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Article

Three-Dimensionalization Mediates the Subjective Experience of Fractal Interior Spaces

by
Joori Suh
1,
William Stalker
2,
Steen Pedersen
3 and
Assaf Harel
2,*
1
School of Architecture and Interior Design, University of Cincinnati, Cincinnati, OH 45221-0016, USA
2
Department of Psychology, College of Science & Mathematics, Wright State University, Dayton, OH 45435-0001, USA
3
Department of Mathematics, College of Science & Mathematics, Wright State University, Dayton, OH 45435-0001, USA
*
Author to whom correspondence should be addressed.
Architecture 2024, 4(3), 651-667; https://doi.org/10.3390/architecture4030034
Submission received: 7 June 2024 / Revised: 6 August 2024 / Accepted: 20 August 2024 / Published: 27 August 2024
Browse Figures
Versions Notes

Abstract

:
A fractal, a self-similar organic or geometric pattern that repeats at varying scales, is one of the most compelling characteristics found in nature. Previous studies on fractal patterns have demonstrated consistent trends in potential psychological benefits, such as stress reduction. However, we fall short of understanding one of the essential properties of fractals found in nature, i.e., the three-dimensionality of their appearance. In this study, we aimed at understanding the role of the three-dimensionalization of fractal patterns in spatial structures on human subjective perceptual experience. Two hundred seventy three-dimensional spatial prototype models were created for this study, spanning two dimensions: (1) the application of spatial depth (shallow; medium; deep) and (2) fractal complexity (low; medium; high). The participants rated each space on six psychological dimensions (bad–good; stressful–relaxing; ugly–beautiful; boring–interesting; leave–enter; ignore–explore). Significant effects of the application depth of fractals were observed primarily for “boring-interesting”, “ignore-explore”, and “leave-enter” dimensions and were primarily manifested within spaces with medium and high D-values. The results suggest that spatial depth plays a significant role in individuals’ experiences of fractal spaces, arguably by making the space more engaging and interesting.

1. Introduction

1.1. Background

The profound characteristics of nature have been sources of perpetual inspiration and motivation. To artists and designers, nature has been used as an attractive source to inspire creativity and imagination, primarily due to nature’s aesthetic quality. Humans’ strong connection with nature was captured by Edward Wilson’s [1] biophilic hypothesis stating each human’s inherent desire to connect to nature. With the discovery of the psychological benefits of exposure to nature’s scenery [2], scholars formed a theory known as the Attention Restoration Theory (ART), which posits that nature’s regenerative power lies in its ability to reverse the negative impacts of everyday stress and mental fatigue [3]. The various psychological benefits embedded in the property of naturalness have been widely reported addressing cognitive gains, attentional restoration, and a positive mood [4,5,6,7,8,9]. This has led to many attempts at identifying the specific principles embedded in nature that may possess positive benefits on human wellbeing. One key principle found in nature is fractal complexity, which refers to self-similar organic, geometric, or abstract patterns that repeat at varying scales. Previous studies exploring humans’ response to fractals were focused primarily on physiological stress response, aesthetic response, preference, and interest [10,11]. One recurrent finding of these studies is that the mid-range of fractal complexity is especially preferred over lower and higher levels of fractal complexity [12,13,14,15] and exhibits significant stress-dampening effects [16,17].
Currently, there is a growing interest within the spatial design community (interior design and architecture) in implementing evidence-based scientific insights to the process of designing spaces. However, despite the reported psychological benefits of fractal patterns, most of the research on their positive impact has been limited to two-dimensional graphics (such as the silhouettes of trees, coastlines, or mountains), thereby neglecting one of the essential properties of fractals in nature, namely their volumetric essence. Indeed, it is still an open question whether the volumetric manifestation of fractal spatial patterns in our three-dimensional real world has the same effect on human behavior as two-dimensional fractals, and since the fractals originally found in nature are mostly three-dimensional (e.g., clouds, tree branches, forests, mountains), a comprehensive account of the impact of fractals on human behavior must determine the role that spatial depth plays when applying fractals in a three-dimensional environment. Moreover, most of the previous research has focused heavily on stress reduction or preference, and research on other perceptual dimensions concerning human response to fractals is still scarce.
To address this gap in the existing literature, in the current study, we examine what role the three-dimensionalization of fractal patterns plays in various human subjective perceptual experiences of fractal spaces and especially focus on cognitive/emotional and behavioral/motivational dimensions. The study paves practical ways to apply fractal patterns in designing spaces in order to accommodate diverse programmatic needs.

