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Edited by: Rita Berto, Università della Valle d’Aosta, Italy

Reviewed by: Caroline M. Hägerhäll, Swedish University of Agricultural Sciences, Sweden; Judith Heerwagen, General Services Administration (GSA), United States

This article was submitted to Environmental Psychology, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Highly prevalent in nature, fractal patterns possess self-similar components that repeat at varying size scales. The perceptual experience of human-made environments can be impacted with inclusion of these natural patterns. Previous work has demonstrated consistent trends in preference for and complexity estimates of fractal patterns. However, limited information has been gathered on the impact of other visual judgments. Here we examine the aesthetic and perceptual experience of fractal ‘global-forest’ designs already installed in humanmade spaces and demonstrate how fractal pattern components are associated with positive psychological experiences that can be utilized to promote occupant wellbeing. These designs are composite fractal patterns consisting of individual fractal ‘tree-seeds’ which combine to create a ‘global fractal forest.’ The local ‘tree-seed’ patterns, global configuration of tree-seed locations, and overall resulting ‘global-forest’ patterns have fractal qualities. These designs span multiple mediums yet are all intended to lower occupant stress without detracting from the function and overall design of the space. In this series of studies, we first establish divergent relationships between various visual attributes, with pattern complexity, preference, and engagement ratings increasing with fractal complexity compared to ratings of refreshment and relaxation which stay the same or decrease with complexity. Subsequently, we determine that the local constituent fractal (‘tree-seed’) patterns contribute to the perception of the overall fractal design, and address how to balance aesthetic and psychological effects (such as individual experiences of perceived engagement and relaxation) in fractal design installations. This set of studies demonstrates that fractal preference is driven by a balance between increased arousal (desire for engagement and complexity) and decreased tension (desire for relaxation or refreshment). Installations of these composite mid-high complexity ‘global-forest’ patterns consisting of ‘tree-seed’ components balance these contrasting needs, and can serve as a practical implementation of biophilic patterns in human-made environments to promote occupant wellbeing.

Driving nature’s aesthetics, fractal patterns are prevalent across both microscopic and global structures in natural environments (

Furthermore, fractal patterns have the prospect of altering more than just the aesthetic experience of a given object (

To utilize the beneficial effects of natural geometry, the

Whereas most studies of nature’s statistical fractals focus on images of individual objects, typical scenes feature ‘fractal composites’ in which individual objects merge to form an overall pattern. In addition to more closely capturing the essence of nature,

Fractal flights.

Fractal ‘trees.’

Fractal ‘forests.’ The forests integrate the flights of

A second motivation for the ‘bird flight’ composition strategy is that when viewing fractal patterns eye movements have been found to follow fractal trajectories (

One challenge remained. For manufacturing demands, the 6ft (15 cm) by 12ft (30 cm) pattern of

Installations. The fractal pattern of

Previous research demonstrates that visual complexity is a key component in the visual impact of fractals. Compared to the simplicity of Euclidean shapes, the fractal repetition of patterns at different scales results in fractal shapes that are inherently complex. The current series of studies expands upon typical measurement of fractal preference or complexity to address broader perceptual judgments (including ratings of complexity, engagement, preference, refreshment, and relaxation) of these “global forest” patterns and their respective local “tree-seed” patterns that are currently installed in multiple settings with the potential to promote viewer wellbeing (see

We will investigate these varied responses to global forest patterns of differing complexity by conducting studies in two laboratories (one at the University of Oregon in the United States [Experiment 1A] and the other at the University of New South Wales in Australia [Experiment 1B]) using slightly different rating scales as a test of the robustness of these effects. The use of both unipolar and bipolar rating scales is employed to ensure that our measurements are both sensitive enough to detect differences in psychological effects related to the fractal design patterns and generalizable across different measurement conditions. It is hypothesized that both of our rating scales will be able to identify consistent variations in the psychological effects of the various fractal design patterns, thus providing evidence of robust response patterns across measurement types. The goal of these studies is to establish an empirical basis for the optimal selection of fractal designs to meet varying psychological and aesthetic needs of a space (see

We first examined the role of physical complexity and pattern arrangement in determining perceived complexity, engagement, preference, refreshment, and relaxation in ‘global forest’ fractal patterns. Experiment 1A used a series of unipolar slider tasks while Experiment 1B used a series of bipolar slider tasks.

We used the pattern’s fractal dimension

Fractal scaling was confirmed from the minimum pattern size of 0.2 inches (0.5 cm) up to 24 inches (61 cm). The box-counting method cannot confirm fractal scaling at scales larger than 24 inches due to a limited number of boxes at these scales (

Example stimuli used in the Experiments. Fractal ‘forest’ stimuli used in Experiment 1 of differing

To address how the addition of global fractal order may impact the perceptual judgments of fractal patterns, 78 participants comprised of undergraduate Psychology students from the University of Oregon were recruited for the current study through the SONA participant pool system (66 females, age ranging between 18 and 30 years old, mean age 20 years old). Informed consent was acquired following a protocol approved by the Institutional Review Board at the University of Oregon and all participants received class credit for their participation.

