Weighted Shannon entropy as a quantitative measure of façade pattern regularity

1 Weighted Shannon entropy as a quantitative measure of façade pattern regularity Michał Malewczyk1* 1Faculty of Architecture, Gdansk University of Technology, Gdansk, Poland *Faculty of Architecture, ul. Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland. Email: [email protected]; ORCID: 0000-0001-9585-1497 The article presents an original measure of facade pattern regularity, SVE (Shannon-Variance-Entropy), based on weighted Shannon entropy, which aims to quantitatively describe the internal diversity of a composition. The analysis showed that simply taking into account the distance between elements does not improve the accuracy of the regularity description. The variance of element proportions and angular relationships proved to be key factors, significantly increasing the indicator's consistency with aesthetic perception. The best results were obtained for the composite measure SVEDAS, which combines all three components. This measure shows high consistency with expert assessments and effectiveness in the algorithmic generation of facade patterns. The use of the designated metric as a design parameter enables quantitative control of aesthetics, supporting sustainable design in line with SDG 11. Keywords: urban planning; smart cities; urban development; city; data-driven design Declarations: not applicable. Funding: not applicable. Conflicts of interest/Competing interests: there are no competing interests. Availability of data and material: on demand. Code availability: not applicable. Ethics approval: not applicable. Consent to participate: not applicable. Consent for publication: sensitive data was not collected. 2 Weighted Shannon entropy as a quantitative measure of façade pattern regularity Introduction The aesthetics of urban space, especially the composition of architectural elements on building façades, play a key role in the perception of the quality of the urban environment (Malewczyk et al. 2022a, 2024; Samalavičius 2021). The regularity and geometric order of façades, especially the arrangement of windows, affect not only the visual experience of residents and users of public spaces, but also the sense of order, security and identity of a place (Ezz et al. 2024; Wang & Munakata, 2024). The pursuit of aesthetically pleasing, orderly and visually friendly urban spaces is directly in line with Sustainable Development Goal 11 (SDG 11), which calls for the creation of cities and neighborhoods that are safe, inclusive, sustainable and of high quality. Aesthetics and architectural legibility, including the geometric regularity of façades, support the perceived quality of urban life, a sense of local identity and comfort in the use of public space (Weber et al. 2008). Despite growing interest in the problem of regularity in the context of the aesthetics of the built environment and its relationship to the broader psychophysical well-being of the users of that environment, there are still objective and unambiguous mathematical tools that would allow quantitative assessment of the geometric ordering of façade pattern. Malewczyk's publication from 2025 responded to this challenge by proposing the first version of an index describing the regularity of patterns, consisting of rectangles (symbolizing windows or façade panels) in a two-dimensional façade space. The author's index Hcenterdist corresponds to the entropy of the metric distance between the centers of adjacent compositional elements. Despite very high efficiency in the prediction of the degree of regularity by Hcenterdist (correlation of about 93%), in the 3 author's opinion, there remains some field that needs improvement. The main objective of this paper is to determine a modified façade pattern regularity measure index (SVE - Shannon-Variance-Entropy), based on the original H(centerdist). The new approach involves the introduction of metric entropy weighting, which should increase the sensitivity of the index to more subtle changes in composition regularity. An additional goal of the paper is to develop a generative algorithm and verify the feasibility of designing a composition by top-down determination of its regularity as a potential tool to support sustainable architectural design in line with SDG 11 goals. The article is based on two research hypotheses. First, the weighted metric entropy index of the distance between the centers of adjacent compositional elements of a façade design describes the regularity of the design better than pure metric entropy. Second, the determined index of weighted metric entropy can be effectively used as an objective function in façade composition pattern generation algorithms. This paper attempts to answer four research questions. First, how to mathematically define an optimal function that combines Shannon entropy with measures of spatial variance? Second, what façade composition parameters (sizes of compositional elements, spacing, location) most strongly influence the developed SVE index? Third, how does the SVE index correlate with the regularity of façade patterns, and is this correlation