1.2. Fractal Complexity

The natural environment is filled with numerous elements that display fractal attributes, such as cloud formations, tree branches, and mountains, to name a few examples. Previous studies consistently report humans’ aesthetic preference of fractals [11,18], regardless of how the fractals were generated, including through nature’s processes, through mathematics, or by human hands [14]. Taylor (2021) explains humans’ preference of fractal visual patterns in accordance with the concept of fractal fluency [17]. According to the fractal fluency model, human vision has become fluent in viewing fractal features within patterns in nature and therefore can process the features effortlessly [17] (p. 2). In mathematics, a fractal is a class of complex regular or irregular shapes that repeat at varying scales. The term fractal, derived from Latin fractus (“fragmented”), was first coined by a Polish-born French-American mathematician Benoit B. Mandelbrot (1989) in his pursuit of theorizing order and beauty in roughness. In explaining the concept of fractal geometry or “fractal geometry of nature and chaos” [19] (p. 4), Mandelbrot considers a fractal to be a middle ground between the Euclidian geometric order and geometric chaos. His initial motivation to develop fractal geometry was to find a theory useful in explaining the complexity found in nature, which “fails to be locally linear” [19] (p. 4). A key characteristic of fractals is self-similarity, which is found in exact fractals, where patterns are repeated at multiple precise scales, as well as in statistical fractals, where only the pattern’s statical qualities repeat. In understanding the fractal nature of the environment, the property of a statistical fractal is crucial, as a greater degree of randomness is prevailing in nature.
To elucidate the complexity of a self-similar pattern, German mathematician Felix Hausdorff (1918) introduced the concept of fractal dimension (D-value), a rational statistical index of the complexity or irregularity of an object. A D-value quantifies the fractal scaling relationship among the patterns within the scene in terms of how the patterns at different scales are accumulated together into the resulting fractal image [20] projected on the retina [21]. The D-value ranges between 1 and 2. As noted above, the mid-range of fractal complexity (about 1.3–1.5) is often rated as more preferrable [12,13,14,15] and also exhibits stress-dampening effects [16,17]. The seminal research conducted by Wise and Rosenberg [22] discovered that an individual’s physiological response to stress is reduced by viewing an abstracted nature-inspired image, which later was identified by Taylor (2006) to be a mid-range fractal [23]. Following research has demonstrated consistent findings on the benefits of fractals in the mid-range [12,13,14,15]. Recently, humans’ response to more specific fractal properties has been an ongoing focus of research. For example, Hägerhäll and colleagues (2015) showed that the natural (statistical) form of fractals induced stronger alpha-band EEG responses (an index of a wakefully relaxed state and internalized attention) compared to exact fractals [16], findings that were in line with ART ideas on the impact of nature on attention. Studies on the preference of varying degrees of fractal complexity have shown that the preference for the mid-range D-value is the highest, although recent studies also highlight the effects of higher fractal dimension on humans. To illustrate, Abboushi and colleagues (2018) found that when fractal images were projected on room surfaces, such as walls, instead of being displayed on a computer monitor, images with medium to medium–high fractal complexity (D: 1.5–1.7) were preferred [24].

1.3. Application of Fractals in Spatial Design

In terms of the way a fractal structure is formed in most cases in nature, the primary unit (i.e., a branch of a tree, a part of a cloud, etc.) is repeated in a three-dimensional space, thus creating a spatial volume as a whole. Likewise, spatial design involves a three-dimensional composition of various components, and therefore, in applying fractal properties to spatial design such as architecture and interior design, it is critical to understand how this unique property of fractal’s three-dimensionality influences human perception. In the field of architecture and interior design, designers have employed fractal characteristics to design spaces, even before fractals were elaborated mathematically. For instance, tracery details of gothic architecture repeat within large and small windows and in furniture, changing the scale of the same or similar patterns. The American architect Frank Lloyd Wright (1867–1959) incorporated multiple variations of geometric shapes collectively within spaces by manipulating and adjusting the selected geometries at varying scales (e.g., Palmer House). Some contemporary examples employ random statistical fractals (e.g., building envelop design at Australian Center for the Moving Image), expressing a greater degree of randomness.
When fractal characteristics are applied to a three-dimensional space, unlike two-dimensional graphics, a dynamic interplay between the spatial components and three-dimensionality of the space is created. In a built environment, the perspectival effect naturally induces perceptual scale variations, causing elements of the same size to gradually appear smaller as they recede from a viewer. The perspectival effect also makes the fractal nature of the overall scene captured in an environment more random and complex by causing the perceptual size of the components to constantly change as the viewer moves within the space. Moreover, spatial dimensions such as openness, closedness, and ceiling height affect human visuospatial perception [25,26,27,28].

1.4. Perceptual Response to Fractals

Previous studies that explored humans’ response to fractals primarily focused on physiological stress responses, aesthetic responses, preference, and interest [10,11]. Recent research on humans’ perception of fractals has expanded from focusing on stress responses and aesthetic preferences to various related psychological dimensions. For instance, a recent emotional preference arousal study on fractal-based design [18] shows the judgment of engagement increases with increasing D-values, whereas the ratings for relaxing showed the reverse tendency. The researchers concluded that fractal preference is formed by “a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment)” [18] (p. 1).
As the attention to psychological wellbeing shifts from a concern for humans’ struggles to an emphasis on helping humans thrive in their environment, a change supported by positive psychology [29], an extensive understanding of the psychological benefits of exposure to fractals in relation to engagement, positive emotion, and motivation will be of great value. With this endeavor, Coburn et al. [30] evaluated perceptual responses to architectural interiors, using the three important domains of psychological processing of cognition, emotion, and behavior [31]. They employed 16 constructs within their aesthetic rating scale, including beauty, interest, valence, stimulation, vitality, comfort, relaxation, uplift, approachability, and explorability. Among them, three constructs (beauty, valence, and relaxation) that are related to the cognitive and emotional dimensions and three constructs (interest, approachability, and explorability) that are related to behavior and motivation are particularly important for the current study. These six constructs were selected as dependent measures for the current study, as they capture what can be described as the fundamental emotional and perhaps even more importantly behavioral–motivational states of the viewers. Beauty, defined as visual attractiveness for this study, is a primary psychological dimension that expresses the viewer’s desire to live in a certain space [32]. Valence is the degree to which the architectural interior space “makes an occupant feel good or bad” [30] (p. 220) and is closely related to preference and liking [33]. Relaxation is the state of being free from tension and has been a main focus of research on individuals’ perceptual response to their environment [34] as well as humans’ response to fractal images [23,35,36]. Interest is the feeling of wanting to know or learn about something and is a primary motivating factor in one’s decision to experience unfamiliar or new things [37]. Approachability (i.e., the intention to approach or avoid) is considered to be a driving factor for consequent behavioral responses occurring within a space: for instance, the action of purchase in a retail space [38]. Because the approach is the first step of coming closer to a space where intended purpose is integrated, understanding this behavioral–motivational dimension in spatial design has a significant value. Explorability is a quality of traveling in or through an unfamiliar space to learn about the space. The quality of explorability is closely related to mystery, one of the most important dimensions of natural elements since it involves a place where new information is inferred in the scene [39] and creates a promising reality that new information could be acquired by moving deeper into the space.
As noted above, 3D fractal spaces evoke a dynamic interplay between the spatial components and three-dimensionality of a space. Introducing perceptual scale variations, the fractal nature of the overall scene is more random and complex, constantly changing the perceptual size of the components as the viewer moves within the space. As an initial step to understand the full spectrum of the effect of the three-dimensional aspects of fractal spaces, the present study aims to examine how the three-dimensionalization of fractal patterns (i.e., spatial depth application of shallow, medium, and deep) modulates various human subjective perceptual experiences of fractal spaces of varying complexity using three-dimensional (3D) fractal-inspired interior space simulated models displayed on a computer screen. We hypothesized that integrating different levels of the spatial depth of fractal patterns in the spatial structure will have an effect on people’s subjective perceptual experience responses to fractal-inspired interior spaces, focusing on the key six psychological constructs mentioned above: the three cognitive/emotional dimensions of beauty, valence, and relaxation and the three behavioral/motivational dimensions of interest, approachability, and explorability.

