Fractal Geometry in Architecture

Fractal geometry is a product of fractal theory, a mathematical approach that describes the way space is filled by figures or objects. A fractal geometric figure is one that can be iteratively subdivided or grown in accordance with a series of rules. The

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractal Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractal Geometry in Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of Fractal Geometry in Architecture and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract Fractal geometry is a product of fractal theory, a mathematical approach that describes the way space is filled by figures or objects. A fractal geometric figure is one that can be iteratively subdivided or grown in accordance with a series of rules. The overall fractal figure then has parts, which under varying levels of magnification tend to look similar – if not identical – to each other, and the figure fills more space than its topological boundaries. While pure mathematical fractal figures can be infinite in their iterations, there are examples of fractal shapes with limited scales that can be found in architecture. This chapter briefly outlines the background of fractal theory and defines fractal geometry. It then looks at the confusion surrounding the claims about fractal geometry in architecture before reviewing the way architecture and fractal geometry can be combined through inspiration, application, or algorithmic generation.

J. Vaughan () The University of Newcastle, Newcastle, NSW, Australia e-mail: [email protected] M. J. Ostwald University of New South Wales, Sydney, NSW, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2020 B. Sriraman (ed.), Handbook of the Mathematics of the Arts and Sciences, https://doi.org/10.1007/978-3-319-70658-0_11-2

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J. Vaughan and M. J. Ostwald

Keywords Fractal geometry · Architecture · Design · Interpretation · Critique

Introduction Benoît Mandelbrot (1982) originally defined fractal geometry as a type of deep geometric phenomena that arises from the application of a system of repetitively applied feedback rules. Today fractal geometry has become a valuable branch of mathematics, with applications in many fields. This chapter examines the way architects and scholars have incorporated fractal geometry into the design and interpretation of the built environment. By the start of the twenty-first century, more