The Fractal Dimension of Architecture
With this book, Birkhäuser starts a new book series Mathematics and the Built Environment. The books in this series will focus on "the complex interaction between mathematics and architecture". The present book fits well into this topic as its title already suggests. It is written by experts who have published a lot on this particular subject on the boundary of architecture and mathematics. Michael J. Ostwald is professor and dean of architecture at the University of Newcastle, Australia. Josephine Vaughan is also working at the same university and specializes in the subject of this book.
Fractal dimensions were introduced by Benoit Mandelbrot. There are however different definitions possible and there are several ways in which to compute it. Here the authors have chosen to use the so called box counting method to compute the fractal dimension of a design of a building. Traditionally this methods looks at some structure at different resolution levels by covering it with smaller and smaller boxes (i.e., squares in a 2D case) from a regular grid. The idea to define the fractal dimension is the following. Count the number N(i) of boxes needed to cover all the details of the picture for grid size s(i). By doing this for different grid sizes, and plotting log(N(i)) versus log(1/s(i)) one gets points of a graph that shows a specific trend. The fractal dimension is then defined as the average slope defined by these points. It is simple and always applicable so that more and more studies (in architecture) become available using this technique.
Next question is on what structure this method should be applied. A choice is made for two types of graphical representations to characterize the construction: the frontal elevation and the top plan view. Each of these can be given with increasing detail by starting with the outline and successively adding the primary, the secondary, and the tertiary forms of the design, and finally the texture. With these successive levels of detail, one can focus the research and zoom in on the different levels of the design aspects and draw conclusions for each of them.
However two more components influence the results and hence the conclusions that can be drawn. The first one is the thickness of the lines. The thickness can vary within one graphic or vary over different graphics that one wants to compare. The second one is how the rectangular grid of boxes is placed over the graphic, more precisely where the relevant structure is located in the grid. Is it at the center, or close to the outer boundary of the grid, for example in a corner of the grid? These two components require a pre- and a post-processing step. In the pre-processing one has to decide on the size of the grid and the position of the structure, the line thickness and the resolution of the image (number of pixels). Only then the box counting method can be applied in a uniform way and allow comparison over different graphics. The post-processing then does the statistical processing of the box counting data. All these aspects are tested so that optimal values can be set for all of the relevant parameters.
Once all this preparatory analysis is done in Part I of the book, Part II does the field work and analyses 85 designs using the technique described. These are carefully chosen, ranging over the period 1901 to 2007, hence covering different style periods, located in different countries, designed by different architects. All the details of the the buildings and the data of the analysis are reported and discussed. Therefore Part II takes about 2/3 of the book. All these data then lead to answers to three hypotheses that the authors have put forward at the beginning: (1) The complexity of the groupings and functions within the home has reduced over time, and this is reflected in a reduction of the fractal dimension in plans and elevations over time. The data do not convincingly support this hypothesis. (2) The specific character of a movement or genre is reflected in the fractal dimension. This is actually rejected by the data. (3) The fractal dimension characterizes different architects. Again this is not confirmed, but it gives some idea what else could be done in this respect.
From the content sketched in the above review, it will be clear that this book is in the first place addressing architect students or researchers, but clearly not the mathematicians. Besides some notes on fractals, and a little bit of statistics, there is no definite mathematical content. All the elements of the research methodology, hence also the fractal dimension, are clearly and extensively explained and motivated. Also the analysis of the results and conclusions are carefully described. So it is easy to read and understand for anyone interested in the topic.