stronger than the original Hcenterdist index? Fourth, is it possible to algorithmically generate façade patterns through a top-down determination of the SVE parameter? This article builds on the earlier study, however, based on an expanded research sample of 343 (original 245) compositions based on 49 residential, multi-family buildings nominated for the 2024 European Union Prize for Contemporary ArchitectureMies 4 Van der Rohe Award (EU Mies Award 2024) in the 'collective housing' category. A new statistical analysis methodology is also proposed. The rest of the article presents an analysis of key concepts in the context of the current state of knowledge. The author focuses on measures of regularity in architecture, entropy in terms of information theory in terms of spatial analysis, and a review of ways to algorithmically generate compositions. Measures of regularity in architecture In the analysis of architectural forms, quantitative methods to objectively assess their regularity and order have become increasingly important. Research on regularity coefficients of façades focuses, among other things, on the analysis of distances between elements, rhythmicity and the occurrence of repetitive geometric patterns (Stiny & Mitchell 1978; Ilgaz et al. 2025; Med'dahi & Boussora 2021). Applications of fractal geometry to façade analysis are also increasingly reported in the architectural literature, indicating a link between geometric complexity and the aesthetic and environmental perception of buildings (Bovill 1996; Salingaros 1998). Measures such as fractal dimension or complexity indices have been used to describe the structure of historic urban complexes and modern façades, demonstrating their potential as a tool for assessing the quality of space (Ali & Mustafa 2024; Lee 2014; Katona 2023). Measures of symmetry and proportion, derived from classical theories of architectural composition (Wittkower 1949), are also an important category of analytical tools. Symmetry, both mirror and translational, plays an important role in the perception of regularity, as confirmed by psychological research and perceptual theory (Jacobsen 2006; Leder et al. 2004). These works indicate that structures with a high degree of 5 regularity can be more easily recognized and positively evaluated by space users. A rich literature analysis of the issue of regularity measures in broader contexts was also made in Malewczyk's 2025 article. Shannon's entropy and its applications Entropy is a concept that was first introduced in the 19th century in thermodynamics to measure the chaotic nature of systems. In 1948, Claude Shannon also began, through entropy (according to Formula 1), to determine the average amount of information in a message, a concept that is particularly relevant in the context of information theory (Shannon 1948; Fourie 2012). 𝐻=−$𝑝!×𝑙𝑜𝑔"𝑝! # !$% Formula 1 Entropy in information theory. Since its introduction, Shannon's entropy has found widespread use in a variety of fields - including the description of the design process (Krus 2013), medical diagnosis (Benish 2020), and research on the relationship between diversity and aesthetic preferences (Redies et al. 2017; Stanischewski et al. 2020; Stamps 2004). The Shannon global entropy coefficient is also widely used in built environment analysis (Güzelci et al. 2020, 2021; Güzelci & Alacam 2019; Crompton 2012; Stamps 2012, 2014). Based on metric entropy, which is independent of the length of information, the Hcenterdist index was developed (Malewczyk, 2025). It is a measure of the metric entropy of a sequence formed from the absolute distances between the centers of compositional elements (e.g., windows, façade panels) on a building façade. A study of 245 compositions based on 50 buildings nominated for the EU Mies Award 2024 in the ‘collective housing’ category found a 93% correlation between the Hcenterdist index and 6 objective visual regularity. However, Malewczyk (2025) points out the limitations of the index, which does not consider the magnitude of disturbance, only the presence of disturbance, due to Shannon's entropy property, which is sensitive to variation but not to the magnitude of change. For example, the strings 1,3,7,3,5 and 2,35,192,79,192 have the same entropy, despite markedly different dispersion. Moreover, this indicator does not take into account the size of the compositional elements. This problem can be solved by weighted entropy, proposed by Guiaşu (1971), which considers not only the probability of events, but also their importance (e.g., significance). An example of its application is in investment risk analysis (Nawrocki and Harding, 1986). Weighted entropy is being developed both theoretically (Ebanks 2010) and practically, including in computer science Kelbert et al. 2017) and urban scale built environment analysis (Boeing 2019; Huynh 2019; Wang 2018). However, no examples of its application in the analysis of architectural forms using metric entropy have yet been found. 𝐻=−$𝑤!×𝑝!×𝑙𝑜𝑔"𝑝! # !