Fractal Dimensions in Architecture: Measuring the Characteristic Complexity of Buildings

  • Michael J. OstwaldEmail author
  • Josephine Vaughan
Living reference work entry


In architectural research, debates about the development, function, or appropriateness of building forms have traditionally been dominated by qualitative approaches. These have been common in the past because the full geometric complexity of a building has proven difficult to encapsulate in any single measurement system. Even simple buildings may be made up of many thousands of separate changes in geometry, which combine together across multiple scales to create a habitable or functional structure. However, since the 1990s architectural scholars have begun to adopt one particular method for mathematically examining the form of a building. This method relies on fractal dimensions, which are measures of the characteristic complexity of an image, object, or set. This chapter introduces fractal dimensions and the primary method used to measure them in architecture, the box-counting approach. The chapter describes key methodological variables and limits that are pertinent to its application in architecture, and then it summarizes the results of past research using this approach. The paper concludes with a tabulated set of typical fractal dimension ranges for sets of plans and elevations of designs by 11 famous architects or practices.


Fractal dimension Architecture Design Box-counting method Measurement Assessment 


  1. Asvestas P, Matsopoulos G, Nikita K (2000) Applications of fractal theory on medical data processing. In: Marsh A, Grandinetti L, Kauranne T (eds) Advanced infrastructures for future healthcare. IOS Press, Amsterdam, pp 425–411Google Scholar
  2. Bechhoefer W, Bovill C (1994) Fractal analysis of traditional housing in Amasya, Turkey. Tradit Dwellings Settlements Work Pap Ser 61:1–21Google Scholar
  3. Bovill C (1996) Fractal geometry in architecture and design. Birkhäuser, BostonCrossRefGoogle Scholar
  4. Burkle-Elizondo G (2001) Fractal geometry in Mesoamerica. Symmetry: Cult Sci 12:201–214zbMATHGoogle Scholar
  5. Burkle-Elizondo G, Valdez-Cepeda RD (2001) Do the Mesoamerican artistic and architectural works have a fractal dimension? In: Novak MM (ed) Emergent nature. Patterns, growth and scaling in the sciences. World Scientific, Singapore, pp 431–432Google Scholar
  6. Çagdaş G, Gözübüyük G, Ediz Ö (2005) Fractal based generative design for harmony between old and new. In: Generative Art 2005, Milan, Italy, 15–17 December, Domus Argenia, Milan, pp 150–159Google Scholar
  7. Capo D (2004) The fractal nature of the architectural orders. Nexus Netw J 6:30–40CrossRefGoogle Scholar
  8. Debailleux L (2010) Complementary approach for vernacular wooden frame structures reconstruction. In: Digital heritage, third international Euro-mediterranean conference, Euromed 2010, Lemessos, Cyprus, 8–13 Nov. Springer, Berlin, pp 441–449CrossRefGoogle Scholar
  9. Ediz Ö, Ostwald MJ (2012) The Süleymaniye Mosque: a computational fractal analysis of visual complexity and layering in Sinan’s masterwork. arq: Archit Res Q 16:171–182CrossRefGoogle Scholar
  10. Feder J (1988) Fractals. Plenum Press, New YorkCrossRefGoogle Scholar
  11. Li J, Du Q, Sun C (2009) An improved box-counting method for image fractal dimension estimation. Pattern Recogn 42:2460–2469CrossRefGoogle Scholar
  12. Lorenz WE (2003) Fractals and fractal architecture. Masters dissertation. Vienna University of Technology, ViennaGoogle Scholar
  13. Lorenz W E (2012) Estimating the fractal dimension of architecture: Using two measurement methods implemented in AutoCAD by VBA. In Digital Physicality, eCAADe 2012, Prague, Czech Republic, 12–14 Sept. ČVUT, Prague, pp 505–513Google Scholar
