Creat. Sp.

Application of Fractal Growth Patterns in Housing Layout Design

Subhadha Battina


Fractals, Computation in Design, Housing Layout Patterns, Organic Growth Patterns

PUBLISHER The Author(s) 2015. This article is published with open access at

In housing layout design, typically, dwelling units are created by the same sequence of rules, based on the form of a generic house. Each of the houses has to meet certain conditions imposed on it by the site topology, its position, its geometry, and so on. Residential projects may also prefer to include predefined dwelling units with specific area requirements to provide choice for the buyers. Repetitions of identical or similar forms create a predictable pattern [21]. This configuration is comparable to that of a typical self-similar fractal.

Bovill, in his book, ‘Fractal Geometry in Architecture and Design’, describes Fractal Geometry as a study of mathematical shapes that display a cascade of never-ending, self-similar, meandering detail as one observes them more closely [6]. For example, the pattern of additive formation of leaves, in which, smaller elements are defined by the same morphogenetic rules as the whole. Here, individual leaves are formed by the interaction between these rules and the local conditions that the leaves are subjected to [1]. Organisation of many natural forms found in everyday life is fractal-like. As seen in organic forms, proportional similarity seen in design is a connective mechanism of our perception [15]. Mitchell (1989), in his work, compares the theory of fractal geometry to a system of rules that can manipulate patterns of recursive elements to create architecture [14]. We respond easily to designs that mirror the patterns of nature as these give a better understanding of relationship between elements. The research starts with a premise that when this theory is applied to housing layout patterns, the resulting variation could capture the recursive and the complex rhythms of fractals. The study also aims at developing a computer application, based on algorithms using fractals, which offers capabilities as a conceptual and organisational tool for a housing layout.

The Sierpinski Gasket, in particular, is one Fractal type in which each of the smaller elements has a scaling similarity with the largest (Figs. 1a, 1b & 1c). Creating housing layouts using principles of a Sierpinski Gasket can make the design more coherent by relating the scale of the overall layout to that of the smaller housing units. Each of the dwelling units can be created recursively with a consistent proportional relationship with the site. The occurrence of the house on the site in the fractal pattern would depend on the organising principles of the Sierpinski Gasket and also the local conditions of the site.


In early phases of design, during the process of form-exploration, architects -- knowingly or unknowingly -- have used mathematics as their guiding tool to evolve a formal methodology of design. Fundamental compositional principles such as symmetry, rhythm and proportion are based on specific mathematical underpinnings. However, very often the designer comes across a situation where these underlying mathematical principles need to be overlapped or interfaced. Applying fractal concepts to the order can accommodate this complex diversity. Fractals allow us to provide a combination of order and surprise in a rhythmic composition using a specific mathematical geometry. Fractals are typically unit-based and, can thus allow exploration in architectural designs which have a ‘unit’ as a fundamental issue or necessity. The design of housing layout stands out prominently among such architectural problems and, can thus be one such instance in which fractals may be used as a design tool. Commonly seen organisational patterns in housing layout designs create rigidity and monotony, while others like clustered groups are too inconsistent and can create disorder. The research tries applying fractal ordering principles to strike a balance between these extremes by creating an orderly arrangement of houses with an underlying variation in the pattern. The traditional processes of creating housing layouts is quite cumbersome. With the mathematical power of computers, fractal ordering principles are used as Iterative functions to generate multiple design options. The research investigates the potential of the emergent patterns of fractals as an organisational principle in designing housing layouts, while limiting it based on site constraints, size and the transforming rules. In doing so, the objective is to explore the computational and mathematical basis of repetitive patterns in architectural order and compositions. The study also aims at developing a computer application, based on algorithms using fractals, which offers capabilities as a conceptual and organisational tool for a housing layout. The application is implemented, tested and its results are demonstrated using a live terrain data.

Page(s) 185–207
ISSN Print : 2321-3892, Online : 2321-7154
DOI 10.15415/cs.2016.32012

The outcome of the study is a computer application, based on the representation of fractal distributions of the Sierpinski Gasket, which offers capabilities as a conceptual and organizational tool for a housing layout. The variation in the transformation rules that can be added to the Sierpinski Gasket makes the application more flexible. The findings of the study provide a good insight of using an optimal range of transformations that are possible in capturing a feasible housing layout.