$% Formula 2 Entropy in information theory. Algorithmic façade generation Over the past two decades, algorithmic façade generation methods have become an important area of architectural research and experimentation, integrating parametric, optimization, environmental and artificial intelligence approaches. Of key importance is parametric design, which enables dynamic control over the geometry, structure and performance of a façade using tools such as Grasshopper, Ladybug or Octopus (Caetano 2020; Narangerel & Stouffs 2016; Shen 2018; Ramadan 2024). This makes it possible 7 to simultaneously test and optimize forms for daylighting, shading, natural ventilation, and energy production (Narangerel & Stouffs 2016; Lahmar et al. 2022; Chen & Tang 2024). In the context of the goals of SDG 11 and the topic of this article, research on the use of algorithmic methods to generate façades with aesthetic quality is particularly relevant. Rezakhani and Kim (2024) used genetic algorithms to optimize façades for aesthetics and visibility while maintaining appropriate sunlight parameters. Alagöz and Jamal (2024) used parametric design strategies in developing kinetic façades, striving for a balance between functionality and aesthetics. Similar issues are also addressed by Mahmoud and Elghazi (2016) and other authors (Engin et al. 2023; Wang, Sun, Shao & He 2024; Wang, Zhang, Zhang, Cui & He 2024; Cudzik 2019; Cudzik & Atasoy 2023; Cesbas et al. 2022). More recently, there have also been attempts to apply artificial intelligence to the design process, including through the use of the Monte Carlo Tree Search (MCTS) algorithm in the early stages of design (Lin et al. 2025), as well as GAN algorithms to generate façades from images (Yu et al. 2020; Shan & Zhang 2022) and other inputs (Wan et al. 2023). Although these algorithms mainly focus on optimizing selected aspects, such as solarization, material consumption or energy efficiency with aesthetics, none of the publications found attempted to quantitatively encode façade aesthetics. Materials and method The present study is an expansion of Malewczyk's 2025 study. To increase the sensitivity of the study, the author partially uses, but also partially expands the original research sample. Modified statistical analysis procedures are also used to improve the quality of the results. 8 Materials The study is based on materials, that is, 245 compositions of rectangles symbolizing windows or façade panels, on which the original Malewczyk (2025) study was based. The compositions were based on 50 photographs of multifamily buildings that were nominated in 2024 for the European Union Prize for Contemporary ArchitectureMies van der Rohe Award (EU Mies Award 2024) in the 'collective housing' category. In the original study, photographs were downloaded from the competition organizer's website (https://miesarch.com/archive). Basing the composition on examples of architecture nominated for such a prestigious award was to ensure the high quality of the research material. Methods Weighted entropy The original Hcenterdist index in Malewczyk's 2025 study was determined based on metric entropy, which was calculated in a similar way to in the works of Stamps (2004, 2014), Cropmpton (2012), Güzelci (2019, 2021) or Stanischewski (2020). However, in order to increase the sensitivity of the determined index, the theoretical basis of Shannon's concept of weighted metric entropy, which was first written about by Guiaşu in 1971, is proposed. Of relevance to this article, from the perspective of the practical application of weighted entropy, are the works of Boeing (2019), Huynh (2019) and Wang and Zhao (2018). These works, although related to urban problems, make use of the author's important idea of weighted entropy with geometric relations. 9 Software The study mainly used Blender version 4.3.2 for macOS. This program was used to create and process the study material and also to run scripts in Python version 3.12. The scripts used in the study used libraries such as bpy, bmesh, random, os, math, csv and io. Creating compositions In the original study, 49 two-dimensional compositions were created from photos of buildings nominated for the EU Mies Award 2024 in the 'collective housing' category. For one building, a composition was not created. In a further step, 4 variants were created based on each of the 49 original compositions, which clearly differed from each other and from the original composition in the degree of visual regularity. This was achieved by using such procedures as breaking symmetry, varying the distance between compositional elements, varying compositional elements among themselves and the like. In the end, 49 groups of compositions were formed, each of which had 5 compositions. A total of 245 compositions were created. The current study uses 245 compositions from Malewczyk's 2025 study (source files available https://doi.org/10.34808/bemn-rd97) however, an additional 2 compositions were created within each of the 49 groups. The purpose of creating the additional compositions was to enrich the study sample with specimens that differed in visual regularity in a more subtle way, in order to be able to verify the sensitivity of the new weighted entropy-based indices. In the end, 343 compositions (245 original and 98 new) were collected for the study. The process of creating compositions using a group of 12 compositions as an example is visualized in Figure 1. 