2. Materials and Methods

2.1. Participants

One hundred sixty-four volunteers were recruited from the Wright State University community. Participants were compensated in exchange for a course credit. All participants reported normal or corrected-to-normal vision and had no history of a current or past neurological or psychiatric illness. They signed an informed consent approved by the Wright State Institutional Review Board. Incomplete, duplicate, and data flagged for insufficient effort responding (IER) (see [40,41] for guidelines) were not kept (see details below). The remaining 57 participants’ data were then further analyzed. Ages ranged from 18 to 51. Participants included 31 females, 24 males, and 2 others (17:12:1 in the first group, 14:12:1 in the second group). While we explicitly avoided collecting additional demographics for privacy reasons, one can safely assume that the participant pool represents a small public university in the Midwest area of the United States.

2.2. Stimuli: Simulated Fractal-Inspired Spatial Structures

To create simulated prototype models as stimuli used for this study, we applied various fractal patterns, including geometric, organic, and abstract patterns, to enclosures of 12′ × 12′ × 12′ spaces. The prototype models were produced using the key characteristics of fractals outlined in the literature reviewed in 1.2. Fractal Complexity: self-similarity, recursion, scale variation, and complexity. The criteria used to generate fractal-inspired patterns in the structure include the following: (1) self-similarity: similar geometric, organic, or abstract patterns are used within the structure; (2) recursion: the patterns are repeated within the structure; (3) scale variation: scale variations of the repeated patterns are observed in the front view of the structure; and (4) complexity: low, medium, and high levels of complexity are created by repeating the patterns. The structures were designed to make them appear as a functioning unit of semi-private reading spaces, resembling an interior shell structure that can be placed in a large open interior space, such as a library or an office (see Figure 1 for examples). In this study, we used application depth (AppDepth for short) as an operational term for the three-dimensionalization of fractal patterns applied within a volumetric space. Unlike the mathematical generation process of 3D fractals, such as Mandelbulb, the three-dimensionalization of fractal patterns used in this study focused on transforming two-dimensional fractal shapes into three-dimensional forms by adding volumetric quality and spatial depth. In this study, we are interested in the practical application of fractal nature in a real-world setting rather than mathematically generating Mandelbulb-like structures that would most likely not be applicable in reality. A total of 270 three-dimensional spatial prototype models were created, spanning two dimensions: (1) application of spatial depth (AppDepth: shallow, medium, and deep) and (2) fractal complexity (D-value: low (1–1.3), medium (1.31–1.55), and high (1.56–2)). For the calculation of fractal value (D-value), each front view image was converted in black and white, and a box counting method programmed in Python was used. The focus of this study was to examine how three-dimensionalization of fractal patterns (i.e., spatial depth application of shallow, medium, and deep) modulates human subjective perceptual experience of fractal-inspired interior spaces. For the fractal structures of shallow spatial depth, fractal patterns were applied to the interior surface of the structure. For the fractal structures of medium spatial depth, various fractal patterns were applied in three dimensions at the perimeter of the structure by adding thickness to the patterns or adding three-dimensional layers to the surface. Less than 6 inches of depth of the shell structure was used to increase the medium level of spatial depth to the patterns. For the deep spatial depth application, various fractal patterns were integrated within the field of the 12′ × 12′ × 12′ space three-dimensionally. To avoid the impression of closure within the shallow application, parts of the walls or ceilings were kept open. Standard functioning interior elements, such as lighting fixtures, tables, and chairs, were placed inside the structure of the shallow and medium spatial depth application. In the simulated models within deep spatial depth application, the functioning elements of lighting fixtures, tables, and chairs were integrated within the structure in a manner that was complimentary to the fractal nature of the overall structure. Shade was applied to each image (ratio: 1080 × 1080 pixels) to make the three-dimensional scene realistic, but shadow was turned off to avoid visual distractions.
Twenty different colors were selected, including three variations of blue, green, yellow, and pink; one purple; two wood tone colors; and five neutral colors (white, grey variations, and black). To avoid color-related confounding factors, each of the above hues was used at least two times in the stimuli set of each condition (i.e., a combination of the two experimental factors). In an effort to equalize the distribution of colors, we conducted a color ratio analysis: All the images in each condition were combined within one field, and the ratio of the colors was calculated based on the selected 20 colors. No single hue exceeded 10% of the overall application in each condition. The resulting images were categorized within nine conditions: 30 images in LS (D-value: low; App-Depth: shallow), 30 images in LM (D-value: low; App-Depth: medium), 30 images in LD (D-value: low; App-Depth: deep), 30 images in MS (D-value: medium; App-Depth: shallow), 30 images in MM (D-value: medium; App-Depth: medium), 30 images in MD (D-value: medium; App-Depth: deep), 30 images in HS (D-value: high; App-Depth: shallow), 30 images in HM (D-value: high; App-Depth: medium), and 30 images in HD (D-value: high; App-Depth: deep). The stimuli were further divided into two equivalent sets (see details below).

