Hexagonal Windows Win Hearts: Golden Ratio Geometry Shapes How Students Feel About School Facades

The facade of a school is often the very first thing a student sees each morning, long before any lesson begins. Yet the emotional weight carried by that first architectural encounter has rarely been measured with any rigor. A new study from Babol, Iran, now offers some of the most detailed evidence to date that the mere geometry of window openings on a school facade can systematically shape how students feel, and that a shape rooted in centuries of Iranian proportional tradition may outperform the celebrated golden ratio rectangle itself.

The research, published in Discover Education, was conducted by Leila Sadat Hamidian Divkolaei of the Technical and Vocational University in Tehran. It asked a deceptively simple question: do high school students respond differently, in emotional and aesthetic terms, to facades whose windows differ only in their geometric proportions? To find out, the study recruited 327 third-grade female high school students in Babol, drawn from three study tracks, mathematics, experimental sciences, and humanities, and from both public and private schools. Participants were selected according to strict criteria, including normal or corrected-to-normal vision and voluntary, informed participation, with written consent obtained from their legal guardians.

The experimental design was carefully controlled. Five facade alternatives were generated from the same base school design, taken from a project under construction by Iran’s School Renovation Organization. Across all five images, the overall building massing, viewpoint, scale, and facade composition were held constant; only the geometry and proportion of the openings varied. The five alternatives comprised a rectangle with proportions of approximately 1.6 to 1, embodying the conventional golden ratio; a golden triangle, an equilateral triangle with an apex angle of 36 degrees; a hexagon, from which the so-called Iranian golden ratio of 1.73 is derived; and two strongly elongated rectangles with 3-to-1 proportions, one horizontal and one vertical. This isolation of a single variable allowed the researchers to attribute differences in emotional response to geometry alone, rather than to materials, color, or form.

Students evaluated the images using a semantic differential questionnaire, a technique that presents bipolar adjective pairs and asks respondents to rate a stimulus between them on a seven-point Likert scale. The 21 attributes spanned several domains: architectural qualities such as beautiful-ugly, traditional-modern, dynamic-static, and light-heavy; educational qualities such as encouraging-discouraging and curious-gloomy; and psychological qualities such as joyful-sad and happy-depressed. The images were projected onto a three-meter screen at a fixed height, and every participant viewed them from a standardized distance of four meters under consistent classroom lighting, with each image displayed for roughly 30 seconds before ratings were completed.

The theoretical scaffolding for the study draws on two complementary strands of environmental aesthetics. The first is processing-fluency theory, which holds that forms the visual system can parse easily, typically coherent and symmetric configurations, tend to be judged more positively. The second encompasses classic environmental-preference models, including the Kaplan and Kaplan preference matrix and Berlyne’s collative-properties framework, which argue that human preference requires a balance between coherence and complexity to avoid monotony. A hexagon, the author argues, occupies an intriguing middle ground: its geometric regularity ensures high perceptual fluency, while its non-orthogonal character introduces a distinctive complexity that pure rectangles lack. In an Iranian cultural context, moreover, repeated historical exposure to hexagon-derived proportions in traditional patterns and spatial modules may lower the cognitive effort needed to process these forms, deepening aesthetic pleasure and emotional comfort.

Because every participant rated all five alternatives, the data were analyzed as repeated measures. Shapiro-Wilk tests revealed that all opening-by-attribute distributions departed significantly from normality, so the researcher employed Friedman tests rather than parametric alternatives, followed by Wilcoxon signed-rank pairwise comparisons with Holm adjustment to control the family-wise error rate. Kendall’s W was reported as the omnibus effect size, and matched-pairs rank-biserial correlations quantified pairwise effects. This nonparametric pipeline reflects a methodological rigor that has often been absent from earlier studies of architectural proportion, many of which relied on small samples and uncorrected comparisons.

