Fractal Dimensional Analysis of Building Facades: The Case of Office Buildings in Erbil City

Abstract

Fractal dimension is a characteristic parameter used to measure the complexity and irregularity of geometric shapes and patterns. It is applied in architecture to explore complexity and irregularity and to assess the aesthetic preferences in architectural design. Office building facade design pattern, as an observation unit, has a positive connection with the aesthetic value. This study aims to evaluate facade design styles in terms of two aesthetic qualities, visual complexity and visual diversity, via applying fractal dimension to three design styles of office building facades in Erbil City. The study uses a combination of qualitative and quantitative evaluations to achieve this goal. It employs box-counting analysis through the ImageJ plugin to FracLac and the mathematical perplexity equation to evaluate visual complexity and diversity. The results indicate that the neo-classical office facade style, with a visual complexity value of 1.7008 and visual diversity of 21.27, presents an elevated level of aesthetics similar to the saccadic pattern facade. This study concluded that a neo-classical architectural style for office building facades is the most aesthetically preferable. Modern facade design is considered a secondary architectural style aimed at achieving aesthetic value. Ultimately, the high-tech style is the least attractive facade style. This study contributes to avoiding designs of unattractive office building facades due to a lack of architectural design vocabulary while avoiding overly complex designs that prove visually upsetting for viewers.

1. Introduction

The fractal dimension visually represents the degree of roughness, indicating the amount of texture that exists in an object. It also shows a sharp increase in the length of a fractal from a single repetition to the subsequent one. Moreover, the fractal dimension is not an integer, contrasting with dimensions in Euclidean geometry. The complex structures of clouds, blood vessels, coastlines, or mountains appear to possess boundless complexity; nevertheless, they exhibit a geometric regularity and scale consistency [1].

Aesthetics is historically recognized as a philosophical branch that addresses the concept of beauty. Moreover, beauty has been a subject of intellectual investigation for Western philosophers since ancient times. The term ‘beautiful’ indicates anything that has generally been considered “aesthetically pleasant” or “(visually) attractive” [2]. The purpose of aesthetics in design is to cause feelings of excitement and creativity in a viewer through their visual experience of a building [3].

A strong positive connection has been seen between the dimensional value of fractal forms and the perceived roughness and complexity of the pattern in building facades. The large number of fractals in nature has inspired many research investigations to explore the correlation between the fractal characteristics of patterns and their related visual characteristics [4,5,6,7]. Interestingly, the classification performance was optimal for fractal characteristics with dimensions near natural physical surfaces, suggesting that the visual system’s sensitivity may align with the statistical distribution of nearby fractal regularity [4].

Additionally, fractal images are generally recognized for their immediate and significant aesthetic appeal [8]. Sprott’s pioneering empirical study involved a set of around 7500 weird attractions (computer-generated fractal pictures plotted on a plane), which were evaluated by eight observers using a five-point scale for aesthetic appeal. The images exhibiting a fractal dimension ranging from approximately 1.1 to 1.5 were deemed the most visually pleasing [9]. Moreover, the 443 photos deemed most aesthetically pleasant by observers exhibited an average fractal dimension of 1.30. The study is consistent with previous research about the aesthetics of fractals, which has concentrated on fractal scaling parameters and their influence on visual attractiveness [10,11].

The formal aesthetics of architectural design are exemplified in the building’s facade [12,13,14]. In addition, the building facade is a significant visual element that indicates the value, including the formal aesthetic value and development of a building. The facade of the building can significantly influence the observer’s perception of its image and architectural superiority [15]. Moreover, it can impact a city’s or even an area’s image even on a larger scale [16,17].

This study focuses on the office building facade as a visual observation unit. Despite multiple studies on the analysis of office building facade design, the investigation of these facades regarding aesthetic preference remains unaddressed. This research aims to examine the fractal dimension analysis of office building facades concerning two aesthetic attributes: complexity and diversity. Accordingly, this study focuses on the use of box-counting analysis to determine the ideal aesthetic beauty via visual complexity. In addition, this study examines the measurable facade design vocabularies of office buildings that express the visual diversity of office buildings’ facade design via a mathematical formula of perplexity.

2. Literature Review

Various studies have addressed the fractal examination of the built environment. First of all, fractal analysis of urban designs has been conducted at a macro scale for nearly two decades. In 1994, Michael Batty and Paul Longley were the pioneers in using Benoit Mandelbrot’s ‘box-counting method’ in urban sciences [18]. Ref. [19] expanded the scope of its possible applications to architecture. Ref. [20] has examined the fractal dimension of several urban layouts while [21] has computed the fractal dimension of street patterns in 20 distinct cities. Furthermore, it has been utilized by [22] to analyze the attributes of streetscapes.

