique beauty of the fractal.

For parameter adjustment, see Eq. (5).

w=w-η×

∂

)

In Eq. (5),

is the learning rate, which represents

the step size corresponding to each parameter

adjustment. A larger learning rate will speed up the

convergence rate, but it may lead to over-adjustment.

Smaller learning rates provide fine-grained

adjustments.

is a loss function that is primarily

used to measure the difference between the generated

art pattern and the desired effect. Based on the

minimization loss function, the generated pattern can

be gradually approached to the ideal fractal shape.

Subsequently, the local optimization should also

be done, as shown in Eq. (6).

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208

x=x+Δ

′

(6)

In Eq. (6),

xΔ

is a local adjustment amount,

which is mainly used to fine-tune a specific part of the

fractal art pattern. With fine local adjustments, the

details of the pattern can be enhanced and made

visually richer and more appealing.

3.4 Optimization of Algorithm

Modeling Technology

To do this, it is necessary to implement the joint

debugging of modules. That is to say, it is necessary

to make joint debugging of each module, including

user interface module and modeling module,

algorithm processing module, data storage module,

rendering and display module, and system

management module, and then ensure the stable data

transmission and normal interaction between each

module. After the module joint debugging is

completed, the system should also be tested on the

functional aspects to ensure that all functions can

work according to the expected requirements.

Specifically, fractal pattern generation and parameter

adjustment, image rendering, and user data

management. After the system integration function is

stable, it is necessary to optimize the performance and

further improve the system response speed and

processing efficiency, so as to ensure that the system

can operate efficiently in the process of processing

complex fractal art patterns. Finally, a comprehensive

integration test is carried out to test the compatibility

test and stability test of the integrated system in

different hardware environments, so as to ensure that

the system can work normally in various conditions.

4 RESULTS AND DISCUSSION

4.1 Introduction to the Case of

Algorithm Modeling Technology

In a high-end laboratory focusing on cutting-edge

graphics technology research, the research team is

exploring algorithmic modeling techniques for

computer-aided fractal art pattern design. The project

aims to provide innovative pattern design tools for the

field of digital art and scientific visualization. The

purpose of this case study is to verify the

effectiveness of the algorithm model in practical

applications, and to optimize the fractal art pattern

generation effect achieved under different parameter

settings. The team selected two specific design tasks,

namely "Complex Natural Landscape" and "Future

Technology Network", to carry out simulation tests

and evaluations under different parameter

configurations. Below are the 2 sets of configuration

data used in the test and their effectiveness

evaluation, The art pattern typing is shown in Figure

Figure 1: The color typing and structure typing samples of

this pattern.

4.2 A Comprehensive Knot for Artistic

Pattern Design

The main purpose of the simulation environment is to

verify the effectiveness of the designed algorithm

model in generating fractal art patterns. The goal is to

verify the stability of the model under the setting of

each initial parameter, to ensure that the algorithm

can stably generate fractal art patterns, and there will

be no crashes and errors. Evaluate the complexity and

aesthetics of the pattern, and iterate to evaluate the

complexity and visual effect of the resulting pattern.

Optimize algorithm parameters. Through the

simulation data, the key parameters in the algorithm,

such as weights and learning rate, can be adjusted and

optimized, and then the generation of high-quality

fractal art patterns can be improved. In the process of

simulation, the core algorithm is tested several times.

These tests are carried out with different parameter

settings and the quality of the resulting fractal art is

evaluated. The simulation data are shown in Table 1.

From the data in Table 1, it can be seen that the

fractal art patterns generated by the algorithm model

show significant differences in complexity and

aesthetic score under different initial parameter

configurations. For example, the patterns generated

by the parameter settings numbered 2-5 have a high

level of complexity and aesthetics, as well as a high

level of stability scores. This fully shows that the

algorithm model can generate high-quality and stable

fractal art patterns under the setting of these

Algorithmic Modeling Technology for Computer-Aided Fractal Art Pattern Design

209

Table 1: Results for this simulation

Simu

lation

numb

Initial

param

eter

config

uration

The

num

ber

itera

tions

Sca

ling

The

comp

lexity

of the

gener

ated

patter

Stab

ility

scor

Aest

hetic

score

1 Weigh

t 0.5,

offset

0.1

100 0.8 middl

high midd

2 Weigh

t 0.7,

offset

0.2

200 0.6 high high high

3 Weigh

t 0.3,

offset

0.3

150 0.9 low mid

dle

low

4 Weigh

t 0.6,

offset

0.15

250 0.7 high high midd

5 Weigh

t 0.8,

bias

0.05

300 0.5 Extre

mely

high

high Extre

mely

high

parameters, The typing results of this pattern are

shown in Figure 2.