This study was generated in PsychoPy3 (

Participants viewed the “global forest” fractal patterns presented in five randomized blocks, with each block consisting of a singular judgment type (complexity, engaging, preference, refreshing, or relaxing). Each block’s stimulus set consisted of 5 unique patterns ranging across 4 levels of complexity or

Data from 78 adult participants (between 18 and 33 years old) were retained from the 130 adults who participated in the experiment. Data were excluded due to: (a) failure to complete the study (6 participants), (b) failure of greater than 3 attention checks (24 participants), or (c) recording the same rating for greater than four consecutive trials. If the same rating was recorded for more than 4 consecutive trials, the entire block of ratings was excluded. Furthermore, if all blocks for a given judgment type were removed, then the participant was excluded (22 participants).

A 3-way repeated measures 4 × 5 × 2 ANOVA [^{2}(5) = 160.41, ^{∗∗}], the interaction between ^{2}(5) = 37.92, ^{∗∗}], ^{2}(77) = 510.44, ^{∗∗}], as well as the three-way interaction between ^{2}(77) = 134.45, ^{∗∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.408, 0.679, 0.365, and 0.672, respectively). Indicated with a double asterisk for significance of ^{∗∗}, 95% CI [0.15, 0.23], η_{p}^{2} = 0.32] and Arrangement emerged [^{∗∗}, 95% CI [0.09, 0.45], η_{p}^{2} = 0.28]. Additional significant interactions were detected between ^{∗∗}, 95% CI [0.43, 0.59], η_{p}^{2} = 0.53], ^{∗∗}, 95% CI [0.06, 0.31], η_{p}^{2} = 0.19], Arrangement and Judgment [^{∗}, 95% CI [0.0, 0.1], η_{p}^{2} = 0.09], as well as ^{∗}, 95% CI [0.0, 0.06], η_{p}^{2} = 0.33]. For illustrative purposes we plot the 3 significant interactions (

Experiment 1A results for “global forest” fractal patterns using a unipolar rating scale. Results show significant 2-way interactions among the experiment’s 3 factors: fractal dimension (

Experiment 1A results for “global forest” fractal patterns for 5 different judgment conditions (how complex, engaging, preferred, refreshing, and relaxing).

Experiment 1A-paired samples

Complex | Engaging | Preference | Refreshing | Relaxing | |

D = 1.2 vs. D = 1.4 | t = −14.31** (d = 0.76) | t = −8.19** (d = 0.66) | t = −3.31* (d = 0.23) | t = 0.71 (d = 0.37) | t = 3.57** (d = 0.24) |

D = 1.2 vs. D = 1.6 | t = −23.26** (d = 2.17) | t = −15.79* (d = 1.7) | t = −6.12** (d = 0.75) | t = 0.42 (d = 0.05) | t = 3.88** (d = 0.51) |

D = 1.2 vs. D = 1.8 | t = −28.47** (d = 3.56) | t = −20.56** (d = 2.77) | t = −6.86** (d = 1.02) | t = 1.24 (d = 0.19) | T = 5.16** (d = 0.79) |

D = 1.4 vs. D = 1.6 | t = −16.51** (d = 1.23) | t = −14.68** (d = 1.2) | t = −6.04** (d = 0.63) | t = 0.01 (d = 0.0) | t = 3.17* (d = 0.34) |

D = 1.4 vs. D = 1.8 | t = −24.42** (d = 2.97) | t = −20.5** (d = 2.4) | t = −6.76** (d = 1.2) | t = 1.25 (d = 0.17) | t = 5.07** (d = 0.69) |

D = 1.6 vs. D = 1.8 | t = −18.38** (d = 1.8) | t = −13.98** (d = 1.25) | t = −5.03** (d = 0.59) | t = 1.91 (d = 0.17) | t = 5.48** (d = 0.45) |

A 2-way 4 × 2 repeated-measures ANOVA [^{2}(5) = 123.06, ^{∗∗}] and the interaction between ^{2}(5) = 15.66, ^{∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.512 and 0.87, respectively). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.82], Arrangement [^{∗}, 95% _{p}^{2} = 0.12], and interaction between ^{∗}, 95% _{p}^{2} = 0.04] were identified. Average complexity ratings (collapsed over pattern arrangement type) ranged from a low of 0.18 (^{∗∗}, 95% ^{∗}, 95%

To determine whether the observed trends could be due to a combination of responses from subgroups of participants, we performed a two-step cluster analysis similar to that used by

