Mathematical beauty and Palladian architecture: Measuring and comparing visual complexity and diversity

Abstract

Palladio’s design principles, including proportion and harmony, have often been associated with mathematical definitions of beauty. However, the geometric and semantic properties of his façades have rarely been analysed in a holistic manner. There is little evidence of how mathematical beauty may be embodied in his architecture. This research investigates complexity (fractal dimension or D) and diversity (perplexity or PP) as aesthetic indices, aiming to capture distinct characteristics of Palladian architecture. The D and PP values of 22 Palladian villa façades are measured and analysed, before being compared with those of three Renaissance facades by Sebastiano Serlio. The combination of D and PP captures the geometric and semantic aesthetic qualities of architectural compositions. Importantly, the developed scatter plot of D and PP results supports the identification of four distinct aesthetic types of Palladian façade designs. The novel combination of D and PP measures contributes to a better understanding of one definition of the mathematical beauty of architecture, wherein “the whole is other than the sum of the parts” in compositional terms (a famous Gestalt principle). Specifically, this research provides new mathematical insights into the visual character of Palladian architecture and compares two important measures thereof.

Keywords

Palladian architecture / Complexity / Diversity / Mathematical beauty / Perplexity / Façade design

References

[1] Alberti, L.B., 1988. On the Art of Building in Ten Books (Trans: Rykwert J, Leach N, Tavernor R). MIT Press, Cambridge MA.

[2] Alexander, C., Ishikawa S., Silverstien M., 1977. A Pattern Language: Towns, Buildings, Construction. Oxford University Press, New York.

[3] Benrós, D., Duarte, J.P., Hanna, S., 2012. A new palladian shape grammar: a subdivision grammar as alternative to the palladian grammar. Int. J. Architect. Comput. 10 (4), 521- 540.

[4] Berlyne, D.E., 1971. Aesthetics and Psychobiology. Appleton-Century-Crofts, New York, NY.

[5] Berlyne, D.E., 1974. Studies in the New Experimental Aesthetics: Steps toward an Objective Psychology of Aesthetic Appreciation. Hemisphere Pub. Corp., Washington.

[6] Dawes, M.J., Ostwald, M.J., Lee, J.H., 2021. Examining control, centrality and flexibility in Palladio’s villa plans using space syntax measurements. Frontiers of Architectural Research 10 (3), 467- 482.

[7] Eilouti, B.H., 2008. A formal language for palladian palazzo façades represented by a string recognition device. Nexus Netw. J. 10 (2), 245- 268.

[8] Fletcher, R., 2015. Golden proportions in a great house: Palladio’s villa Emo. In: Williams, K., Ostwald, M.J. (Eds.), Architecture and Mathematics from Antiquity to the Future: Volume II: the 1500s to the Future. Springer International Publishing, Cham, pp. 121-138.

[9] Frijda, N.H., 1986. The Emotions. Cambridge University Press, Cambridge, UK.

[10] García-Salgado, T., 2008. A perspective analysis of the proportions of Palladio’s villa Rotonda: making the invisible visible. Nexus Netw. J. 10 (2), 269- 282.

[11] Grasl, T., 2012. Transformational palladians. Environ. Plann. Plann. Des. 39 (1), 83- 95.

[12] Howard, D., Longair, M., 1982. Harmonic proportion and palladio’s "Quattro libri". J. Soc. Archit. Hist. 41 (2), 116- 143.

[13] Jost, L., 2006. Entropy and diversity. Oikos 113 (2), 363- 375.

[14] Klinger, A., Salingaros, N.A., 2000. A pattern measure. Environ. Plann. Plann. Des. 27 (4), 537- 547.

[15] Lee, J.H., Ostwald, M.J., 2020. Grammatical and Syntactical Approaches in Architecture: Emerging Research and Opportunities. IGI Global, Hershey, PA.