Figure 2: Color and shape classification of artistic patterns

The color and shape of artistic patterns are

classified as a whole and the results are relatively

remarkable. The simulation results show that under

the reasonable parameter configuration, the

computer-aided intelligent algorithm model in this

study can be Effectively generate complex and

beautiful fractal art patterns, and can provide great

support and verification basis for subsequent system

applications, The typing of the verification art pattern

was recorded, and the results are shown in Table 2.

Table 2: Parameters and evaluation of fractal pattern

generation for complex natural landscape design tasks

Parameter

configurati

The

number

iteratio

Scalin

Complexi

ty score

Visual

beauty

score

Weight

0.65, bias

0.12

180

times

0.75 high Extreme

ly high

Weight

0.7, offset

0.1

220

times

0.8 Extremel

y high

high

Weight

0.6, offset

.18

140

times

0.65 middle middle

As can be seen from Table 2, in the design task of

complex natural landscape, when the weight is 0.7,

the offset is 0.1, the number of iterations is 220, and

the scale reaches 0.8, the resulting fractal art pattern

has a very high complexity and aesthetic score.

4.3 Computer-Aided Fractal Art

Pattern Effects

It can be seen that different parameter skin positions

have a significant impact on the complexity and

visual beauty of fractal art patterns. Moreover, in the

design of "complex natural landscape", the

configuration with a weight of 0.7 and an offset of 0.1

has the best performance when the number of

iterations is 220 times and the scale is 0.8, and the

complexity and visual beauty of the pattern have

extremely high scores, the specific typing structure

and modeling design are shown in Figure 3.

Figure 3: The contrast design results of three art pattern

typing

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The data analysis in Figure 3 can know the overall

contrast design of artistic patterns, which is relatively

reasonable, indicating that computer-aided

technology can achieve the corresponding calculation

in one sentence. The artistic pattern is

comprehensively judged, and the results are shown in

Table 3, but the shape and color contrast are in

contrast. In the task of designing the fractal art pattern

of the "Future Technology Network", the high-

weight, multi-iteration configuration brings the most

complex and aesthetic patterns. This shows that in

fractal pattern design, a higher number of iterations

and a lower offset value can effectively enhance the

complexity of the pattern and its visual impact. This

shows that increasing the number of iterations

moderately and configuring the parameters

reasonably will significantly improve the artistic

effect of the fractal pattern design, the contrast and

data analysis results during the analysis process are

shown in Table 3.

Table 3: Data for the design tasks of the future technology

network

Parameter

configurati

The

number

iteratio

Scalin

Complexi

ty score

Visual

beauty

score

Weight

0.8, offset

0.05

300

times

0.5 Extremel

y high

Extreme

ly high

Weight

0.75, offset

0.08

260

times

0.6 high high

Weight

0.7, offset

0.1

220

times

0.7 middle middle

As can be seen from Table 3, in the fractal art

design of the "Future Technology Network", when

the parameters are configured to be equal to 0.8, the

bias is equal to 0.05, the number of iterations is 300,

and the scale is 0.5, the generated pattern has a very

high complexity and aesthetic score. A higher number

of iterations and a lower offset value will produce a

complex fractal art pattern with a strong visual impact

in the style of modern science and technology.

5 CONCLUSIONS

In this paper, intelligent algorithms are used to

successfully solve many challenges in the design of

fractal art patterns, such as the lack of complexity and

artistic expression. The research in this paper shows

that intelligent algorithms, as a computer-aided

technology, have significant advantages in generating

complex and beautiful fractal art, and can effectively

improve the design details and overall visual effect of

fractal art patterns. In the process of experiments, this

paper finds that the application of reasonable

parameter configuration and optimization will

significantly improve the complexity and artistic

beauty of the pattern, and provide powerful intelligent

algorithm support for the further development of

modern art and science visualization. In addition, the

results of the study further prove that intelligence

The energy algorithm has practicability in

different design tasks, showing its wide application

prospects in high-end art pattern design. However,

although the research in this paper has achieved

positive results, it still has limitations, so it can be

further optimized in the future.

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