A 2-way 4 × 2 repeated-measures ANOVA [^{2}(5) = 114.57, ^{∗∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.521). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.73], Arrangement [^{∗∗}, 95% _{p}^{2} = 0.21], and significant interaction between ^{∗}, 95% _{p}^{2} = 0.12] were identified. Collapsed over pattern arrangement, the mean engagement ratings ranged from a low of 0.22 (^{∗∗}, 95% ^{∗∗}, 95%

A 2-way 4 × 2 repeated-measures ANOVA [^{2}(5) = 159.69, ^{∗∗}] and the interaction between ^{2}(5) = 23.54, ^{∗∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.445 and 0.795, respectively). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.26], Arrangement [^{∗∗}, 95% _{p}^{2} = 0.21], and interaction between ^{∗}, 95% _{p}^{2} = 0.09]. Collapsed over pattern arrangement, average ratings of preference ranged from a low of 0.31 (^{∗∗}, 95% ^{∗∗}, 95%

A 2-step cluster analysis identified and separated individuals into 2 subgroups (^{2}(5) = 58.05, ^{∗∗}] and the interaction between ^{2}(5) = 21.95 ^{∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.676 and 0.805, respectively). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.2] and Arrangement emerged in the analysis [^{∗∗}, 95% _{p}^{2} = 0.22], as well as a significant interaction between ^{∗∗}, 95% _{p}^{2} = 0.62] as well as ^{∗∗}, 95% _{p}^{2} = 1.0]. The first cluster accounts for 66% of the sample and is most reflective of the overall perceptual trend with preference ratings increasing with higher

A 2-way 4 × 2 repeated-measures ANOVA [^{2}(5) = 213.96, ^{∗∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.399). Both the main effect of pattern Arrangement [^{∗}, 95% _{p}^{2} = 0.14] and interaction between ^{∗∗}, 95% _{p}^{2} = 0.1], but not _{p}^{2} = 0.01]. Between non-random and random arrangements, significant differences exist for the mid-range ^{∗}, 95% ^{∗∗}, 95%

A 2-step cluster analysis identified and separated individuals into two subgroups with respect to ratings of pattern refreshment (^{2}(5) = 73.86, ^{∗∗}] and interaction between ^{2}(5) = 11.49, ^{∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.580 and 0.893, respectively). Whereas the main effect of _{p}^{2} = 0.02], a significant main effect of Arrangement emerged in the analysis [^{∗}, 95% _{p}^{2} = 0.15], as well as a significant interaction between ^{∗∗}, 95% _{p}^{2} = 0.67] and between ^{∗∗}, 95% _{p}^{2} = 0.11]. The first cluster encompassed 51% of participants and produces a trend that increases with

A 2-way 4 × 2 repeated-measures ANOVA [^{2}(5) = 239.32 ^{∗∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.388). A significant main effect of ^{∗∗}, 95% CI [0.03, 0.29], η_{p}^{2} = 0.15] and Arrangement [^{∗}, 95% CI [0.01, 0.25], η_{p}^{2} = 0.11], and interaction between ^{∗} 95% CI [0.01, 0.13], η_{p}^{2} = 0.01]. Collapsed over pattern arrangement, average ratings of pattern relaxation ranged from a low of 0.34 (SD = 0.27) for ^{∗∗}, 95% CI [0.0, 0.07], ^{∗}, 95% CI [0.03, 0.12], ^{∗}, 95% CI [0.0, 0.1],

A 2-step cluster analysis identified and separated individuals into two subgroups with respect to ratings of pattern relaxation. Mauchly’s test indicated a violation of the assumptions of sphericity for ^{2}(5) = 93.78, ^{∗∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.554). A significant main effect of ^{∗}, 95% _{p}^{2} = 0.1] and Arrangement [^{∗∗}, 95% _{p}^{2} = 0.17] were identified, as well as significant interactions between ^{∗∗}, 95% _{p}^{2} = 0.72], Arrangement and Cluster membership [^{∗}, 95% _{p}^{2} = 0.12], and ^{∗}, 95% _{p}^{2} = 0.08]. The first cluster encompassed 64% of participants and produces a trend in which ratings of pattern relaxation steeply decrease with higher

Experiment 1A explored broad psychological effects of fractal patterns used in installations of multiple mediums including carpets, wall patterns, and screensavers. Overall, we find that perceptions of fractal pattern complexity, engagement, and preference, increase with greater

The current experiment used the same stimuli as described in Experiment 1A.

81 participants (69 females), comprised of undergraduate Psychology students from the UNSW Sydney volunteered to participate in the current study through the SONA participant pool system in exchange for course credit. The mean age of participants was 20.42 years (ranging between 18 and 47 years). All study protocols, including obtaining Informed Consent were approved by the UNSW Human Research Advisory Panel (Reference ID: HREAP-C 2349).