16 Results Analysis of five different formulas of the author's Shannon-Variance-Entropy (SVE) index showed significant differences in their ability to predict the regularity of building façade patterns. The five different weighting combinations were tested against the original Hcenterdist index (Malewczyk, 2025) in 49 composition groups, each of which contained seven patterns ranked according to the REG scale (from 1 to 7, with 1 indicating the greatest regularity and 7 indicating the greatest irregularity). Three complementary measures-Spearman's rank correlation coefficient, the percentage of correctly ordered pairs, and the sum of squares of rank differences-were used to assess the performance of the indicators. These measures were chosen to account for the ordinal nature of rank order regularity (REG) and to avoid the limitations of linear correlation assumptions (as in the case of the r-Pearson correlation coefficient). Table 1 Summary of results of statistical analyses. Measure Spearman ρ Correct Pairs (%) Sum of Squared Rank Differences Hcenterdist .929 88.7% 5.3 SVED .871 90.0% 6.9 SVEA .919 92.1% 4.8 SVES .946 93.2% 3.3 SVEDS .940 94.4% 3.1 SVEDA .949 95.3% 2.8 SVEAS .958 96.2% 2.5 SVEDAS .964 96.8% 2.0 A comparison of the performance of the SVE coefficient variants against the original Hcenterdist coefficient is shown in Table 1. The results show that SVEDAS variance weighting for distance, proportion and angles) achieves the highest performance in all statistical indicators with a Spearman correlation of .964, a correct ordering of 96.8% and the lowest sum of squares of rank differences (2.0). 17 Correlation of individual variance components The analysis revealed differential effects of individual variance components on the effectiveness of the SVE index. SVE(D) (ρ=.871) weighted by distance variance (CVdistance) shows the lowest effectiveness, even lower than the baseline Hcenterdist index (ρ=.929). An aspect that was completely ignored in the original study by Malewczyk (2025) is the variability of the angles formed between the connections of the centers of adjacent rectangles. The SVEA weighted CVangle obtained a Spearman correlation coefficient of ρ=.919 and is clearly superior to the baseline Hcenterdist in terms of the percentage of correctly ordered pairs (92.1% vs. 88.7%). However, the most significant component is the variation in the proportion of rectangles (CVshape). SVES, weighted by CVshape obtained a Spearman correlation coefficient of ρ=.946. Verification of research hypotheses The statistical analyses performed on the data collected in the conducted study made it possible to verify both research hypotheses and at the same time answer all four research questions. Regarding hypothesis one, which assumed that weighted metric entropy better describes the regularity of the elevation pattern than unweighted entropy, it was confirmed for some of the variants of the SVE index. The combined SVEDAS index performed 3.5 points better than the baseline Hcenterdist index and ordered 8.1 % points more pairs correctly. Thus, it was shown that all aspects-variability of distances, proportions and sizes of compositional elements and angles in the connection grid-are important in regularity analysis. The high efficiency of the SVEDAS index (ρ=.964), the very high percentage of correctly ordered pairs (96.8%) and its simplicity indicate the great potential of this 18 measure as an objective function in generative algorithms. In the course of the research, such a generative tool was developed, whose mechanism of operation was based directly on the SVEDAS indicator . The tool was developed as an add-on to Blender and made available under an open-source license. The tool definitions of six composition patterns created by Malewczyk, Taraszkiewicz and Czyż (2022b) as a composition typology of façades of residential, multi-family buildings. The tool allows you to select the specific type of composition you want to generate (according to the pattern definition), specify the regularity by providing expected numerical values for the interval of the SVEDAS index and other parameters, such as the number of windows in rows and columns, window types, distances between window rows, heights of interstory strips and other basic physical aspects. Screenshots showing the user interface in various configurations are shown in Figure 3. Fig. 3 User interface of the developed facade composition generator. 