[16] Lee, J.H., Ostwald, M.J., 2021. Fractal dimension calculation and visual attention simulation: assessing the visual character of an architectural façade. Buildings 11 (4), 163

[17] Lee, J.H., Ostwald, M.J., 2022. The ‘visual attractiveness’ of architectural facades: measuring visual complexity and attractive strength in architecture. Architect. Sci. Rev. 1- 11.

[18] Lee, J.H., Ostwald, M.J., 2023a. Complexity and diversity in palladian facades. Nexus Netw. J. 25, 15- 21.

[19] Lee, J.H., Ostwald, M.J., 2023b. The ‘visual attractiveness’ of architectural facades: measuring visual complexity and attractive strength in architecture. Architect. Sci. Rev. 66 (1), 42- 52.

[20] Lee, J.H., Ostwald, M.J., Dawes, M.J., 2022. Examining visitorinhabitant relations in palladian villas. Nexus Netw. J. 24 (2), 315- 332.

[21] Manning, C., Schutze, H., 1999. Foundations of Statistical Natural Language Processing. MIT press, Cambridge, MA.

[22] Millon, H.A., 1972. Rudolf wittkower, "architectural principles in the age of humanism": its influence on the development and interpretation of modern architectur. J. Soc. Archit. Hist. 31 (2), 83- 91.

[23] Ostwald, M.J., Vaughan, J., 2016. The Fractal Dimension of Architecture. Birkhäuser, Cham, Switzerland.

[24] Palladio, A., 1570. I Quattro Libri Dell’architettura. Domenico de’ Franceschi, Venetia.

[25] Palladio, A., 1715. The Architecture of A. Palladio, in Four Books. John Watts, London.

[26] Palladio, A., 1965. The Four Books of Architecture. Dover Publications, New York.

[27] Payne, A.A., 1994. Rudolf wittkower and architectural principles in the age of modernism. J. Soc. Archit. Hist. 53 (3), 322- 342.

[28] Quattrini, R., Baleani, E., 2015. Theoretical background and historical analysis for 3D reconstruction model. Villa Thiene at Cicogna. J. Cult. Herit. 16 (1), 119- 125.

[29] Rowe, C., 1947. The mathematics of the ideal villa. Palladio and le corbusier compared. Architect. Rev. 101, 101- 104.

[30] Saalman, H., 1958. Filippo Brunelleschi: capital studies. Art Bull. 40 (2), 113- 137.

[31] Salingaros, N.A., 1997. Life and complexity in architecture from a thermodynamic analogy. Phys. Essays 10, 165- 173.

[32] Salingaros, N.A., 2020. Symmetry gives meaning to architecture. Symmetry: Culture and Science 31 (3), 231- 260.

[33] Serlio, Sebastiano, 1619. Tutte l’opere d’architettura, et prospetiua. Appresso Giacomo de’ Franceschi, Venetia.

[34] Shannon, C.E., 1948. A mathematical theory of communication. Bell System Technical Journal 27 (3), 379- 423.

[35] Stamps, A.E., 2003. Advances in visual diversity and entropy. Environ. Plann. Plann. Des. 30 (3), 449- 463.

[36] Stiny, G., Mitchell, W.J., 1978. The Palladian grammar. Environ. Plann. B 5 (1), 5- 18.

[37] Vitruvius, 1960. The Ten Books on Architecture (Trans: Morgan MH). Dover Publications, New York.

[38] Wassell, S.R., 2008. Andrea Palladio (1508-1580). Nexus Netw. J. 10 (2), 213- 226.

[39] Williams, K., March, L., Wassell, S.R., 2010. Introduction. In: Williams, K., March, L., Wassell, S.R. (Eds.), The Mathematical Works of Leon Battista Alberti. Springer, Basel, Basel, pp. 1-8.

[40] Wittkower, R., 1971. Architectural Principles in the Age of Humanism. W. W. Norton & Company, New York.

RIGHTS & PERMISSIONS

2024 The Author(s). Publishing services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd.