Abstract
In the late 1970s Mandelbrot argued that natural systems frequently possess characteristic geometric or visual complexity over multiple scales of observation, suggesting that systems which have evolved over time may exhibit certain local visual qualities that also possess deep structural resonance. In mathematics this led to the formulation of fractal geometry and was central to the rise of the sciences of non-linearity and complexity. This concept was developed in relation to architectural design and urban planning, and architectural scholars have suggested that such approaches might be used in the analysis of historic buildings. At the heart of this approach, in both its theoretical and computational forms, is a set of rules for analysing buildings. However, the assumptions implicit in this method have never been adequately questioned. This chapter returns to the origins of the conventional “box counting” method of fractal analysis for historic buildings to reconsider the initial interpretations of the architecture of Le Corbusier and Frank Lloyd Wright.
First published as: Michael Ostwald , Josephine Vaughan and Chris Tucker, “Characteristic Visual Complexity: Fractal Dimensions in the Architecture of Frank Lloyd Wright and Le Corbusier ”. Pp. 217–231 in Nexus VII: Architecture and Mathematics, Kim Williams, ed. Turin: Kim Williams Books, 2008.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
While Bovill ’s Robie House façade is a relatively close approximation of Wright’s original elevation, the elevation Bovill uses of the Villa Savoye lacks the same level of detail and it is less consistent in its relationship with the original.
- 2.
In its default mode, Benoit determines starting grid size by dividing the shortest input boundary by four. The grid is then extrapolated across the image until the last complete grid-line fits on the page. Any additional space in the source image is deleted. Thus, if the input image was 500 × 1020 pixels, then Benoit identifies a 4 × 8 grid of 125 × 125 pixels and it discards a 20 pixel wide slice of space at the boundary of the input image.
- 3.
The settings of Benoit were modified to match Archimage’s. Thus scaling coefficients, grid angles, and line widths are consistent between the two programs.
- 4.
Despite its designation, original working drawings record that this project was for Raymond W. Evans.
- 5.
Archimage and Benoit produce data to three or more decimal places but present it rounded to, respectively, two and three decimal points. In the present research the D (Arch + Benoit) values are rounded to two decimal places while D (Comp) results are left at three (because they are an average of two results).
References
Ando, Tadao, ed. 2001. Le Corbusier: Houses. Tokyo: Toto Shuppan.
Batty, Michael and Paul Longley. 1994. Fractal Cities: A Geometry of Form and Function. New York: Academic Press.
Bechhoefer, William and Marilyn Appleby. 1997. Fractals, Music and Vernacular Architecture: An Experiment in Contextual Design. In Critical Methodologies in the Study of Traditional Environments (Working Paper Series, vol. 97), Nezar Al Sayyad, ed. Berkeley: University of California at Berkeley. (Unpag.)
Bovill, Carl. 1996. Fractal Geometry in Architecture and Design. Boston: Birkhäuser.
———. 1997. Fractal Calculations in Vernacular Design. In Critical Methodologies in the Study of Traditional Environments (Working Paper Series, vol. 97), Nezar Al Sayyad, ed. Berkeley: University of California at Berkeley. (Unpag.)
Burkle-Elizondo, Gerardo, Nicoletta Sala and Ricardo David Valdez-Cepeda. 2014. Geometric and Complex Analyses of Maya Architecture: Some Examples. Pp 113–126. in Kim Williams and Michael J. Ostwald eds. Architecture and Mathematics from Antiquity to the Future: Volume I Antiquity to the 1500s. Cham: Springer International Publishing.
Cooper, J. 2003. Fractal Assessment of Street-level Skylines: a Possible Means of Assessing and Comparing Character. Urban Morphology: Journal of the International Seminar on Urban Form 7, 2: 73–82.
———. 2005. Assessing Urban Character: The Use of Fractal Analysis of Street Edges. Urban Morphology: Journal of the International Seminar on Urban Form 9, 2: 95–107.
Curtis, William J. 1986. Le Corbusier: Ideas and Forms. New York: Rizzoli.
Feder, Jens. 1988. Fractals. New York: Plenum Books.
Foroutan-Pour, K., P. Dutilleul and D. L. Smith. 1999. Advances in the Implementation of the Box-counting Method of Fractal Dimension Estimation. Applied Mathematics and Computation 105, 2: 195–210.
Gans, Deborah. 2000. The Le Corbusier Guide. Princeton: Princeton Architectural Press.
Hillier, Bill. 1996. Space is the Machine: A Configurational Theory of Architecture. Cambridge: Cambridge University Press.
Kakei, Hidekazu and Setsuko Mizuno. 1990. Fractal Analysis of Street Forms. Journal of Architecture, Planning and Environmental Engineering 8, 414: 103–108.
Lorenz, Wolfgang. 2003. Fractals and Fractal Architecture. Masters Diss. Vienna University of Technology. http://www.iemar.tuwien.ac.at/modul23/Fractals/.
Makhzoumi, Jala and Gloria Pungetti. 1999. Ecological Landscape Design and Planning: The Mediterranean Context. London: E and FN Spon.
Mandelbrot, Benoit. 1977. Fractals: Form, Chance, and Dimension. San Francisco: W. H. Freeman and Company.
———. 1982. The Fractal Geometry of Nature. New York: W. H. Freeman and Company.
Meier, Richard. 1972. Le Corbusier: Villa Savoye, Poissy, France, 1929–31. Global Architecture, vol. 13. Tokyo: A.D.A. Edita.
Oku, T. 1990. On Visual Complexity on the Urban Skyline. Journal of Planning, Architecture and Environmental Engineering 8, 412: 61–71.
Ostwald, Michael J. 2001. Fractal Architecture: Late Twentieth Century Connections Between Architecture and Fractal Geometry. Nexus Network Journal 3, 1: 73–84.
———. 2003. Fractal Architecture: The Philosophical Implications of an Iterative Design Process. Communication and Cognition 36, 3/4: 263–295.
Pfeiffer, Bruce Brooks and Yukio Futagawa. 1987. Frank Lloyd Wright Monograph 1907–1913, Vol. 3. Tokyo: A.D.A Edita.
Rodin, Vladimir and Elena Rodina. 2000. The Fractal Dimension of Tokyo’s Streets. Fractals 8, 4: 413–418.
Schroeder, Manfred. 1991. Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. New York: W. H. Freeman and Company.
Storrer, William Allin. 1993. The Frank Lloyd Wright Companion. Chicago: University of Chicago Press.
Thomson, Iain. 1999. Frank Lloyd Wright: A Visual Encyclopedia. London: PRC.
Yamagishi, R., S. Uchida and S. Kuga. 1988. An Experimental Study of Complexity and Order of Street-Vista. Journal of Architecture, Planning and Environmental Engineering 2, 384: 27–35.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Ostwald, M.J., Vaughan, J., Tucker, C. (2015). Characteristic Visual Complexity: Fractal Dimensions in the Architecture of Frank Lloyd Wright and Le Corbusier. In: Williams, K., Ostwald, M. (eds) Architecture and Mathematics from Antiquity to the Future. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-00143-2_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-00143-2_22
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-00142-5
Online ISBN: 978-3-319-00143-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)