The study was generated with Inquisit (by Milliseconds) software and run via the Inquisit Web Platform. The participants completed the study on their personal computers with program stimuli scaled to the individual computer’s respective full-screen dimensions.

Like in the Experiment 1A, participants viewed the “global forest” fractal patterns presented in separate randomized blocks, with each block consisting of a singular judgment type (complexity, engaging, preference, refreshing, or relaxing). Each block’s stimulus set consisted of 4 unique patterns ranging across eight levels of complexity or

Data from 75 adult participants were analyzed with 6 participants excluded due to a failure to complete the study (4 participants), or technical error with data recording (2 participants).

A 3-way 8 × 5 × 2 repeated-measures ANOVA [^{2}(27) = 638.83, ^{∗∗}], Judgment [χ^{2}(9) = 39.25, ^{∗∗}], the interaction between ^{2}(27) = 63.65, ^{∗∗}], ^{2}(405) = 2571.82, ^{∗∗}], Judgment and Arrangement [χ^{2}(9) = 64.32, ^{∗∗}], as well as the three-way interaction between ^{2}(405) = 647.13, ^{∗∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.205, 0.789, 0.749, 0.121, 0.692, and 0.542, respectively). Indicated with a double asterisk for significance of ^{∗∗}, 95% _{p}^{2} = 0.68] and Arrangement emerged [^{∗∗}, 95% _{p}^{2} = 0.3]. Additional significant interactions were found between ^{∗∗}, 95% _{p}^{2} = 0.3], ^{∗∗}, 95% _{p}^{2} = 0.06], Arrangement and Judgment [^{∗∗}, 95% _{p}^{2} = 0.12], as well as ^{∗∗}, 95% _{p}^{2} = 0.04]. For illustrative purposes we plot the 3 significant interactions (

Experiment 1B. Results for “global forest” fractal patterns using a bipolar rating scale. Results show significant 2-way interactions among the experiment’s 3 factors: fractal dimension (

Experiment 1B results for “global forest” fractal patterns for 5 different judgment conditions (complex, engaging, liking, refreshing, and tense).

Experiment 1B- Paired Samples

Simple-Complex | Indifferent-Engaging | Dislike-Like | Tired-Refreshing | Relaxing-Tense | |

D = 1.1 vs. D = 1.2 | t = −3.43* (d = 1.3) | t = −1.93** (d = 0.93) | t = −1.35 (d = 0.17) | t = 0.39 (d = 0.01) | t = −3.95** (d = 0.15) |

D = 1.1 vs. D = 1.3 | t = −10.05** (d = 0.51) | t = −16.01** (d = 0.32) | t = −3.03* (d = 1.42) | t = 0.99 (d = 0.04) | t = −6.63** (d = 0.34) |

D = 1.1 vs. D = 1.4 | t = −14.18** (d = 1.02) | t = −10.61** (d = 0.8) | t = −3.51* (d = 0.25) | t = 1.10 (d = 0.08) | t = −11.23** (d = 0.78) |

D = 1.1 vs. D = 1.5 | t = −18.17** (d = 1.51) | t = −13.01** (d = 1.1) | t = −4.32** (d = 0.36) | t = 0.58 (d = 0.06) | t = −12.84** (d = 1.18) |

D = 1.1 vs. D = 1.6 | t = −19.81** (d = 2.09) | t = −14.87** (d = 1.7) | t = −4.82** (d = 0.52) | t = 0.19 (d = 0.08) | t = −14.55** (d = 1.61) |

D = 1.1 vs. D = 1.7 | t = −21.02** (d = 2.61) | t = −16.62** (d = 1.9) | t = −4.05** (d = 0.47) | t = 0.72 (d = 0.10) | t = −17.25** (d = 2.25) |

D = 1.1 vs. D = 1.8 | t = −24.54** (d = 3.03) | t = −17.54** (d = 2.14) | t = −3.48** (d = 0.44) | t = 0.38 (d = 0.06) | t = −18.89** (d = 2.61) |

D = 1.2 vs. D = 1.3 | t = −6.51** (d = 0.35) | t = −4.72** (d = 1.15) | t = −1.71 (d = 0.08) | t = 0.78 (d = 0.03) | t = −3.52* (d = 0.17) |

D = 1.2 vs. D = 1.4 | t = −11.02** (d = 0.83) | t = −11.13** (d = 1.71) | t = −2.75* (d = 0.18) | t = 1.04 (d = 0.06) | t = −9.41** (d = 0.61) |

D = 1.2 vs. D = 1.5 | t = −15.53** (d = 1.28) | t = −13.45** (d = 2.06) | t = −3.69** (d = 0.3) | t = 0.46 (d = 0.04) | t = −12.04** (d = 1.0) |