Furthermore, In the study by [23], an approach that can guide architectural design and the production of architectural forms in a computer environment is proposed by using the principles in the fractal configuration of the elements in the form dictionary of a certain architectural language. More recently, ref. [24] conducted interdisciplinary research on the significance attributed to visual recognition systems and architectural facades. Visual recognition and facades were analyzed using fractal analysis. In addition, ref. [25] conducted a study on the fractal texture of cities and asserted that modernist cities lost their fractal traditions and significance. A work area named City Laboratory has been established in his research, and a formulation was built through fractal analysis. Moreover, ref. [26] indicated that a study on perception indicates visual interpretation. Indeed, ref. [27] uses an advanced computational fractal analysis method to describe the visual layering (the hierarchical relationship between form, ornamentation, and materiality) of the Süleymaniye Mosque’s four facades and two facade features. The research gives measurable facts for the first time to validate the theorized qualities of this historic architecture. This is done as a reflection of many notable experts who believe Sinan’s elevations have a single formal layering design.

The study by [28] investigates possibilities for supporting through two measuring techniques based on the first appearing in architecture by Bovill (1996). The box-counting approach quantifies the fractal dimension of facades, analyzing the rhythm of design and the scaling of buildings. In the same vein, ref. [29] examines the facade of the Berlin–Baghdad railway station buildings in Turkey using the fractal dimension approach through their program for subjective analysis. Upon completing the studies, it is shown that despite differences in design, scale, and architectural style, an identical proportion exists in the facades of the station buildings. Following a recent study on architecture, ref. [30] harmonized golden ratios with Hausdorff dimensions to determine the existence of a beauty code within logarithmic formulas on building facades or internationally pixelated architectural drawings. As a result, the expected dimensions were less than 2. Ref. [31] does not contest the validity of the planar evaluation; nonetheless, he has consistently replaced geometrical proportion with scaling proportion, claiming that the latter enhances complexity, whilst the first lacks evidence to support it. Similarly, the study by [32] shows the trend of a building’s fractal dimension concerning observation distance to enhance the understanding of its visual complexity. The fractal dimension of a building, as a general multifractal pattern, fluctuates with different observation scales. A standard method for calculating a building’s fractal dimension pattern includes image-processing software for the background extraction and recognizing edges, as well as a box-counting computation tool. The technique uses a visual complexity map to evaluate a building’s visual complexity. On the other hand, ref. [33] believes that fractal patterns in nature may cause a sense of creative points in people, suggesting a comparison to Euclidean patterns. It analyzes pattern-oriented cognition in historical buildings. The study by [34] investigates the possibility of using two computer techniques to analyze potential links between visual stimuli and sensations in facade design. The first approach, fractal analysis, is utilized to comprehensively assess the visual stimuli of a design. The findings indicate that the fractal dimension values of a facade considered correspond with pre-attentive processing and attention pattern simulations. Comparably, ref. [35] highlight questions on the usefulness of fractal geometry in characterizing architectural designs. This study examines complexity-related aesthetic characteristics in architecture, utilizing the measurability of fractal dimension by concentrating on two-dimensional elevation drawings. This research indicates that future systems must incorporate many ways to assess balanced aesthetic complexity in design.

In a recently updated work, the study by [36] investigates facades developed with fractal geometry, which mathematically defines complexity, self-similarity, and the transition from micro to macro scales. In summary, a facade system may adjust to changes by applying a fractal design and permitting daylight and solar regulation within the interior. Interestingly, ref. [37] examines the fractal geometry of the facade level in historically public buildings inside the city center of Afyonkarahisar using the box-counting method. Fractal analysis was conducted using the box-counting approach to examine occupancy/vacancy ratios created by building components, including windows, doors, jambs, floor erasures, buttresses, and eaves that form the facade design. In the same vein, ref. [38] argues that fractals can be employed to differentiate visual beauty and provide objective analysis. Fractals, particularly in facades, have been employed to evaluate complexity and aesthetic appeal in intricate geometry. They focus on the application of fractal geometry and dimension index to analyze the aesthetic appeal of traditional building facades. This is due to their strong faith that the possible application of fractals in classical architecture deserves further investigation. Lastly, ref. [39] in their latest research, examine the aesthetic classification of architecture by analyzing the interplay of complexity and diversity, thereby uncovering mathematical and compositional principles that characterize Palladian architecture.

In the breadth of studies on building facades, several studies have been done on building facades in Erbil City. Interestingly, ref. [40] aimed to examine the optimal double-skin facade arrangement for enhancing the energy efficiency of office buildings in Erbil City. In the same vein, ref. [41] attempted to determine the degree to which contemporary mosques in Erbil include the golden ratio in the aesthetics of their facades, a combination of quantitative and qualitative analysis methods. Moreover, ref. [42] attempted to evaluate the influence of facade design on visual pollution by examining which design factors most contribute to visual pollution on Peshawa-Qazi Street (100 m) in Erbil City. Figure 1 provides more clarification on the previous work about fractals in architecture.