Unlike the commonly used organisational principles in housing layouts, this paper offers a design rule for ordering of houses in which the distribution of sub-units are done according to the sub divisive mathematical rules of the Sierpinski Gasket that are not pre-conceived by the user. This research is carried out on the premise that rhythmic or repetitive architectural design can be based on scientific mathematical principles that are analogous to some of the structural laws in biology.

The final outcome is a fractal layout in which there is a consistent relationship between whole-to-part, part-to-whole and part-to-part. The layout is flexible can accept growth without changing the overall character of the pattern. The results of the case studies were closely observed to draw inferences of the impact of the changing parameters and transformation rules. Different patterns are created with changing reference points and orientation of child objects. Creating child units that are far apart from each other creates a more sprawling layout. In case vice versa, it creates a layout that has houses that are closer to one another and the overall layout is denser. The directions of the vectors of the Units can be changed in order to flip or rotate the configuration in parts. By not limiting the fractal to the parent unit, the designer can see the complete layout. These cases may be used to phase a design project for a future development. The paper offers an alternate simpler option to automate an initial phase of housing-layout design. The study provides a method to study and experiment with the concept of imitating nature’s growth patterns.

  • ALEXANDER, C. et al (1969) Houses generated by patterns . Berkley, California: Center for Environmental Structure.
  • ALEXANDER, C. et al (1977) A Pattern Language: Towns, Buildings, Construction. New York: Oxford University Press.
  • ARNHEIM, R. (1977) The Dynamics of Architectural Form . Berkeley: University of California Press.
  • BATTINA, S. (2004) Exploring non-deterministic growth patterns to generate housing layout designs: Fractals as a framework . Unpublished thesis (M.Arch.) Arizona State University, Tempe, USA.
  • BATTY, M. and LONGLEY, P. (1994) Fractal cities: a geometry of form and function , San Diego: Academic Press, London.
  • BOVILL, C. ( 1996) Fractal Geometry in Architecture and Design . Boston: Birkhauser.
  • CHING, F.D.K. (1996) Architecture: Form, Space and Order . New York: Van Nostrand Reinhold,
  • DEVANEY, R. L. (1995) Mathematics and Statistics . Boston University. Available from: [Accessed: April 30, 2016]
  • EASTMAN, C.M. (ed.) (1975) Spatial Synthesis and Computer-aided Building Design. London: Applied Sciences.
  • EMDANAT, S., STINY, G. and VAKALÓ, E-G. (1999) Generative Systems in Design. Artificial Intelligence for Engineering Design Analysis & Manufacturing , 13(4). p. 239-240.
  • KALAY, Y. (ed.) (1987) Computability in Design, New York: Wiley.
  • KARMILOFF-SMITH, A. (1995) Beyond Modularity: A Developmental Perspective On Cognitive Science: Learning, Development, and Conceptual Change. Cambridge, Massachusetts: MIT Press.
  • MITCHELL, W., LIGGETT, R. and KVAN, T. (1987) The Art of Computer Graphics Programming. New York: Van Nostrand Reinhold.
  • MITCHELL, W.J. (1989) The Logic of Architecture: Design, Computation, Cognition. Cambridge, Massachusetts: MIT Press.
  • PEITGEN, H.O., JURGENS, H. and SAUPE, D. (2004,1992) Chaos and Fractals -- New Frontiers of Science. New York: Springer Science.
  • SALINGAROS, N. (1998) A Scientific Basis for Creating Architectural Forms. Journal of Architectural and Planning. [Online] 15(4). p. 283-293. Available from: [Accessed: April 30, 2016]
  • THOMSON, D.W. (1945) On Growth and Form . New York: Cambridge University Press. Available from: [Accessed April 30, 2016]
  • WEBER, R. (1995) On the Aesthetics of Architecture: A Psychological Approach to the Structure and the Order of Perceived Architectural Space. Brookfield, USA: Avebury, Aldershot.
  • William, J.M. (1990) The Logic of Architecture: Design, Computation and Cognition . Cambridge and London: The MIT Press.
  • WRIGHT, F. L. (1992) The machine and the unit system. Unknown Binder
  • ZELANSKI, P. and FISHER, M.P. (1984) Design Principles and Problems. New York: Harcourt School Publisher