  14. Mandelbrot BB (1977) Fractals: form, chance and dimension. W.H. Freeman, New YorkzbMATHGoogle Scholar
  15. Mandelbrot BB (1982) The fractal geometry of nature. W.H. Freeman, San FranciscozbMATHGoogle Scholar
  16. Oleschko K, Brambila R, Brambila F, Parrot J-F, López P (2000) Fractal analysis of Teotihuacan, Mexico. J Archaeol Sci 27:1007–1016CrossRefGoogle Scholar
  17. Ostwald MJ (2001) Fractal architecture: late twentieth century connections between architecture and fractal geometry. Nexus Network Journal 3:73–84CrossRefGoogle Scholar
  18. Ostwald MJ (2003) Fractal architecture: the philosophical implications of an iterative design process. Commun Cogn 36:263–295Google Scholar
  19. Ostwald MJ (2013) The fractal analysis of architecture: calibrating the box-counting method using scaling coefficient and grid disposition variables. Environ Plan B: Plan Des 40:644–663CrossRefGoogle Scholar
  20. Ostwald MJ, Ediz Ö (2015) Measuring form, ornament and materiality in Sinan’s Kılıç Ali Paşa Mosque: an analysis using fractal dimensions. Nexus Netw J 17:5–22CrossRefGoogle Scholar
  21. Ostwald MJ, Vaughan J (2009) Visual qualities in early modern and late modern architecture: a mathematical comparison of formal complexity in the houses of gray and Sejima. In: Gu N, Ostwald MJ, Williams A (eds) Computing, cognition and education: recent research in the architectural sciences. ANZAScA, Sydney, pp 9–32Google Scholar
  22. Ostwald MJ, Vaughan J (2013) Representing architecture for fractal analysis: a framework for identifying significant lines. Archit Sci Rev 56(3):242–251CrossRefGoogle Scholar
  23. Ostwald MJ, Vaughan J (2016) The fractal dimension of architecture. Birkhauser, ChamCrossRefGoogle Scholar
  24. Rian IM, Park J-H, Ahn HU, Chang D (2007) Fractal geometry as the synthesis of Hindu cosmology in Kandariya Mahadev temple, Khajuraho. Build Environ 42:4093–4107CrossRefGoogle Scholar
  25. Vaughan J, Ostwald MJ (2008) Approaching Euclidean limits: a fractal analysis of the architecture of Kazuyo Sejima. In: Innovation inspiration and instruction: new knowledge in the architectural sciences: ANZAScA 08, Newcastle, Australia, 26–28 Nov. ANZAScA, Newcastle, pp 285–294Google Scholar
  26. Vaughan J, Ostwald MJ (2010) Using fractal analysis to compare the characteristic complexity of nature and architecture: re-examining the evidence. Archit Sci Rev 53:323–332CrossRefGoogle Scholar
  27. Vaughan J, Ostwald MJ (2011) The relationship between the fractal dimension of plans and elevations in the architecture of Frank Lloyd Wright: comparing the prairie style, textile block and Usonian periods. ArS Archit Sci 4:21–44Google Scholar
  28. Voss RF (1986) Characterization and measurement of random fractals. Phys Scr T13:27–32CrossRefGoogle Scholar
  29. Voss RF (1988) Fractals in nature: from characterization to simulation. In: Peitgen HO, Saupe D, Barnsley MF (eds) The science of fractal images. Springer, New York, pp 21–70CrossRefGoogle Scholar
  30. Wen K-C, Kao Y-N (2005) An analytic study of architectural design style by fractal dimension method. In: 2005 Proceedings of the 22nd ISARC, ISARC, Ferrara, Italy, pp 1–6Google Scholar
  31. Zarnowiecka J C (2002) In search of new computer tools: what does Bovill really measure in architecture? In: Connecting the real and the virtual – design education 20th eCAADe conference proceedings, eCAADe, Warsaw, pp 342–345Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The University of NewcastleNewcastleAustralia

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