2.3. Experimental Design

The experiment was delivered using Qualtrics, an online research study platform (Qualtrics.com, Provo, UT, USA). The experiment was administered on the participants’ personal computers via Wright State University’s research participation system. No mobile devices, such as smartphones or tablets, were allowed, and they were disabled by the Qualtrics software (Qualtrics, Provo, UT, USA, version 2021). Participants were randomly assigned to two groups, and each group was presented with one of two sets of images derived from the full image set. The reason the initial set of 270 images was divided into two equivalent sets was to reduce participant fatigue and the likelihood of careless responses [41]. Accordingly, each set consisted of 135 images spanning all nine conditions (15 individual exemplars in each of the nine conditions). Participants viewed all 135 images in succession and were asked to rate each individual image on each one of the six subjective perceptual dimensions: relaxation, valence, beauty, interest, approachability, and explorability. The stimuli order within each set was pseudorandomized for each administration to minimize potential order and sequence effects.

2.4. Subjective Measure of Perceptual Experience

Six subjective measures were selected for the experiment based on the 16 dimensions of psychological responses to architectural interior spaces introduced by Coburn et al. [30]. The selected variables for this study obviously do not encompass the full spectrum of human emotions related to the experience of a space. Furthermore, additional cognitive or cultural factors influence the interpretation of symbols and patterns. However, the rationale for selecting the current measures was to establish some limits to the study, and hence, we selected three scales related to cognitive and emotional dimensions (beauty, valence, and relaxation) and three scales related to motivation and potential behavior (interest, approachability, and explorability). For each scale, end anchors that best represent the concept in each spectrum were used with appropriate rating prompts on a 1-7 Likert scale (Table 1).

2.5. Experimental Procedures

The participants began the experiment upon signing a digital consent form. They were then instructed on their task, followed by their first trial. In each trial an exemplar image was presented, and below it appeared a 7-point Likert scale prompt. After the participant gave his/her rating, the next scale automatically appeared below the image where the previous scale had been. A new image was presented only after the participant had completed all six of the scales for a given image. This continued until all 135 images within the set were rated across all six scales. The participants were then thanked for their time and instructed to close the browser tab.
The survey presentation was designed so that the image and scale were always presented together, preventing the participants from ever having to scroll. The automatic progression after question completion also reduced the number of actions/clicks required, saving time and reducing the potential for fatigue. Participants, however, were allowed to go back to a previous question to change their answer if they desired to do so. Each scale was set to forced choice to reduce the amount of incomplete data. The order of scale presentation was consistent throughout the experiment to reduce potential confusion.

2.6. Data Analysis

A three-way Analysis of Variance (ANOVA) was conducted separately for each of the six subjective perceptual dimensions with the following independent variables: fractal dimensionality (D-value: low/medium/high), AppDepth (shallow/medium/deep), and version (first or second version of the image sets). No significant effects (either main effects or interactions; p < 0.05) of version were noted on the ratings in any of the scales and therefore were not further analyzed.
Incomplete, duplicate, and data flagged for IER were not included in the analysis beyond the step of initial data cleaning. Insufficient effort responding (IER), also commonly known as content non-responsivity, characterizes the practice of responding without regard to item content. Potential indicators of IER include consistently responding with the same answer (“1, 1, 1, 1, 1, 1, 1”) or a patterned approach (“2, 3, 2, 3, 2, 3, 2, 3”). High ratings of IER warrant potentially removing participants [40,41]. Participants whose data IER score was 50% or more were removed in order to avoid methodological confounds and preserve research survey integrity.

3. Results

3.1. Relaxation

Relaxation ratings of the fractal spaces (“This space makes me feel…” with Stressed and Relaxed at the opposite ends of the continuum) were significantly impacted by the fractal complexity of the designed spaces (the main effect of the D-value: F(2,110) = 42.72, p < 0.001, partial eta squared = 0.44). Post hoc comparisons showed participants found the spaces with a low D-value (M = 4.36, SE = 0.10) as equally relaxing as the medium level (M = 4.23, SE = 0.08), with the latter significantly more relaxing than the spaces with a high D-value (M = 3.39, SE = 0.12) (all significant pairwise comparisons between consecutive levels). The AppDepth also had a significant impact on relaxation ratings (F(2,110) = 3.82, p < 0.03, partial eta squared = 0.06), with the shallow level rated (M = 4.06, SE = 0.09) as equally relaxing as the medium level (M = 4.02, SE = 0.08), with the latter significantly more relaxing than the deep application level (M = 3.90, SE =0.09) (all significant pairwise comparisons between consecutive levels). However, the impact of AppDepth was moderated by the D-value (significant interaction effect between the D-value and AppDepth: F(4,220) = 7.13, p < 0.001, partial eta squared = 0.11). Post hoc pairwise comparisons between consecutive AppDepth levels (p = 0.05) conducted separately for each D-value level revealed a more subtle and complex impact of the AppDepth on relaxation ratings (see Table 2 for the full descriptive statistics and Figure 2 for graphical representation). At the low D-value level, participants rated the shallow applications as more relaxing compared to the medium level, which was not rated statistically different from the deep level. At the medium D-value level, no significant difference was observed between the shallow and the medium AppDepth levels, but now, the latter was rated as more relaxing than the deep AppDepth level. Finally, at the high D-value level, participants rated the shallow applications as less relaxing compared to the medium level, which did not statistically differ from the deep level.