D = 1.2 vs. D = 1.6 | t = −16.89** (d = 1.83) | t = −15.58** (d = 2.55) | t = −4.51** (d = 0.47) | t = 0.08 (d = 0.01) | t = −13.53** (d = 1.43) |

D = 1.2 vs. D = 1.7 | t = −18.69** (d = 2.36) | t = −17.18** (d = 1.68) | t = −3.65** (d = 0.42) | t = 0.67 (d = 0.09) | t = −16.21** (d = 2.07) |

D = 1.2 vs. D = 1.8 | t = −21.79** (d = 2.75) | t = −18.29** (d = 2.9) | t = −3.19* (d = 0.39) | t = 0.31 (d = 0.04) | t = −18.15** (d = 2.43) |

D = 1.3 vs. D = 1.4 | t = −7.79** (d = 0.52) | t = −7.0** (d = 0.45) | t = −1.4 (d = 0.10) | t = 0.57 (d = 0.03) | t = −7.62** (d = 0.46) |

D = 1.3 vs. D = 1.5 | t = −14.22* (d = 1.01) | t = −11.16** (d = 0.82) | t = −2.83* (d = 0.21) | t = 0.06 (d = 0.01) | t = −11.44** (d = 0.89) |

D = 1.3 vs. D = 1.6 | t = −16.31** (d = 1.61) | t = −13.35** (d = 1.37) | t = −3.75** (d = 0.39) | t = −0.37 (d = 0.03) | t = −14.24** (d = 1.35) |

D = 1.3 vs. D = 1.7 | t = −19.33** (d = 2.19) | t = −15.86** (d = 1.61) | t = −3.06* (d = 0.34) | t = 0.48 (d = 0.33) | t = −17.24** (d = 2.03) |

D = 1.3 vs. D = 1.8 | t = −22.58** (d = 2.51) | t = −16.61** (d = 1.84) | t = −2.64* (d = 0.32) | t = 0.11 (d = 0.02) | t = −19.23** (d = 2.42) |

D = 1.4 vs. D = 1.5 | t = −7.38** (d = 0.49) | t = −6.59** (d = 0.41) | t = −2.01 (d = 0.13) | t = −0.50 (d = 0.02) | t = −6.49** (d = 0.45) |

D = 1.4 vs. D = 1.6 | t = −12.26** (d = 1.4) | t = −11.68** (d = 1.01) | t = −3.79** (d = 0.33) | t = −0.78 (d = 0.08) | t = −10.29** (d = 0.92) |

D = 1.4 vs. D = 1.7 | t = −16.81** (d = 1.77) | t = −13.51** (d = 1.28) | t = −2.78* (d = 0.28) | t = 0.30 (d = 0.04) | t = −14.57** (d = 1.63) |

D = 1.4 vs. D = 1.8 | t = −20.82** (d = 2.21) | t = −15.08** (d = 1.54) | t = −2.32* (d = 0.26) | t = −0.11 (d = 0.01) | t = −17.68** (d = 2.04) |

D = 1.5 vs. D = 1.6 | t = −8.49** (d = 0.67) | t = −8.47** (d = 0.61) | t = −2.64* (d = 0.21) | t = −0.59 (d = 0.05) | t = −6.11** (d = 0.47) |

D = 1.5 vs. D = 1.7 | t = −14.74** (d = 0.67) | t = −12.28** (d = 0.91) | t = −1.95 (d = 0.17) | t = 0.74 (d = 0.07) | t = −13.35** (d = 1.18) |

D = 1.5 vs. D = 1.8 | t = −19.12** (d = 1.83) | t = −13.59** (d = 1.67) | t = −1.58 (d = 0.18) | t = 0.13 (d = 0.01) | t = −17.27** (d = 1.61) |

D = 1.6 vs. D = 1.7 | t = −10.34** (d = 0.75) | t = −5.87** (d = 0.35) | t = 0.38 (d = 0.02) | t = 1.69 (d = 0.11) | t = −10.3** (d = 0.73) |

D = 1.6 vs. D = 1.8 | t = −17.44** (d = 1.2) | t = −9.82** (d = 0.6) | t = 0.33 (d = 0.02) | t = 0.57 (d = 0.05) | t = −15.23** (d = 1.18) |

D = 1.7 vs. D = 1.8 | t = −7.99** (d = 0.41) | t = −5.86** (d = 0.25) | t = 0.04 (d = 0.0) | t = −0.90 (d = 0.04) | t = −9.9** (d = 0.47) |