Although many research have addressed the architectural design characteristics of office building facades, studies related to fractal dimensions, especially exploring the design of office building facades and the architectural characteristics that reflect the aesthetic quality of different design styles of these building facades, have not yet been addressed in Erbil City. This study aims to evaluate the aesthetic preferences of three styles of office building facades. The evaluation involves two aesthetic attributes, visual complexity and visual diversity, grounded in fractal dimension theory. This will be achieved by a mixed-method analysis, quantitatively via applying the box-counting technique using the ImageJ version 2024 plugin to the FracLac simulation program. Moreover, it evaluates the architectural facade design aspects of office buildings qualitatively via architectural drawings to achieve the framework of architectural vocabularies that will be used in the calculating perplexity mathematical formula to evoke a visually attractive response to the observer in Erbil City.

Hence, this study will raise multiple questions as follows:

• To what level of complexity in architectural design compositions for office building facades is aesthetically acceptable?
• To what extent is diversity in the architectural composition of the office building facade design more aesthetic?
• What aspects of architectural facade design are visually attractive to the viewer?

Research Objectives:

• To evaluate the degree of complexity in facade compositions for obtaining the level of aesthetic preferences.
• To explore the level of aesthetic preference through diversity in the facade architectural design composition of office buildings in Erbil City.
• To find which type of architectural facade design aspects is more or less visually attractive to the viewer.

3. Theoretical Aspects of Fractal Aesthetics in Architecture

3.1. Fractal Geometry in Architecture

The interest in fractals has emerged in the fields of physics, geography, artwork, and philosophy since the late 1970s. The French mathematician Mandelbrot introduced the concept of fractals in the 1970s [43]. In his article “How Long is the Coast of Britain?”, through statistical self-similarity and fractional dimension, he highlighted that most natural objects are so complex and unpredictable that they cannot be adequately characterized by basic primitives [43]. In their perspective, the coastline represents a fractal, showing exact or statistical self-similarity [44,45,46]. In the 1980s, it was examined as an aesthetic geometric theory [47,48]. Currently, fractals are utilized across diverse domains, including music, fine arts, literature, and architecture, where they are employed by innovative architects to develop novel design methodologies [43,49].

Mandelbrot’s fractal geometry may clarify the complex details present in various natural forms. Fractal geometry examines mathematical shapes that show an endless, self-similar complexity upon closer study [50]. Architecture, while based on simple geometric shapes such as squares and circles, typically shows up as a complex overall form that is frequently challenging to describe using simplistic vocabulary [19]. Accordingly, fractal geometry seeks to understand nature’s complicated phenotypic systems, and fractal analysis can also express aesthetic complexity in architecture [33]. Moreover, fractals effectively represent nature due to their fundamental mathematical patterns and their capacity to include basic characteristics such as roughness, self-similarity, and difficult complexity. In addition to various geometric types generated from the continuous replacement of the starting segment, new fractal objects have developed, such as the Cantor set, Koh curve, Seirpinski triangle, Koch snowflake, Seirpinski square, and Mandelbrot set [51,52], as shown in Figure 2 to provide more clarification.

3.2. Fractal Geometry and Its Relation to Aesthetics in Building Facades

The concept of beauty in architecture and its discussion on aesthetic preferences about building design has been examined since ancient times, with Vitruvius’ book on architecture serving as the earliest example. Numerous theoretical studies in recent decades have examined aesthetics in architectural design, seeking to discover fundamental elements and resources for creating aesthetically pleasing or harmonious buildings and facades.

The facade design is a fundamental architectural component that includes more than simply the building’s exterior appearance. Facades can influence the overall aesthetics, functional aspects, and environmental performance of a building [53]. The architectural facade, or external face of a building, is the first physical interface most people engage with as they approach a building, providing important visual clues about its functionalities. People unconsciously see and continuously experience building facades and their elements (geometry, patterns, textures, and details) [34,54].

Consequently, it can be argued that the fractal analysis of architecture is appropriate for categorizing architectural forms based on visual attributes. In addition, this represents a somewhat new perspective on architectural quality [31]. Consequently, it is predictable that the relationship between the beauty of nature and iconic buildings can be discovered by examining their fractal properties [55]. In a similar vein, the aesthetic quality in architecture necessitates uniform design principles across various scales; thus, analytical methods associated with fractal geometry serve as effective tools for quantifying and visualizing characteristic relationships, considering both the overall form and its complex parts; this includes the uniformity of roughness and the degree of repetition in proportions [55]. Consequently, the aesthetic quality of architecture is associated with fractal attributes of roughness, self-similarity, and scale invariance [35]. In architecture, fractals have been used successfully to create interesting patterns for building facades, especially when viewed from a distance [56]. Fractal analysis can be utilized to articulate aesthetic complexity in architecture, suggesting its suitability for defining architectural visual aspects [36]. Consequently, it is unsurprising that the relationship between the beauty of nature and iconic buildings such as Islamic architecture can be discovered by examining their fractal properties [55,57].