19 Thus, a tool has been obtained that allows the creation of an infinite number of elevation patterns with a given value of regularity, understood precisely as the value of the SVE index, which confirms its usefulness not only as a diagnostic measure, but also as an objective function in composition algorithms. Discussion The analysis of SVE (Shannon Variance-Entropy) variants reveals the complexity of the perception of regularity in facade patterns and the potential of advanced measures in their quantitative description. The results confirm that although simple entropy (Hcenterdist) is highly effective, its performance can be significantly improved by considering the internal variability of the pattern. It has been shown that distance variance (CVdistance) as a standalone component not only fails to improve the accuracy of regularity description but may even worsen it. This suggests that information about the distances between elements (e.g., the centers of gravity of facade panels) does not constitute a sufficiently clear perceptual signal. This may be since local symmetry or rhythm often occur despite significant deviations from equal distances. In turn, the variance of angles between lines connecting the centers of compositional elements, although moderately effective as a standalone component, has a strong effect when combined with other features. This points to the importance of directional relationships in pattern perception, especially in more complex and irregular arrangements. Angular differences can act as ‘visual disturbances’ of rhythm and controlling them affects the perception of order. The strongest component was found to be the variance in the proportions of elements (CVshape). Its high effectiveness, both on its own and in combination, confirms the 20 intuitive understanding of regularity as the repetition of forms with similar dimensions, which is consistent with previous qualitative studies on rhythm, proportion, and scale in architectural compositions. The best results were obtained for multi-component variants, especially SVEDAS, which combines all three components: distance, angle, and proportion. Each of them describes a different aspect of perception: spatial arrangement, directional relationships, and similarity of forms, which ensures high analytical resolution and the ability to capture subtle differences in the order of the composition. The method of evaluating the effectiveness of the measures is also worth noting. The use of three indicators (Spearman's rank correlation, pair order accuracy, and sum of rank differences) allowed for a multifaceted evaluation that better reflects the nature of ordinal data and avoids overinterpretation of linear correlation. As a result, SVEDAS can serve as an analytical tool and indicator supporting the design process. Its high consistency with expert assessments confirms its operational potentialboth in the analysis of existing designs and in the automatic generation of new variants, which has been tested during the research. The algorithm, which uses the SVEDAS value as a control parameter, confirmed the functionality of the formula and its usefulness in design applications. Examples of compositions generated by the algorithm are shown in Figure 4. The ability to define regularity through numerical values or to optimize the facade design in relation to a minimum regularity index (understood as the SVEDAS value) opens a new chapter in architectural aesthetics, in line with the idea of data-driven design. The key point here is that SVEDAS does not describe indirect factors influencing aesthetics (such as sunlight), but aesthetics itself. By taking this index into account, 21 designers gain the ability to objectively compare aesthetic aspects, which can contribute to the creation of architecture that is more inclusive, sustainable, and supportive of SDG 11. Conclusion In conclusion, in the course of the conducted study based on the materials and method of the 2025 Malewczyk study, statistical analyses showed that the SVEDAS index , that is, the metric entropy of the lengths of the connections between the centers of the compositional elements weighted by the indicators of the variation of these lengths, the angles in the grid of the connections of the centers of the compositional elements and the variation of the proportions of the compositional elements, proved to be extremely effective in predicting the visual regularity of the compositional designs. A correlation of the author's index with expert assessments of compositional regularity of more than 96% was obtained. For nearly 97% of the pairs correctly ordered the regularity of the composition. The original Hcenterdist index (Malewczyk 2025) admittedly also achieved high results (93% correlation and 89% correctly ordered pairs), but most importantly, the new SVEDAS index shows much greater sensitivity to subtle differences in regularity. In the course of the work, an add-on for Blender was also developed to generate façade composition patterns, based on the typology of composition patterns by Malevich, Taraszkiewicz and Czyż (2022b). The add-on has the ability to generate compositions with a predefined range of SVEDAS factor . Thus, the possibility of using the determined compositional regularity coefficient as an input function in façade generative algorithms was verified. Optimizing façade design for regularity promotes the goals of SDG 11 through architectural design. The optimization of regularity serves to improve 22 architectural aesthetics and furthersupports the creation of safe, inclusive, sustainable and high-quality cities and neighborhoods. 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