3.2. Valence

Ratings of the valence of the fractal spaces were significantly impacted by the complexity of the fractal patterns of the designed spaces (the main effect of the D-value: F(2,110) = 16.87, p < 0.001, partial eta squared = 0.23). Post hoc comparisons showed that participants ranked the medium (M = 4.40, SE = 0.07) level of D-value as higher in valence compared to either the low (M = 4.19, SE = 0.10) or high (M = 3.82, SE = 0.11) D-value.
While the AppDepth on its own did not have an impact on valence ratings (the main effect of the AppDepth: (F(2,110) < 1.00), it did have an impact once the separate levels of the D-value were considered (significant interaction effect: (F(4,220) = 11.49, p < 0.001). Post hoc pairwise comparisons between consecutive AppDepth levels (p = 0.05) conducted separately for each D-value level revealed a significant yet varied effect of the AppDepth on valence ratings (see Table 2 for the full descriptive statistics and Figure 3 for graphical representation). At the low D-value, participants rated the shallow applications as higher compared to the medium level, which did statistically differ from the deep level. At the medium D-value level, no significant difference was observed between either of the AppDepth levels. Finally, at the high D-value, participants rated the shallow applications as lower compared to the medium level, which did not statistically differ from the deep level.

3.3. Beauty

Beauty judgments of the designed spaces were also influenced by the complexity of the fractal patterns applied to them (the main effect of the D-value: F(2,110) = 12.00, p < 0.001, partial eta squared = 0.18). Post hoc comparisons showed participants ranked the spaces with a medium (M = 4.41, SE = 0.10) D-value level as more beautiful than spaces with both the low (M = 3.95, SE = 0.11) and high (M = 3.81, SE = 0.14) levels of fractal dimensionality (all significant pairwise comparisons between consecutive levels). While the AppDepth on its own did not have a significant impact on beauty judgments (the main effect of the AppDepth: F(2,110) = 1.60, p < 0.21, partial eta squared = 0.03), its impact was revealed once the D-value was taken into account (interaction effect between the D-value and the AppDepth: F(4,220) = 11.22, p < 0.001, partial eta squared = 0.17). Post hoc pairwise comparisons between consecutive AppDepth levels (p = 0.05) conducted separately for each D-value level showed the following: At the low D-value level, participants judged the shallow applications as more beautiful than the medium-level images, which did statistically differ from the deep-level images. At the medium D-value level, no significant difference was observed between the different AppDepth levels. At the high D-value level, participants rated the shallow applications as less beautiful compared to the medium-level images, which were not judged differently than the deep-level images (see Table 2 for the full descriptive statistics and Figure 4 for graphical representation).

3.4. Interest

Participants’ interest level in the images was influenced by both the D-value (the main effect of the D-value: F(2,110) = 74.71, p < 0.001, partial eta squared = 0.58) and the AppDepth (the main effect of the AppDepth: F(2,110) = 45.40, p < 0.001, partial eta squared = 0.45). Post hoc comparisons showed participants ranked the high (M = 5.22, SE = 0.13) D-value as significantly more interesting than the medium D-value (M = 5.06, SE = 0.09), which in turn was rated as significantly more interesting than the low (M = 3.81, SE = 0.11) D-value (all significant pairwise comparisons between consecutive levels, p < 0.05). The deep AppDepth level was rated as significantly more interesting (M = 5.13, SE = 0.12) than the medium level (M = 4.61, SE = 0.09), with the latter significantly more interesting than the shallow AppDepth spaces (M = 4.35, SE = 0.10). The two dimensions had a joint effect on participants’ interest ratings (the interaction effect between the D-value and the AppDepth: F(4,220) = 9.13, p < 0.001, partial eta squared = 0.14), showing that the impact of the AppDepth was not entirely uniform across the three D-value levels: At the low D-value, participants evaluated the deep applications as more interesting than the medium applications, which in turn were considered to be of equal interest as the shallow applications. Conversely, in both the medium and high D-value, the deep application was more interesting than the medium application, which was judged to be more interesting than the shallow application (see Table 2 for the full descriptive statistics and Figure 5 for graphical representation; all post hoc pairwise comparisons between consecutive levels, p = 0.05).

3.5. Approachability

The decision of participants for whether to leave or enter the presented spaces was significantly influenced by the fractal complexity of the spaces (the main effect of the D-value: F(2,110) = 9.21, p < 0.001, partial eta squared = 0.14) and the depth by which these fractals were applied (the main effect of the AppDepth: (F(2,110) = 4.95, p < 0.02, partial eta squared = 0.08). Participants voiced a greater interest in entering the medium complexity spaces (M = 4.58, SE = 0.11) compared to the low complexity spaces (M = 4.23, SE = 0.13) on the one hand and the high complexity spaces on the other (M = 4.02, SE = 0.15) (all significant pairwise comparisons between consecutive levels, p < 0.05). The shallow AppDepth spaces were rated as less approachable (M = 4.15, SE = 0.12) than the medium AppDepth spaces (M = 4.27, SE = 0.11), albeit this preference was not different from their preference to approach the deep AppDepth spaces (M = 4.40, SE = 0.13). Again, the two dimensions interacted in their influence on approach judgments (the D-value and the AppDepth interaction effect: F(4,220) = 10.71, p < 0.001, partial eta squared = 0.16). In the low fractal dimensionality spaces, the effect of the AppDepth was such that the medium level of application was deemed less approachable than either the shallow or the deep spaces. In contrast, in the high D-value level, the medium level of application was deemed to be more approachable than either the shallow or the deep spaces. Interestingly, in the medium D-value level, there was no significant difference between either of the three AppDepth levels (see Table 2 for the full descriptive statistics and Figure 6 for graphical representation; all post hoc pairwise comparisons between consecutive levels, p = 0.05).