3.3. Fractal Dimension and Its Characteristics

The fractal dimension (Db) is a quantitative measurement of the amount of complexity in the illustrated texture. Natural forms and rhythms, including leaves, tree branching, mountain ridges, river flood levels, wave patterns, and nerve impulses, show this evolution of self-similar structure [28]. Architecture and design focused on rhythm regulation may derive advantages from this relatively novel mathematical tool [58]. Consequently, an increasing number of scientists are adopting the fractal dimension (or Hausdorff dimension [58,59]) as the metric to characterize the extent of self-similarity and complexity in geometry. The fractal dimension indicates the dimension of an object’s self-similarity [60]. Additionally, it provides an opportunity to compare buildings based on their features of visual complexity. C. Bovill (1996) presented two specific approaches for assessing fractal dimensions in architecture.

3.4. Most Effective Fractal Aesthetic Variables in Building Facades

Additional theoretical studies of relevance have concentrated on fractal geometry design principles that range across emotions, including the pleasure, excitement, and tranquility of people. Moreover, it can be enhanced by specific architectural design elements such as complexity, order, style, human scale, diversity, and invariance [61]. In addition, the expressive aspects of architecture suggest that buildings contain social values, which can be perceived through the reflection of our personal experiences [37].

Aesthetic preferences in high-rise buildings have reflected multiple design characteristics, and two contrasting sets of factors have been addressed: (a) primary factors, which include balance, regularity, simplicity, unity, emphasis, scale, flatness, proportion, repetition, color, materials, and style; and (b) distinctive factors, which includes asymmetry, complexity, looseness, movements, silence, confidence, transparency, consistency diversity, scale, depth, randomness, color, materials, and solids/voids [14,62]. In accordance, the aesthetic quality of architecture is associated with fractal attributes such as roughness, self-similarity, and scale invariance [33]; in addition to golden ratios, it is proportional systems and geometric complexity [63,64]; moreover, irregularity, self-similarity, scale-invariance, and iterated function systems refer to the repeating of a set of geometric transformations such as scaling, rotating, and translating; and lastly, geometric patterns. All were defining the fractal geometry properties [64,65]. Figure 3 will provide more detail.

Hence, the current research will focus on the architectural facade aesthetic quality, which is linked to the fractal characteristics of (a) visual complexity and (b) visual diversity.

1.

Visual complexity: Visual complexity in a facade architectural design that relates to the complexity and multitude of architectural components. Multiple variables may contribute to the complexity of facades, including the arrangement of shapes, materials, colors, scale, textures, and patterns [66]. Numerous research has highlighted the influence of environmental visual complexity on aesthetic preferences. A study on urban facades identified complexity as the primary element affecting preferences for building facades [35,67,68]. In addition, the complexity of an architectural facade might be increased or decreased by changing many features, such as the quantity, form, relationships, and surface materials of building components. Also, complexity, as a subcategory of facade shape, has been identified as an indicator of visual preference [69,70]. In sum, visual complexity represents a design variable that determines the preferences for building facades. The most effective approach is referred to as the box-counting method [19]. This approach allowed C. Bovill to illustrate the differences in complexity between the principal facade of Robie House by Frank Lloyd Wright and that of Villa Savoye by Le Corbusier [71].

▪

What is Box-Counting?

Box-counting is an approach for data analysis aimed at evaluating complex patterns by subdividing the image into progressively smaller, often box-shaped segments and then examining these segments at each decreasing scale [19,72]. The concept was first applied to digital arts by Mandelbrot, but it may now be utilized to examine certain architectural patterns. Natural items have greater fractal dimensions than manmade objects; hence, architecture with a larger fractal dimension is more suitable for human habitation than architecture with a lower fractal dimension [73,74,75,76].

In addition, this method originates in and is utilized for fractional and multifractal analyses. The facade of a building changes its ‘effective’ dimension, transforming from a two-dimensional outline facade to a dynamic reality facade after closeness. The building looks like a point or line from afar, while details such as windows and facade elements become visible with a closer look [36]. From a theoretical perspective, the objective of counting squares is to determine the fractional scale; however, from a practical viewpoint, this requires that the divisions of squares follow a pre-established scale, as the square count is computed mathematically through the following iterative procedure:

• Overlaying a grid of square boxes onto the image (s1);
• Enumerating the number of boxes that include a portion of the image (n(s1));
• Iterating this technique by changing (s1) to a smaller grid size (s2);
• Re-evaluating the quantity of boxes containing the image (n(s2));
• Iterating this technique by progressively reducing the grid sizes of s.

The formula for calculating the fractal dimension is the following [74,75,77]:

$$\mathrm{Db}=\frac{\left[\log \right(N\left(s2\right))-\log (N\left(s1\right)\left)\right]}{\left[\log \right(\frac{1}{s2})-\log (\frac{1}{s1}\left)\right]}$$

(1)

Db: fractal dimension

• N—is the number of boxes in each box grid which contains part of the structure;
• 1/s—is the number of boxes across the bottom of the grid, such as the unity size;
• s1—The first boxes cover parts of the image;
• s2—The second smaller boxes cover parts of the image.