The results were striking. On the overall 21-attribute score, the opening geometries differed significantly, with a chi-squared statistic of 104.29 across four degrees of freedom and N equal to 327, Holm-adjusted p below .001, and a Kendall’s W of 0.080. The hexagonal opening achieved the highest overall mean rating of 4.421, followed by the vertical 3:1 rectangle at 3.902, the horizontal 3:1 rectangle at 3.735, the golden triangle at 3.684, and the conventional golden-ratio rectangle at 3.680. In Holm-adjusted pairwise comparisons, the hexagon was rated significantly higher than each of the other four openings, with rank-biserial correlations ranging from .437 to .571. Perhaps most tellingly, the hexagonal opening took the highest mean on 18 of the 21 individual attributes, including beautiful, classy, light, distinctive, harmonic, modern, creative, encouraging, curious, ideal, joyful, peaceful, and memorable. The conventional golden-ratio rectangle led only on three attributes: identified, purity, and safe, where it posted its single strongest mean of 4.575.

Secondary, exploratory analyses added texture to the picture. Comparing public and private school students on the hexagonal opening using Mann-Whitney U tests with Holm correction, five attributes differed significantly, classy, beautiful, harmonic, identified, and safe, with public-school students rating the hexagon higher on all five. Across the three study tracks, only 8 of 105 opening-by-attribute tests remained significant after correction, and no attribute differed by track for the golden-ratio rectangle. Comparisons by height category found no significant differences, while weight categories produced only isolated effects. The author is careful to flag all subgroup findings as exploratory, cautioning that the data do not support prescribing particular geometries for specific school types or student subgroups. Attribute-level effect sizes were mostly small, with Kendall’s W ranging from 0.018 to 0.137, indicating consistent but modest relative differences rather than dramatic swings in sentiment.

Why might the hexagon triumph over the golden ratio rectangle, a proportion long mythologized as the pinnacle of beauty since Gustav Theodor Fechner’s pioneering psychophysical experiments in the nineteenth century? The study situates the answer in both perception and culture. Iranian proportional tradition, exemplified by monuments such as the Taq-i Kisra at Ctesiphon and the Sassanid Palace of Sarvestan, is built on rectangles with ratios of the square root of 2, the square root of 3, and the rarer 1.118, alongside the Peymoun, a measurement system derived from human body proportions. The hexagon, composed of three rectangles each with the 1.73 ratio of the Iranian golden proportion, has historically shaped courtyards, ponds, and the celebrated five-door rooms of traditional Iranian architecture. Prior work cited in the study lends convergent support: research on consumer behavior found hexagonal forms received the highest preference in packaging design, and a Kansei-method study of bank facades found triangular openings received comparatively unfavorable emotional ratings, echoing the golden triangle’s middling performance here.

The author is candid about the study’s limits. The sample came from a single city, a single grade, and female students only, restricting generalizability, and some adjective sets showed low internal consistency, particularly within the ethical-identity and two-item spiritual categories. Crucially, the study measures self-reported aesthetic and emotional responses to static facade images, not learning outcomes, behavior, motivation, or well-being, and the author explicitly warns against inferring such outcomes from these results. Still, the design implications are clear. Within this sample, the hexagonal opening emerges as the strongest candidate for user-centered facade development, though any final choice must weigh daylighting, privacy, energy performance, constructability, cost, and cultural context. Future directions include extending the approach to other facade parameters and interior spaces, replicating with broader and more diverse samples, complementing self-report with physiological measures such as eye-tracking or EEG, and testing longitudinally whether aesthetic preference translates into tangible benefits. For now, the message for architects and school planners is quietly provocative: the shapes we cut into a wall may speak to students more powerfully than we assumed, and the oldest proportional wisdom of Persian architecture may deserve a seat at the drafting table alongside the golden ratio.

Subject of Research: Emotional and aesthetic responses of high school students to school facade opening geometries based on the golden ratio and Iranian proportional systems

Article Title: Emotional responses to school façade opening geometries associated with the golden ratio among high school girls in Babol, Iran

Article References: Divkolaei, L. S. H. (2026). Emotional responses to school façade opening geometries associated with the golden ratio among high school girls in Babol, Iran. Discover Education, 5(1), Article 1140. https://doi.org/10.1007/s44217-026-02143-z

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02143-z

Keywords: golden ratio, hexagonal openings, school facade design, environmental psychology, processing fluency, semantic differential, Iranian architecture, aesthetic preference, emotional response, architectural geometry, user-centered design, high school students