Josephine Vaughan

Characteristic visual complexity: fractal dimensions in the architecture of Frank Lloyd Wright and Le Corbusier

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.

Using fractal analysis to compare the characteristic complexity of nature and architecture: re-examining the evidence

Since the late 1970s, Complexity Scientists have shown that many natural systems possess similar geometric patterns that are repeated over multiple scales of observation. This observation lead to the development of fractal geometry, as a means for determining, over progressive scales, the characteristic visual complexity of natural systems and forms. More recently it has been demonstrated that fractal geometry can also be useful for measuring the visual complexity of the constructed environment. As a result of this, it became possible, in the late 1990s to compare the fractal dimensions of both natural and constructed forms and thereby investigate the extent to which buildings are a reflection of their natural setting. The most famous examples of this provided evidence that a formal connection could be traced between the city of Amasya in Turkey and the landforms of its natural setting, and a similar argument has been made about Sea Ranch in California. With only minimal evidence, a range of conclusions were drawn from this early work about the connection and influence of a local ecology on local architecture. The present article re-tests these previous results using a more refined, computational version of the fractal analysis method.

Representing architecture for fractal analysis: a framework for identifying significant lines

Over the last two decades, a range of computational techniques have been developed for measuring the formal characteristics of architecture. One of the most widely used methods, fractal analysis, measures the typical or characteristic spread of visual information (form and texture) present in a plan or elevation. However, when preparing a plan or elevation for this method, several critical decisions must be made about which architectural features should be included in the representation and why. Without a consistent, reasoned way of making such decisions, isolated fractal dimension results for buildings are potentially meaningless. Therefore, the present paper draws on postpositivist reasoning to propose a framework for deciding which lines in an architectural representation are significant for a study and why. The framework contains five cumulative levels of representation that are defined and mapped against comparable research agendas. These levels are described and demonstrated using a plan and an elevation from Le Corbusiers Villa Jaquemet. In each case, the results of the fractal analysis of different representations of the Villa are used to demonstrate how decisions about significant lines have a direct impact on measures derived from fractal analysis.

Measuring the significance of façade transparency in Australian regionalist architecture: A computational analysis of 10 designs by Glenn Murcutt

Historians and critics argue that a key characteristic of late twentieth century Australian regionalist architecture is the close visual connection it creates between the interior and the landscape. While various design properties are allegedly responsible for this connection, one of the most tangible of these is associated with the use of transparent and layered elements in a buildings façade. Indeed, as exemplified in the work of Glenn Murcutt, the importance of façade transparency is a recurring theme in Australian architecture. But is it really that significant? In this paper, computational fractal analysis is used to measure the difference between the visual complexity of opaque and transparent depictions of façades. By comparing these two façade conditions, first in sets of elevations derived from 10 of Murcutts houses and then in a detailed review of one of Murcutts most iconic works, the Marie-Short House, this paper calculates the visual impact of transparency on the characteristic complexity of Murcutts architecture.

Fractal Dimensions in Architecture: Measuring the Characteristic Complexity of Buildings

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 M. J. Ostwald ( ) • J. Vaughan The University of Newcastle, Newcastle, NSW, Australia e-mail: michael.ostwald@newcastle.edu.au; josephine.vaughan@uon.edu.au

The comparative numerical analysis of nature and architecture: a new framework

Maintaining or creating a visual relationship between the form of a building and its surrounding natural landscape is often cited as a crucial factor in producing designs that support psychological comfort or environmental sustainability. While multiple methods for the analysis of nature and architecture have developed over time, only a handful of past studies have ever attempted to quantitatively compare the geometric properties of nature to those of architecture. Fractal analysis provides one of the very few methods available to analyse and compare the geometry of diverse objects. The fractal dimension (D) of an object is a numerical value which reflects the volume and distribution of detail in an item. Of the many subjects analysed using this method, the forms of nature (such as coastlines, rivers and plant elements) have been successfully measured, as have built forms (such as houses, public buildings and cityscapes). However, despite the method’s application to each subject area, few examples exist where fractal dimension data derived from nature are compared with equivalent architectural data. A primary reason cited for this situation is the disparity of methodological variables, in particular, representational approaches to the images used for comparison are presently disparate and uncategorised. This paper responds to the existing lack of a comparable basis by analysing and categorising methodological examples from applications of fractal analysis to both natural and architectural cases. Specifically, the type of image delineation and the level of information contained in it are compared and ranked. Through this process, the paper provides a critical overview of the past application of fractal analysis to images, and thereby provides a starting framework for how the built and natural environments might be rigorously compared in the future.

Analysing the Twentieth-Century House

Whereas Part I of this book provides a detailed review of the intricacies of the box-counting method, Part II presents an application of this method to a specific project: a review of the formal complexity of eighty-five canonical designs. This chapter describes the method used for this larger research project, its rationale, limitations and scope. It includes a detailed discussion of the stages in the research process and the use of calculated and derived measures to characterise and compare the properties of individual designs and sets of designs. It concludes with a short, hypothetical example of the way the results of this research will be visualised and interpreted throughout the later chapters.

Introducing the Box-Counting Method

This chapter presents three worked examples of the most basic variation of the box-counting method for calculating the fractal dimension of an image.

Fractals in Architectural Design and Critique

This book is about the analysis of architecture using fractal dimensions. This method and its application are described in detail in the coming chapters, but it must also be acknowledged that the relationship between fractals and architecture has traditionally been both more diverse and more controversial than the scope of this book might imply. For thirty years architectural scholars and designers have opportunistically appropriated images and ideas from fractal geometry along with concepts broadly related to fractal dimensions and non-linear dynamics, and used them for a wide variety of purposes.

The Avant-Garde and Abstraction

The works of three architects who have variously been described as Late Modernists or Neo-Modernists are examined in this chapter.