The scaling range of an object, and consequently the grid size for the box-counting method, is associated with the characteristics of visual perception. Detailed features can be focused within a two-degree range (2°) from the center of the studied item, called foveal vision. However, notable details are also recognized from angles of 10, 15, and 20 degrees [78]. Figure 4 provides more detail.

Each scale preserves a specific distance from the observed item, which the equation can articulate: the distance from the building (viewing distance) multiplied by the tangent of the angle equals the size of the measuring unit [77]. However, there is a difference between focusing on a specific element and observing the entirety from the same distance. In the first instance, examining certain details, such as a brief line in an abstract picture or the doorknob of an entrance, will also draw our focus to the surrounding things. However, the second one focuses on greater detail or the entirety, such as the building’s height, which diminishes the perception of small details [19]. Figure 5 provides more detail.

The fractal value obtained by this method is always between 1 and 2, and it is stated that as you approach 1, the fractal is simple, plain, and based on Euclidean geometry, and as the fraction dimension is close to 2, the fractal is complex and irregular [19]. Accordingly, the human scene and the fractal dimension have a very close relationship and a five-point classification rating for human perception types has been confirmed by studies on the value of the fractal dimension, more detail in

Table 1. Details on the type of human scenes and their fractal dimension range value.

Type of Perception	Fractal Dimension Range Value in (Box-Counting)
Not attractive	Less than 1 represents very poor facade design elements [79]
Stress reduction	The best facade designs for stress reduction have fractal dimensions between 1.3 and 1.5 [80]
Aesthetical	1.5–1.7 [80]
Pattern of saccadic	Rapid eye movement during object evaluation forms a fractal dimension; the range is between 1.5 and 1.9 [80]
Visual complex and not stable	More than 2 [79]

.

2.

Visual Diversity: This refers to the richness and variety of architectural design elements, patterns, shapes, appearance, and visual characteristics in the building facades, as well as the observational diversity of elements reflecting as an aesthetic concept in facade designs [61,81,82]. Entropy can quantify an object’s visual diversity or variety, indicating the unpredictability level involved. Moreover, entropy can be employed to mathematically represent visual diversity and entropy [83]. Theoretically, the perplexity of a probability distribution might be a better index for the diversity of a system.

The perplexity of probability (PP) can be determined by the frequency of a design component or element [39,83]. Consequently, every relevant detail regarding identifiable design features in a facade image is computed by determining the perplexity of probability, as defined by the exponentiation of entropy [84,85], through the product of each probability raised to the power of its negative value.

The formula for calculating the perplexity probability (PP) [39,84] is as follows:

$$PP=\prod _{i-1}^n{Pi}^{-pi}$$

(2)

where (p) is the probability of observing a specific event (i). (n) is the number of vocabularies.

4. Materials and Method

4.1. Case Study Selection Strategy

The current study selected three office buildings in Erbil City to fulfill its objectives and address its research questions. These buildings are highly comparable to one another in terms of built-up area, elevation length, and building height. Likewise, specific case studies were chosen based on the differences in their facade architectural design style. The architectural facade design was clarified as follows:

1.

Modern Facade Style

The modernist style of building facades is characterized by the use of geometrically motivated clean facades made of materials like glass, steel, and concrete [86,87].

2.

High-Tec Facade Style

Glazed cladding refers to a modern, high-tech, and technologically advanced facade style. The core values of the high-tech facade style use large glass panels to create transparency and natural light besides aesthetic performance [86,87].

3.

Neo-Classic Facade Style

The neo-classical style of building facades draws inspiration from classical architecture to meet the current requirements [88].

Hence, the primary information and the basic drawings of three different styles of office buildings in Erbil City is illustrated in

Table 2. Basic information of case study.

Information	Plan
Elevation	Image
#C1 Korek Telecom HQ Center Net Floor Area: 2450 m2 Elevation Length: 70 m Building Height: 49.5 m Modern Facade Style
#C2 Asia Cell Office Center Area: 1580 m2 Elevation Length: 41 m Building Height: 39 m Modern close to High-Tec Facade Style
#C3 Phoenix Office Building Area: 1610 m2 Elevation Length: 52 m Building Height: 58 m Neo-Classical Facade style

.

4.2. Research Method

The approach to achieving the objectives of the current research depends on the mixed-method research method and utilizing various approaches, including drawings, graphs, tables, and photographs. Figure 6 shows the research methodology choices.

Hence, the current study combines qualitative and quantitative methods to broaden our proof and increase the credibility of our findings. It is based on a convergent design, which is a form of mixed-methods research in which the authors collected quantitative and qualitative data simultaneously and analyzed them separately. Figure 7 clarifies the qualitative and quantitative method sections.

In addition, the details of the mixed-methods research have been outlined below:

4.2.1. Qualitative Method

This approach provides information to the case study through a detailed drawing analysis using the Auto-Cad version 2023-drawing tool and photos to illustrate the chosen facades’ patterns, shapes, and architectural components.