3.6. Explorability

The decision of participants on how much they would be willing to explore the presented spaces was also significantly influenced by the fractal complexity of the spaces (the main effect of the D-value: F(2,110) = 11.93, p < 0.001, partial eta squared = 0.18) and the depth by which these fractals were applied (the main effect of the AppDepth: (F(2,110) = 18.56, p < 0.001, partial eta squared = 0.25). Participants were more inclined to explore the medium D-value spaces (M = 4.50, SE = 0.11) compared to spaces with both low and high D-values (M = 3.89, SE = 0.14; M = 4.13, SE = 0.14, respectively) (all significant pairwise comparisons between consecutive levels, p < 0.05). As for the impact of AppDepth in regard to exploration, participants preferred to explore the deep spaces (M = 4.44, SE = 0.12) more than the medium spaces (M = 4.16, SE = 0.12), which in turn were preferred over the shallow spaces (M = 3.93, SE = 0.12). The two dimensions had a joint effect on participants’ exploration judgments (the interaction effect between the D-value and the AppDepth: F(4,220) = 10.73, p < 0.001, partial eta squared = 0.36). In both the medium and high D-value levels, the exploration judgments were higher for the deep compared to the medium and the shallow AppDepth levels. However, in the low D-value level, the shallow level and medium level were judged equally, which in turn were judged as less explorable than the deep level (see Table 2 for the full descriptive statistics and Figure 7 for graphical representation; all post hoc pairwise comparisons between consecutive levels, p = 0.05).

4. Discussion

Recent years have witnessed a growing interest in implementing evidence-based insights to the process of designing spaces. Research on the nature of fractal-related information has especially attracted spatial designers given its biophilic characteristics that have been shown to have beneficial effects on health and wellbeing. Despite this recent interest, however, the exact relationship between the properties embedded in fractal patterns and people’s perception and appreciation of these patterns remains to be determined. The current study aimed to minimize this gap by asking not only how varying the “fractalness” (fractal dimensionality, measured by the D-value) of a fractal pattern impacts its perception but also how the spatial application of these patterns contributes to it. Our rationale was that in future applications of fractals in spaces, designers should not consider fractals as merely flat, ornamental 2D images but rather conceptualize them in the context of design spaces, in which they could vary in the extent of their “spatialization”. We found that both fractal dimensionality and their 3D application play a role in how people perceive fractal-inspired spaces. In fact, our findings raise the following question: when applying fractal qualities to a physical environment, should medium fractal complexity be the goal of the design since it is traditionally believed to be the most desirable for viewers? The results of this study, in line with some of the recent discoveries of fractal response research, imply that the multidimensional characteristics embedded in the fractal nature of different levels of D-values as well as their spatial depth should be considered to maximize the rich potentials of fractal patterns for designing spaces.

4.1. Spatial Depth Application and Behaviroal, Motivational Response to Fractal Spaces

The results of the current study show that people’s subjective perceptual experience of fractal-inspired interior space structures is dependent on their level of spatial depth integration, especially in the behavioral–motivational dimensions, i.e., interest, explorability, and approachability constructs. Significant effects of the application depth of fractals were observed primarily for the behavioral and motivational dimensions and were manifested within the medium and high spatial depth applications. These results imply that the spatial depth embedded in fractal spaces has a direct impact on people’s interactions within fractal spaces, with the depth of application making the spaces more engaging than surface-deep applications. In other words, when applying medium-to-high D-values of fractal patterns, increased depth integration of fractal spatial patterns plays an important role in making the viewers more interested in exploring the surrounding space. This increased level of interest means that the occupants feel motivated to know or learn more about the subject, which results in them choosing to experience unfamiliar or new things [37]. What makes the application of depth contribute to making the spatial fractal structure more interesting and enhancing the desire of exploration among viewers? Arguably, this may be due to the quality of mystery created when the patterns are spatially integrated rather than applied to the surface. In explaining mystery, one of the four components of the environmental preference model, Kaplan (1987) focuses on the way information is displayed and predicted by the viewers [39]: Mystery implies that “there was new information acquired by going deeper into the space… venturing further into the scene would indeed yield additional information” [39] (p. 9). A deeper application or extensive spatial integration of fractal patterns within the spatial structure causes occlusion generated by multiple layering of the segments of the three-dimensional shapes of the pattern and thus may increase such quality as mystery. Because of this quality of mystery, these deep fractal structures may encourage exploration by increasing anticipation for new information that could be acquired by going deeper into the scene. Thus, this quality of mystery created within the spatial fractal structures with increased depth integration is different from one created simply by incorporating fractal patterns on a flat surface. Salimpoor and colleagues’ (2011) study on anticipation and dopamine release found that in listening to music, pleasant anticipation of abstract reward (“a sequence of tone unfolding over time” [42], p. 261) can result in dopamine release. Likewise, the anticipation of a pleasant visual experience created by the quality of mystery in a deep fractal structure may induce pleasure. Our findings thus point to the significant contribution of the spatial depth dimension’s ability to create more interesting fractal-inspired spaces, and it further suggests the importance of spatial designers’ creative role of volumetrically applying fractals when designing real-world environments. Also, spatializing fractal principles may be useful for creating spaces where improving motivation, interest, and curiosity among viewers is a priority.
Compared to the shallow application of fractal patterns, when medium and deep spatial depth application were used, the structure naturally increased the frequency of open gaps between small components in the envelope of the spatial boundary, thereby increasing the properties of prospect and refuge, the concepts introduced by Appleton (1975). The quality of prospect is achieved when the space provides an unimpeded view over a distance for surveillance and planning of action. Clearwater and Coss’s (1991) research shows that the quality of unimpeded view reduced boredom, and pictures with the “greatest apparent depths of field engendered the strongest aesthetic appeal” [43] (p. 343). Medium and deep spatial applications created an open screen-like effect that signaled to viewers that they were able to see their surroundings while staying in a protected area. A refuge is a place for withdrawal from activities with a feeling of protection. According to Mumcu et al. (2010), the quality of prospect is related to people’s preferences, and the concept of refuge is related to the sense of safety [44].