A 2-way 8 × 2 repeated-measures ANOVA [^{2}(27) = 521.02, ^{∗∗}] and interaction between ^{2}(27) = 76.63, ^{∗∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.232 and 0.764, respectively). A significant main effect of ^{∗∗}, 95% CI [0.65, 0.79], η_{p}^{2} = 0.73], however, no significant effect of Arrangement [_{p}^{2} = 0.01], nor interaction between _{p}^{2} = 0.01] were identified. Average complexity ratings (collapsed over pattern arrangement type) ranged from a low of 2.09 (SD = 1.16) for

A 2-way 8 × 2 repeated-measures ANOVA [^{2}(27) = 443.38, ^{∗∗}] and interaction of ^{2}(27) = 88.06, ^{∗∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.260 and 0.661, respectively). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.64], Arrangement [^{∗∗}, 95% _{p}^{2} = 0.19], and interaction between ^{∗}, 95% _{p}^{2} = 0.05]. Collapsed over pattern arrangement, the mean engagement ratings ranged from a low of 2.15 (^{∗}, 95% ^{∗}, 95% ^{∗}, 95% ^{∗}, 95% ^{∗∗}, 95%

A 2-step cluster analysis identified and separated individuals into 3 subgroups (^{2}(27) = 214.33, ^{∗∗}] and the interaction between ^{2}(27) = 82.31, ^{∗∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.446 and 0.673, respectively). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.62], and Arrangement emerged in the analysis [^{∗∗}, 95% _{p}^{2} = 0.24], as well as significant interactions between ^{∗∗}, 95% _{p}^{2} = 0.47] and ^{∗∗}, 95% _{p}^{2} = 0.07]. All three clusters of engagement ratings increase with

A 2-way 8 × 2 repeated-measures ANOVA [^{2}(27) = 534, ^{∗∗}] and the interaction between ^{2}(27) = 108.05, ^{∗∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.232 and 0.608, respectively). A significant main effect of ^{∗}, 95% _{p}^{2} = 0.08], Arrangement [^{∗∗}, 95% _{p}^{2} = 0.21], and the interaction between ^{∗∗}, 95% _{p}^{2} = 0.08] were identified. Collapsed over pattern arrangement, average ratings of preference ranged from a low of 3.21 (^{∗}, 95% ^{∗∗}, 95% ^{∗∗}, 95% ^{∗}, 95% ^{∗∗}, 95%

Two subgroups were identified in ratings of pattern liking with the two step cluster analysis (^{2}(27) = 397.62, ^{∗∗}] and interaction between ^{2}(27) = 109.84, ^{∗∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.286 and 0.603, respectively). A significant main effect of ^{∗∗}, 95% CI [0.06, 0.26], η_{p}^{2} = 0.16] and Arrangement [^{∗∗}, 95% CI [0.06, 0.36], η_{p}^{2} = 0.21], as well as interactions between ^{∗∗}, 95% CI [0.16, 0.38], η_{p}^{2} = 0.29] as well as Arrangement and Clusters were identified [^{∗∗}, 95% CI [0.02, 0.18], η_{p}^{2} = 0.08]. Cluster 1, comprising 57% of the sample, shows similar ratings of pattern liking for low and moderate patterns then decreases with higher

A 2-way 8 × 2 repeated-measures ANOVA [^{2}(27) = 853.96, ^{∗∗}] and the interaction between ^{2}(27) = 46.53, ^{∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.174 and 0.847, respectively). The only significant effect was for Arrangement [^{∗∗}, 95% _{p}^{2} = 0.18]. Ratings for non-randomized patterns (_{p}^{2} = 0.002], or interaction between _{p}^{2} = 0.02] were found. Between non-random and random arrangements significant differences exist for ^{∗}, 95% ^{∗}, 95% ^{∗}, 95%

A 2-way 8 × 2 repeated-measures ANOVA [^{2}(27) = 560.25, ^{∗∗}] and interaction between ^{2}(27) = 52.99, ^{∗}], thus degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.218 and 0.842, respectively). A sole significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.66]. Thus no main effect of Arrangement [_{p}^{2} = 0.03] or significant interaction between _{p}^{2} = 0.02]. Average ratings of pattern relaxation ranged from a low of 2.28 (

Three subgroups of participant perceptions of tension/relaxation were identified through two step cluster analysis (^{2}(27) = 259.97, ^{∗∗}] and interaction between ^{2}(27) = 52.56 ^{∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.367 and 0.839, respectively). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.58] and interaction between ^{∗∗}, 95% _{p}^{2} = 0.54] emerged in the analysis. Cluster 1 containing 64% of participants as well as cluster 2 containing 19% of participants both produced a perceptual trend in which ratings of tension increased with pattern complexity. Cluster 3 containing the remaining 17% of the sample, produces a flat trend in ratings of pattern tension/relaxation.