4.2.2. Quantitative Method

Firstly, the visual complexity calculation: This approach will depend on the new mathematical techniques named box-counting for obtaining fractal dimensions in the facade.

This method aims to address the primary objective of the current research, which relates to the visual complexity of facades and its direct correlation with aesthetics in facade design. In addition, the process of box-counting will be in three conservations steps or three different distances for facade observation. The steps are as follows:

• Applying the 8 × 8 grid in the first step over the facade (Figure 8a);
• By using a scale factor of 0.5 or √2, the grid will be 4 × 4 (Figure 8b);
• Repeating the grid with a scale factor of 0.5 or √2 with dimensions 2 × 2 (Figure 8c).

The current study integrates ImageJ software with the FracLac plugin to compute the fractal dimension, consequently clarifying the visual complexity of the facades.

Secondly, the visual diversity calculation: This approach will depend on the mathematical formula named perplexity for obtaining diversity in the vocabulary and architectural elements in the facade via applying mathematical formula (2), as mentioned before.

In the first step of analyzing the visual diversity of the facades, the current study extracted an arrangement of architectural elements utilized in the three selected facades as architectural vocabulary. Moreover, the architectural elements identified in the case study facades as architectural vocabulary include arcature, cornice, balcony, column, windows and openings, cantilever, and various facade decorations [36], excluding colors and textures.

Table 3. Description of the most available facade vocabulary encodings in the case studies. (Authors).

Category	Code	Vocabulary Description
Entablature	En.	An entablature is a horizontal element in classical architecture placed above columns.
Arcature	Ar.	It involves a series of connected arches, all supported by columns or piers, adding elegance to structures without serving a structural purpose.
Cornice	Cr.1	It indicates a horizontal mass, which may be simple, external, or placed behind the glass.
	Cr.2	It is the fully decorated edge of the roof where it connects with the outer wall.
Balcony	B.1	Large-size suspended balcony.
	B.2	Smaller size suspended balcony.
Column	Clm.	Engaged columns are partially constructed facing a wall, not freestanding, and extending more than half from the wall.
Walls	W.1	Wall with large-size arched windows that are used as a gate, too.
	W.2	Walls with fenestration pattern awning windows.
	W.3	Walls with casement windows.
	W.4	Walls with single windows.
	W.5	Walls without windows.
	W.6	Glazed cladding wall.
Shapes	GS	Geometric shapes to create a standout facade.
	PS	Parametric shapes to create a standout facade.
Cantilever	Cn	A cantilever is a rigid part of construction that extends horizontally over doors and windows.
	N = 16

Note: Red box related to the codes of vocabulary.

offers further details for interpreting the architectural vocabularies.

5. Results and Discussion

5.1. Visual Complexity Results and Discussion

The current study is based on using ImageJ with its plugin FracLac for the fractal dimension analysis (Db). In length, the analysis was done in three series with different box sizes: (1) large boxes with 8 × 8 units for each pixel; (b) medium-sized boxes with 4 × 4 units for each pixel; (3) small sizes with 2 × 2 units for each pixel. In addition, one grid position is randomly added to the orientation around the corner location by the FracLac simulation program by using building facade images with a resolution of 498 × 485. Moreover,

Table 4. Box-counting analysis results for fractal dimension (Db).

	Series 1	Series 2	Series 3
#C1: Korek Telecom HQ Center
	Db = 1.2732	Db = 1.5234	Db = 1.6138
	Fractal Dimension (Db) Average = 1.4702
#C2: Asia Cell Office Center
	Db = 1.1928	Db = 1.3004	Db = 1.5665
	Fractal Dimension (Db) Average = 1.353
#C3: Phoenix Office Building
	Db = 1.5438	Db = 1.7466	Db = 1.8202
	Fractal Dimension (Db) Average = 1.7008

provides a detailed investigation of the result of the box-counting analysis.

5.2. Visual Diversity Results and Discussion

A vocabulary for office building facades was defined, and the results of encoding the architectural composition and design attributes of a facade have been developed to assess visual diversity. The probability of each code is utilized to calculate the diversity score via the perplexity (PP) calculation. The total architectural vocabulary derived from the three chosen samples was n = 16. The complexity of the three facades was examined and measured, as expressed in

Table 5. Case studies perplexity analysis.