4.2. Spatial Depth Application and Judgement of Beauty

Interestingly, although no significant effect of the application of depth was found in the beauty construct, a significant interaction effect between the D-value and application depth was found. The effect of application depth was especially evident in the high fractal dimension, suggesting that the subjective experience ratings for ugly–beautiful were altered by spatial depth integration for the structures with high D-values: when the highly complex fractal patterns were applied to a spatial structure (with medium and deep spatial depth), participants’ judgments of beauty were higher compared to fractal patterns being applied to the surface. What caused the deep spatial integration of high fractal patterns to be more appealing? In applying fractal principles in three-dimensional volumes of a space, more effort is required to integrate fractal patterns in three dimensions while organizing all the spatial components within the space. This integration may have enhanced an immediate understanding of the space, thereby evoking a sense of coherence, one of the four components of the environmental preference model introduced by Kaplan (1987). Compared to the surface-deep application of highly complex fractal patterns, the comprehensive integration of intricate patterns within the volume of the structure necessitated considerable effort to organize the components in a way that made the space livable and functional. This enhanced the organized appearance of the complex components, consequently increasing aesthetic preference. This result aligns with Lavdas and Schirpke’s (2020) study on aesthetic preference for organized complexity, as defined by a “hierarchy of scales, the presence of local contrast, and an overall coherence” [45] (p. 2).

4.3. Paradox in Subjective Perceptual Response to Fractal Complexity

Notably, we found a significant main effect of the D-value in all six psychological constructs, with different trends in each construct. The medium D-value was rated high in valence, beauty, explorability, and approachability. The judgment of interest increased with a corresponding increase in the D-value, whereas the ratings for relaxation showed a slight reverse tendency. This trend is supported by Robles et al.’s (2021) recent study on fractal-based design, which also demonstrated this paradoxical relationship [18]. Abboushi and colleagues’ research (2018) also confirmed that the participants’ preference shifted from medium to medium–high fractal complexity (D: 1.5–1.7) when fractal images as light–shadow patterns are projected on room surfaces [24]. These results suggest that applying low fractal complexity would be beneficial in spaces where the users desire to relax; however, medium to medium–high fractal complexity can be used to make the space appear more interesting. Providing spaces with differing levels of fractal complexity and giving choice and control to the users within the environment, especially in a public space, would empower the users and allow them to find appropriate spaces to satisfy their multifarious needs. It is also worth noting that the consistent finding of a reverse relationship between interest and relax states (in relation to fractal complexity) can be interpreted that exposure to low to medium fractal complexity may be useful to reduce stress; however, it may also decrease physiological arousal [18].

5. Limitations

While the current study provides novel insights into the question of how spatial fractals are evaluated, it also contains a number of limitations. First, as a preemptive measure, a rigorous data cleaning process was applied to the dataset given the highly repetitive nature of the task, which might have led to a withdrawal effort in some of the participants (see [40,41] for guidelines). This has the undesired consequence of ultimately reducing the sample size. Second, our study used static scenes displayed on a computer screen. However, when a person moves within a spatial structure, the fractal dimension of the scene seen by the viewer may change. Therefore, the fractal dimension used for this study was limited to the front view of the spatial structure. Third, although a color distribution analysis was conducted to ensure that no hue exceeded 10% within each category, the impact of colors was not fully controlled in this study. Further research with a more controlled use of color would be of significant importance. Fourth, the survey was conducted by displaying the scene on the computer screen. Future fractal spatial depth studies that use virtual reality for viewing of the spatial structure inside the space would be of great value. Future studies assessing the temporal dynamics of fractal perception (e.g., EEG studies examining how the human brain responds to varying degrees of the spatial depth application of fractals according to the sequence of exposure to differing fractal dimensions) would provide more detailed information in terms of the effects of applying fractals in spatial design. Lastly, we acknowledge that additional cognitive and cultural aspects involved in interpreting symbols and patterns were beyond the scope of this study. Further studies examining how individual differences in cognitive styles and cultural backgrounds impact the experience of fractal space would offer significant value.

6. Conclusions

The prevalence of mental health concerns urges us to investigate the potential benefits extracted from nature that can be applied to the design process to promote wellbeing in a holistic way. The findings of this study point to one unique aspect of fractals, namely the contribution of the spatial depth dimension. This finding should encourage designers to create more engaging fractal-inspired spaces and suggests the importance of spatial designers’ creative role of applying fractals in three-dimensional real-world everyday space design. The study was meaningful in that it used realistic interior spaces for the stimuli instead of simplified 2D fractal images and graphics. The results show the significant role of depth application in shaping various positive subjective perceptual experiences of the fractal spaces, especially in the motivational and behavioral dimensions. Together, the findings of the current study expand our understanding of fractal complexity by highlighting the multifarious characteristics encompassing a varying range of psychological properties. The study points to the contribution of three-dimensionalization of fractals and spatial designers’ creative role of curating psychological effects pertaining to fractal-inspired design to choreograph a variety of different emotional and behavioral modes of spaces that range from relaxing to explorative. It is our hope that this study inspires designers and manufacturers to proactively apply three-dimensional spatial fractals to spaces in order to ultimately enhance people’s health and wellbeing.