Overall, we find that bipolar ratings of fractal ‘global-forest’ pattern complexity, preference, and engagement increase with additional

Experiment 1B expands our investigation of psychological effects of these installed patterns but incorporates a different population of viewers and bipolar rating design. In this iteration of the perceptual rating task, participants are still recruited from a college population but from a different continent in the opposing global hemisphere (Australia) with a very different natural landscape. The rating task is also altered such that participants are instructed to rate their perception of images on a larger sliding scale between two opposing descriptors. Even with a new population and expanded study design results are highly similar to Experiment 1A. Similar to Experiment 1A, complexity, engaging, and preference ratings of ‘global-forest’ patterns all increase with increasing

Experiment 2 isolates the local components of the ‘global-forest’ patterns. These local ‘tree-seed’ patterns represent a local fractal pattern composed of rectangular ‘seeds’ with locations determined by the generated flightpath (see the description of the generation method in the Introduction and Experiment 1A). The stimuli consisted of a total of 20 patterns, with 5 examples each of 4

To identify the locus of these perceptual trends, 39 participants comprised of undergraduate Psychology students from the University of Oregon were recruited for the current study through the SONA participant pool system (22 females, age ranging between 18 and 29 years old, mean age 20 years old). Informed consent was acquired following a protocol approved by the Institutional Review Board at the University of Oregon and all participants received class credit for their participation.

Experiment 2 was programmed in PsychoPy3 and presented using the online research study platform of Pavlovia (

Participants viewed a series of fractal “tree-seed” patterns presented in five randomized blocks. Each block’s stimulus set consisted of 5 unique patterns ranging across 4 levels of complexity or

Before each block, participants were instructed to make a single randomly ordered judgment (complexity, preference, engaging, refreshing, or relaxing) for each stimulus presented in that block. Specifically, they were asked to answer one of 5 questions for each block: “How _______ is the image?” with one of 5 different words placed in the blank (complex, engaging, preferable, refreshing, relaxing). They were told to indicate their rating of each given pattern on a slider ranging between 0 and 1 located below the image, with the “0” end of the slider indicating “not at all” and the “1” end of the slider indicating “completely.” They were asked to use the full range of the slider and to click on the slider to indicate their rating. Periodically, an attention check trial appeared in which participants were instructed to select either “0” or “1.” The images remained on the screen until participants selected their rating. Upon completion of the experiment, participants completed a demographic questionnaire and were debriefed according to the protocols approved by the Institutional Review Board at the University of Oregon.

Data from 39 adult participants (between 18 and 29 years old) were retained from the 60 adults who participated in the experiment. Data were excluded due to: (a) failure to complete the study, (b) failure of greater than 3 attention checks, or (c) recording the same rating for greater than four consecutive trials.

A 2-way repeated-measures 4 × 5 ANOVA [^{2}(5) = 59.58, ^{∗∗}], Judgment [χ^{2}(9) = 25.24, ^{∗}], and the interaction of ^{2}(77) = 216.75, ^{∗∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.525, 0.732, and 0.463, respectively). Indicated by a double asterisk for significance of ^{∗∗}, 95% _{p}^{2} = 0.25] and Judgment [^{∗}, 95% _{p}^{2} = 0.13], as well as an interaction between ^{∗∗}, 95% _{p}^{2} = 0.32]. For the

Experiment 2 results for ‘tree-seed’ fractal patterns using a unipolar rating scale. Results show a significant interaction between fractal dimension (

Experiment 2-paired samples

Complex | Engaging | Preference | Refreshing | Relaxing | |

D = 1.2 vs. D = 1.4 | t = −10.01** (d = 1.29) | t = −4.49** (d = 0.8) | t = −0.36 (d = 0.05) | t = 2.11* (d = 0.33) | t = 2.13* (d = 0.38) |

D = 1.2 vs. D = 1.6 | t = −11.69** (d = 2.35) | t = −6.02** (d = 1.61) | t = −0.71 (d = 0.15) | t = 1.42 (d = 0.37) | t = 1.98 (d = 0.53) |

D = 1.2 vs. D = 1.8 | t = −13.39** (d = 3.21) | t = −7.33** (d = 1.94) | t = −1.77 (d = 0.49) | t = 1.36 (d = 0.39) | t = 0.90 (d = 0.29) |

D = 1.4 vs. D = 1.6 | t = −6.30** (d = 1.03) | t = −3.67** (d = 0.77) | t = −0.71 (d = 0.12) | t = −0.20 (d = 0.0) | t = 0.66 (d = 0.13) |

D = 1.4 vs. D = 1.8 | t = −8.41** (d = 1.71) | t = −5.38** (d = 1.15) | t = −2.17* (d = 0.51) | t = 0.27 (d = 0.09) | t = −0.42 (d = 0.05) |

D = 1.6 vs. D = 1.8 | t = −5.22** (d = 0.77) | t = −3.45** (d = 0.47) | t = −2.47* (d = 0.44) | t = 0.60 (d = 0.10) | t = −1.37 (d = 0.15) |

A one-way repeated measures ANOVA was completed on the effects of ^{2}(5) = 28.03, ^{∗∗}], thus, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.659). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.74] was detected. Average complexity ratings ranged from a low of 0.29 (

Experiment 2 results for ‘tree-seed’ fractal patterns for 5 different judgment conditions (how complex, engaging, preferred, refreshing, and relaxing).