Cases	Facade Visual Diversity Analysis
	Perplexity Calculation
#C1: Korek Telecom HQ Center
	Total architectural vocabulary (n = 16); The number of the available architectural vocabulary in the Phoenix office is (p = 4) as follows: GS., Cr.1, W2, W3 The number of repeated vocabularies is (i). According to Formula (2), the perplexity will be calculated as follows: $PP=P_{\mathit{GS}}^{-\mathit{pGs}}\times P_{\mathit{Cr}.1}^{-\mathit{pCr}.1}\times P_{W2}^{-pW2}\times P_{W3}^{-pW3}$ $PP={\frac{2}{16}}^{-\frac{2}{16}}\times {\frac{18}{16}}^{-\frac{18}{16}}\times {\frac{2}{16}}^{-\frac{2}{16}}\times {\frac{12}{16}}^{-\frac{12}{16}}$ PP = 2.08
#C2: Asia Cell Office Center
	Total architectural vocabulary (n = 16); The number of the available architectural vocabulary in the Phoenix office is (p = 4) as follows: W2, W5, Cr.1, Cn The number of repeated vocabularies is (i) According to Formula (2), the perplexity will be calculated as follows: $PP=P_{\mathit{Cr}}^{-\mathit{pCr}}\times \times P_{W2}^{-pW2}\times P_{W5}^{-pW5}\times P_{\mathit{Cn}}^{-\mathit{pCn}}$ $PP={\frac{16}{16}}^{-\frac{16}{16}}\times {\frac{1}{16}}^{-\frac{1}{16}}\times {\frac{1}{16}}^{-\frac{1}{16}}\times {\frac{2}{16}}^{-\frac{2}{16}}$ PP = 1.84
#C3: Pheonix Building Office
	Total architectural vocabulary (n = 16); The number of the available architectural vocabulary in the Phoenix office is (p = 11) as follows: Ar., W4, W3, W2, W1, Cr.1, Cr.2, B.1, B.2, En., Clm. The number of repeated vocabularies is (i) According to Formula (2), the perplexity will be calculated as follows: $PP=P_{\mathit{Ar}}^{-\mathit{pAr}}\times P_{W4}^{-\mathit{pW}4}\times P_{W3}^{-\mathit{pW}3}\times P_{W2}^{-\mathit{pW}2}\times P_{W1}^{-\mathit{pW}1}\times P_{\mathit{Cr}.1}^{-\mathit{Pcr}.1}\times P_{\mathit{Cr}.2}^{-\mathit{pcr}.2}\times P_{B.1}^{-\mathit{pB}.1}\times P_{B.2}^{-\mathit{pB}.2}\times P_{\mathit{En}}^{-\mathit{pEn}}\times P_{\mathit{Clm}}^{-\mathit{pclm}}$ $PP={\frac{6}{16}}^{-\frac{6}{16}}\times {\frac{2}{16}}^{-\frac{2}{16}}\times {\frac{2}{16}}^{-\frac{2}{16}}\times {\frac{1}{16}}^{-\frac{1}{16}}\times {\frac{6}{16}}^{-\frac{6}{16}}\times {\frac{2}{16}}^{-\frac{2}{16}}\times {\frac{2}{16}}^{-\frac{2}{16}}\times {\frac{2}{16}}^{-\frac{2}{16}}\times {\frac{2}{16}}^{-\frac{2}{16}}\times {\frac{1}{16}}^{-\frac{1}{16}}\times {\frac{6}{16}}^{-\frac{6}{16}}$ PP = 21.27

.

6. Findings

The objective of this research is to evaluate the aesthetic preferences of office building facades regarding two aesthetic attributes, complexity and diversity, and is grounded in fractal dimension theory. In addition, this study attempts to evaluate and extract the most attractive type of facade design based on the mixed-method analysis. Moreover, this study focuses on the use of box-counting analysis to determine the ideal aesthetic beauty via visual complexity. In addition, this study examines the measurable facade design vocabularies of office buildings that express the visual diversity of office buildings’ facade designs. The research raised the following three questions to achieve its objectives.

Firstly, to what extent is the degree of complexity in office facade compositions aesthetically preferred? The research answered this subject through quantitative analysis utilizing the box-counting method via ImageJ with the FracLac plugin for fractal dimension analysis (Db). The average fractal dimension or visual complexity range for the Korek Telecom Quarter Center, which exemplifies modern facade design, is 1.4720. This result indicates stress reduction, according to Table 1, which defines the fractal dimension range values related to facade perception. Similarly, the average fractal dimension of Asia’s modern high-tech office center facade complexity is 1.353, which contributes to a sense of stress reduction for the observer. Finally, the Phoenix office building is classified as neo-classical, built at a very late period, having a facade fractal dimension value of approximately 1.7008. According to Table 1, the average (aesthetical) range in facade design is between 1.5 and 1.7; hence, the neo-classical office facade design achieves a heightened level of aesthetics, similar to the saccadic pattern facade Figure 9.

Secondly, this study asked to what extent is the visual diversity in office building facade architectural design vocabularies considered more aesthetic. As a response, this study extracted a set of architectural features from the three selected facades. Additionally, the architectural components recognized in the case study facades as architectural vocabularies include arcature, cornice, balcony, column, windows and openings, cantilever, and diverse facade decorations. The total number of architectural vocabularies was n = 16. The visual diversity of the chosen case study was computed using the perplexity mathematical formula. Accordingly, having four of the sixteen architectural vocabularies available, the Korek Telecom HQ Center had a perplexity level of 2.08. Similarly, the Asia Cell Office Center uses four different vocabularies in its facade design, resulting in a perplexity level of 1.84 out of a total of 16. Notably, both the Korek Telecom headquarters and the Asia Cell center, which are characterized by modern and modern high-tech exterior styles, exhibit a simplistic facade design lacking visual diversity, as both contain minimal architectural vocabulary according to the overall vocabulary. On the contrary, the Phoenix office building, classified as neo-classical, incorporates eleven out of sixteen various vocabularies, giving a perplexity score of 21.27, suggesting considerable visual diversity in the exterior design.