Author Contributions

Conceptualization, J.S., S.P. and A.H.; methodology, J.S. and A.H.; software, S.P.; validation, A.H.; formal analysis, W.S. and A.H.; investigation, J.S., W.S. and A.H.; data curation, W.S.; writing—original draft, J.S. and A.H.; writing—review & editing, J.S., W.S. and A.H.; visualization, J.S. and W.S.; supervision, J.S. and A.H. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Institutional Review Board Statement

The study was conducted in accordance with the Declaration of Helsinki, and approved by the Institutional Review Board of Wright State University (protocol #07054, approved 2021).

Informed Consent Statement

Informed consent was obtained from all subjects involved in the study.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

The authors thank Callie Forsythe, Jenna Weber, and Ngoc Luong who participated in the production of spatial fractal structures for the study.

Conflicts of Interest

The authors declare no conflict of interest.

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Architecture 04 00034 g001
Figure 1. Examples of simulated fractal-inspired interior space structures arranged according to the three levels (shallow, medium, and deep) of spatial depth and three levels (low, medium, and high) of fractal dimension.
Figure 1. Examples of simulated fractal-inspired interior space structures arranged according to the three levels (shallow, medium, and deep) of spatial depth and three levels (low, medium, and high) of fractal dimension.
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Figure 3. Observed mean ratings on the construct of valence organized according to the different levels of fractal dimension.
Figure 3. Observed mean ratings on the construct of valence organized according to the different levels of fractal dimension.
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Figure 4. Observed mean ratings on the construct of beauty organized according to the different levels of fractal dimension.
Figure 4. Observed mean ratings on the construct of beauty organized according to the different levels of fractal dimension.
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Figure 5. Observed mean ratings on the construct of interest organized according to the different levels of fractal dimension.
Figure 5. Observed mean ratings on the construct of interest organized according to the different levels of fractal dimension.
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Figure 6. Observed mean ratings on the construct of approachability organized according to the different levels of fractal dimension.
Figure 6. Observed mean ratings on the construct of approachability organized according to the different levels of fractal dimension.
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Figure 7. Observed mean ratings on the construct of explorability organized according to the different levels of fractal dimension.
Figure 7. Observed mean ratings on the construct of explorability organized according to the different levels of fractal dimension.
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Figure 2. Observed mean ratings on the construct of relaxation organized according to the different levels of fractal dimension.
Figure 2. Observed mean ratings on the construct of relaxation organized according to the different levels of fractal dimension.
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Table 1. Subjective perceptual experience scales and end anchors.
Table 1. Subjective perceptual experience scales and end anchors.
Subjective Perceptual
Experience Dimension		Rating Prompt
(On a Scale of 1–7, …)		Low Anchor (1)High Anchor (7)
RelaxationThis space makes me feel…stressedrelaxed
ValenceThis space makes me feel…badgood
BeautyThis space looks…uglybeautiful
InterestThis space looks…boringinteresting
ApproachabilityIf I saw this space, I would…leaveenter
ExplorabilityIf I saw this space, I would…ignoreexplore
Table 2. Mean rankings in the nine experimental conditions for each of the six subjective perceptual experience dimensions (standard errors of the mean, SEMs, are in parentheses).
Table 2. Mean rankings in the nine experimental conditions for each of the six subjective perceptual experience dimensions (standard errors of the mean, SEMs, are in parentheses).
Subjective Perceptual Application Depth
Experience DimensionsD-ValueShallowMediumDeep
1. RelaxationLow4.55 (0.10)4.31 (0.12)4.21 (0.10)
Medium4.34 (0.10)4.28 (0.08)4.07 (0.10)
High3.28 (0.13)3.48 (0.13)3.41 (0.13)
2. ValenceLow4.34 (0.10)4.05 (0.11)4.17 (0.10)
Medium4.44 (0.08)4.40 (0.08)4.36 (0.09)
High3.62 (0.12)3.93 (0.12)3.89 (0.12)
3. BeautyLow4.03 (0.12)3.86 (0.13)3.95 (0.11)
Medium4.39 (0.10)4.46 (0.11)4.37 (0.12)
High3.52 (0.14)4.00 (0.15)3.91 (0.16)
4. InterestLow3.58 (0.13)3.48 (0.15)4.38 (0.13)
Medium4.70 (0.09)5.03 (0.10)5.46 (0.12)
High4.78 (0.13)5.31 (0.14)5.56 (0.15)
5. ApproachabilityLow4.27 (0.14)4.05 (0.15)4.35 (0.14)
Medium4.52 (0.12)4.63 (0.12)4.58 (0.14)
High3.65 (0.15)4.12 (0.16)4.27 (0.18)
6. ExplorabilityLow3.86 (0.15)3.74 (0.18)4.07 (0.14)
Medium4.28 (0.12)4.51 (0.11)4.72 (0.14)
High3.65 (0.15)4.21 (0.16)4.55 (0.16)
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Suh, J.; Stalker, W.; Pedersen, S.; Harel, A. Three-Dimensionalization Mediates the Subjective Experience of Fractal Interior Spaces. Architecture 2024, 4, 651-667. https://doi.org/10.3390/architecture4030034

AMA Style

Suh J, Stalker W, Pedersen S, Harel A. Three-Dimensionalization Mediates the Subjective Experience of Fractal Interior Spaces. Architecture. 2024; 4(3):651-667. https://doi.org/10.3390/architecture4030034

Chicago/Turabian Style

Suh, Joori, William Stalker, Steen Pedersen, and Assaf Harel. 2024. "Three-Dimensionalization Mediates the Subjective Experience of Fractal Interior Spaces" Architecture 4, no. 3: 651-667. https://doi.org/10.3390/architecture4030034

APA Style

Suh, J., Stalker, W., Pedersen, S., & Harel, A. (2024). Three-Dimensionalization Mediates the Subjective Experience of Fractal Interior Spaces. Architecture, 4(3), 651-667. https://doi.org/10.3390/architecture4030034

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