A one-way repeated measures ANOVA was completed on the effects of ^{2}(5) = 28.01, ^{∗∗}], thus, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.660). A significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.47]. Average engagement ratings ranged from a low of 0.32 (

A one-way repeated measures ANOVA was completed on the effects of ^{2}(5) = 42.16, ^{∗∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.586). No significant main effect of _{p}^{2} = 0.06]. Paired samples

Two subgroups emerged in the 2-step cluster analysis (^{2}(5) = 11.89, ^{∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.861). Both a significant main effect of ^{∗∗}, 95% _{p}^{2} = 0.2], and significant interaction between ^{∗∗}, 95% _{p}^{2} = 0.53] emerged. Cluster 1 comprised 56% of the sample and represents a trend of fractal preference peaking at the lowest

A one-way repeated measures ANOVA was completed on the effects of ^{2}(5) = 54.02, ^{∗∗}]. Thus, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.554). No significant main effect of _{p}^{2} = 0.04]. Paired samples

A one-way repeated measures ANOVA was completed on the effects of ^{2}(5) = 64.07, ^{∗∗}]. Therefore, degrees of freedom were corrected using Greenhouse-Geisser estimates of sphericity (ε = 0.489). No significant main effect of _{p}^{2} = 0.05]. Paired samples

Experiment 2 maintains the same methodological structure and perceptual decisions as Experiment 1A but replaces the ‘global-forest’ pattern with fractal ‘tree-seed’ patterns. Similar to Experiment 1, judgments of complexity and engagement increase with

Evaluations of Euclidean human-made space can be altered by integrating the aesthetics of nature (

Across 3 experiments that vary in stimulus pattern composition, participant population, and rating scale we find similar trends in fractal perception. Experiment 1 used ‘global-forest’ fractal designs to demonstrate that ratings of pattern complexity, engagement, and preference increase with fractal complexity or

Across both studies subgroupings have a significant impact on overall trends, supporting previous findings of individual differences in preference for fractal complexity (

Both studies also demonstrate an effect of pattern randomization, whereby ratings of engagement, preference, refreshing, and relaxing qualities are slightly higher for non-randomized compared to randomized patterns. Fortunately, for many of the installations (e.g., carpets and projected light patterns) the visibility of the edges defining the randomly positioned tiles is much less apparent in the installations than in the randomized patterns presented here.

Lastly, the similarity between findings of Study 1A and 1B are not impacted by the geographical location of participants. Our findings suggest that perceptions of fractal patterns are not altered by the diverse natural environments where participants reside. This result supports the finding that preference for fractal complexity forms early in human development (sometime prior to three years of age) and is not further altered by life experience in western participants (

The “global-forest” patterns with

Future studies will further explore the ways in which these fractal designs impact occupants’ perceptions by expanding our studies to assess the extent to which our findings apply to broader populations of participants, additional changes in pattern design (including different local components, global flight-path arrangements, and global design), and can be directly identified with changes in physiological and verbal measures of stress and arousal. Further replications will be conducted utilizing Virtual Reality (VR) to assess responses to these patterns installed in 3-dimensional architectural spaces in order to more directly manipulate participant experience and measure changes in psychological effects in an immersive environment. By balancing perceptual factors, patterns can be produced and installed to maximize aesthetic experiences of particular spaces. The collaboration of design, physics, psychology, and technology provides a vital opportunity to test for and determine visual patterns that produce optimal perceptual responses and experiences in occupants of human-made structures. By selecting fractal patterns with

The datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below:

The studies involving human participants were reviewed and approved by Institutional Review Board at the University of Oregon and UNSW Human Research Advisory Panel. The patients/participants provided their written informed consent to participate in this study.

KR, RT, BS, and MS contributed to the study design. RT, JS, CR, SM, SS, AL, and ML contributed to stimulus generation. KR and CV contributed to programming the experiments. KR and BS contributed to testing and data collection. KR, MS, and BS contributed to the data analysis and interpretation. KR and MS drafted the manuscript. All authors contributed to the manuscript editing and approved the final version of the manuscript for submission.

SS, AL, and ML were employed by company 13&9 Design, Austria. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

We thank 13&9 Design and Fractals Research, Mohawk Group, and INNOCAD Architecture for the images used in