Thirdly, this study inquired as to which type of architectural facade design aspects are more or less attractive visually to the viewer. In response to this query, this study integrated the results of both visual complexity and visual diversity across different case study types.

Table 6. Sequence of most attractive facade styles.

Case Studies	Facade Style	Visual Complexity	Visual Diversity	Level of Attractive Facade
#C3	Neo-Classical style	1.7008	21.27	1	Most attractive
#C1	Modern style	1.4702	2.08	2	Moderate attractive
#C2	Modern High-Tec	1.353	1.84	3	Least attractive

indicates the hierarchy of the most attractive and aesthetically pleasing facade designs for office buildings in Erbil City.

It is interesting to note that the Phoenix office building, which is of a neo-classic office style and has a high degree of aesthetic preferences, has the most attractive facade design when it comes to the idea of utilizing a variety of architectural elements without creating visual complexity. The second level of fractal aesthetics, notably different from the neo-classical style, is exemplified by the modern facade design of the Korek Telecom headquarters office center. Lastly, the Asia Cell office building, which has modern architecture that is strongly linked to high-tech concepts, was chosen as the least attractive aesthetic preference in this study.

In sum, the analysis of the data obtained from three distinct facade design styles in Erbil City indicates a variety of architectural characteristics. This study emphasizes that the fractal dimension, relating to visual complexity and diversity, could reflect the aesthetic value of the facades at various levels for observers.

7. Conclusions

The fractal dimension, which significantly increases in extent with repetition, represents the complexity and roughness of items. Despite the complexities displaying scale consistency and geometric regularity, it differs from Euclidean geometry dimensions. Moreover, the term ‘aesthetics’ indicates anything that has generally been considered visually attractive. The literature indicates a strong positive correlation between the dimensional value of fractal forms and the perceived visual complexity and diversity of patterns in building facades. The facade of a building can greatly affect the observer’s perception of its aesthetic and architectural beauty. This study aimed to assess the aesthetic preferences of office building facade design styles concerning two aesthetic attributes, such as complexity and diversity and was based on fractal dimension theory. The objective was to evaluate the visual complexity rating value and to explore the visual diversity value in facade architectural design compositions to indicate the level of aesthetic preferences of different facade design styles, as well as to find which type of architectural facade design style aspects are more or less visually attractive to the observer.

Three questions arose from the study to achieve the goals. To obtain these goals, the current study used mixed-method techniques. Firstly, architectural drawings were presented to describe the selected case study qualitatively. Secondly, the study used the quantitative approach, such as the box-counting technique via the ImageJ plugin for the FracLac simulation program. Furthermore, it analyzed the architectural facade design elements of office buildings in Erbil City by utilizing drawings to show variations in visual diversity through the mathematical perplexity formula.

The results show that the average fractal dimension or visual complexity range of the Korek Telecom Quarter Center, which represents the modern facade design style, leads to reduced stress. The typical fractal dimension of the facades of Asia’s modern high-tech office centers leads to a sensation of stress relief for the observer. The Phoenix office building is categorized as a neo-classical design style of facade design, attaining a higher visual appeal that is close to a saccadic pattern facade. Moreover, regarding visual diversity, the Korek Telecom headquarters exhibited a moderate degree of perplexity. Likewise, the Asia Cell office center received an extremely low visual diversity score. Accordingly, both the Korek Telecom headquarters and the Asia Cell center, distinguished by their modern and modern high-tech exteriors, exhibit a basic facade design devoid of visual diversity since both facades have minimal architectural vocabulary concerning their total design vocabulary. Conversely, the Phoenix office building, categorized as a neo-classical facade style, shows a high degree of complexity, indicating significant visual diversity in its exterior design.

Furthermore, the Phoenix office building, distinguished by its neo-classical architectural style and exceptional aesthetic value, displays the most attractive facade design by effectively incorporating diverse architectural components without reaching visual complexity. The second level of fractal aesthetics is illustrated by the modern facade design of the Korek Telecom headquarters office center. Ultimately, the Asia Cell office building, characterized by modern architecture, was closely associated with high technology and was identified as the least attractive choice in this study.

This study uncovers that integrating varied architectural characteristics may enhance the facade’s aesthetic value, even if it does not achieve a high level of complexity. Similarly, applying diversity and keeping a range of visual complexities will influence aesthetic choices in the design and pattern of office building facades. Conversely, the simplicity of design facades will indicate a lack of sensitivity to the perceptions of viewers. Furthermore, this study proposes further research on the multifractal analysis of building masses to address the entirety of the building’s geometry to demonstrate how the complexity of geometry masses influences receivers’ perceptions.