European Mathematical Society - Michael J. Ostwald
https://euro-math-soc.eu/author/michael-j-ostwald
enThe Fractal Dimension of Architecture
https://euro-math-soc.eu/review/fractal-dimension-architecture
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
With this book, Birkhäuser starts a new book series <em>Mathematics and the Built Environment</em>. 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.</p>
<p>
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.</p>
<p>
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.</p>
<p>
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.</p>
<p>
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) <em>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.</em> The data do not convincingly support this hypothesis. (2) <em>The specific character of a movement or genre is reflected in the fractal dimension.</em> This is actually rejected by the data. (3) <em>The fractal dimension characterizes different architects.</em> Again this is not confirmed, but it gives some idea what else could be done in this respect.</p>
<p>
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.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
The technique of box counting to compute the fractal dimension is applied to drawings of the elevation and the top plan of buildings, which are represented at successive levels of detail. In part I the methodology is explained and tested. In part II this is applied to 85 designs to test three hypotheses relating the fractal dimension to the complexity of functionality, the stylistic genre, and the individual architects.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/michael-j-ostwald" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Michael J. Ostwald</a></li><li class="vocabulary-links field-item odd"><a href="/author/josephine-vaughan" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Josephine Vaughan</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-verlag" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Birkhäuser Verlag</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2016</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-32424-1 (hbk)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">116.59 € (hbk)</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">422</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/geometry" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Geometry</a></li><li class="vocabulary-links field-item odd"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319324241" title="Link to web page">http://www.springer.com/gp/book/9783319324241</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a67" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a67</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-other field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/28a80" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">28A80</a></li></ul></span>Fri, 02 Dec 2016 11:36:37 +0000Adhemar Bultheel47306 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/fractal-dimension-architecture#commentsArchitecture and Mathematics from Antiquity to the Future
https://euro-math-soc.eu/review/architecture-and-mathematics-antiquity-future
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
There has been a growing interest in the interaction between mathematics and architecture. That was the reason that the first Nexus conference on this topic was organized in 1996 and these conferences are being organized on a biennial basis ever since. The need for a platform to discuss this interaction on a more continuous basis grew and soon the <em>Nexus Network Journal</em> (NNJ) was started by Kim Williams. It is available since 1999. She and Michael Ostwald, who is the current editor in chief of the NNJ, have collected in this period almost 100 papers on the subject that they have collected in this comprehensive two volume set.</p>
<p>
The ambitious goal is to describe both the intimate relation but also the alienation between mathematics and architecture and between mathematicians and architects. The issue is covered both thematically and chronologically which means that concepts such as measurements, composition, skills, etc. are discussed and that such groups of papers are interlaced with sets of papers that deal with a certain period. Of course, as time evolves, new concepts were born, changing the way people look at ideas causing the rise and decline of tools and techniques and the way architecture is realized.</p>
<p>
The two volumes consist of 9 parts, each collecting an average of around 10 papers written by experts. That includes architectural historians, designers, mathematicians, engineers, philosophers, and computer scientists. This diversity already illustrates the broadness and the interdisciplinary aspects of the material. Each volume starts with a paper that gives an extensive description of the contents of the books. An obvious start is to try to answer the question what the common or binding concepts between the two disciplines are. This is followed by a discussion of the first period 2000 BC - 1000 AD. Then it is explained how mathematics became important in developing methods of measurements and to guarantee stability of constructions. That is explored in the historical section 1000 - 1400 (from Medieval to Romanesque with early Gothic churches). Next the concepts of proportion, symmetry and periodicity are introduced. While Vitruvius referred to the human body, now ratios were related to a music scale. These concepts of course became important in the Gothic buildings and even more so for the Renaissance in the period 1400 - 1500. Of course several famous buildings are discussed in detail with ample illustrations. The examples cover geographically not only the obvious regions that influenced Western civilization like Egypt, Greece, Rome, Persia, etc., but include also Mayan and Indian elements.</p>
<p>
The second volume continues in the same style. There are two historical parts covering the periods 1500 - 1800 and 1800 to the present time respectively. The first period is connected with the introduction of perspective and work by Palladio, Borromini, Michelangelo, Wren,... After 1800, we meet Frank Loyd Wright, Le Corbusier, Louis Kahn, Niemeyer, Gerhy, and many others with all the variants of Modernism and Post-Modernism. In the later parts of this volume also architecture from North America and Oceania enters the picture. It is in the nineteenth-twentieth century that mathematics and architecture somewhat diverged, being in the different continents of science and arts that were drifting away from each other like tectonic plates. Although of course many architectural constructions were only possible because of technical evolution and there were definitely interactions of how mathematics and architecture developed. For example new mathematical findings such as aperiodic tiling were applied by architects and computer systems were used to analyze ancient amphitheaters or laser scans were used for 3D modeling. Towards the end the thematic and historical subdivision becomes more fuzzy. The concluding part of this volume even sheds some light on the future, highlighting opportunities and challenges that arise by increased application of computers in architecture.</p>
<p>
Even if you start reading, convinced that mathematics and architecture are related and amply interact, you will still be surprised in how many aspects they are interwoven. Even though sciences and arts have separated, the mutual love between mathematics and architecture has never died, and as you read the later chapters in the book, it will be clear that the mutual attraction is nowadays still very strong, like the Nexus platform, the Nexus journal, and these volumes and several other books illustrate. It is an important and highly inspiring collection of papers that will be of interest to researchers from as many disciplines as illustrated by the diversity of the background of the authors. It revives some of the former polymath idea that has been gradually lost in the 20th century. Highly recommended for readers who do not want to drown or hide in their own abyss of specialization.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">Adhemar Bultheel</div></div></div><div class="field field-name-field-review-desc field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>
This is a collection of papers that highlight the interaction between mathematics and architecture from 2000 BC to the present time. Volume I covers the period till 1500, and the rest is in the second volume. Besides a survey paper that starts each volume, there are ninety contributions by a wide variety of authors (historians, designers, philosophers, mathematicians, engineers, computer scientists). The papers are grouped partially chronologically by period, and partially thematically when it concerns concepts and tools.</p>
</div></div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/kim-williams" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Kim Williams</a></li><li class="vocabulary-links field-item odd"><a href="/author/michael-j-ostwald" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Michael J. Ostwald</a></li><li class="vocabulary-links field-item even"><a href="/author/eds-1" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">(eds.)</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/birkh%C3%A4user-basel" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">birkhäuser basel</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2015</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">978-3-319-00136-4 (hbk) (2 volume set)</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">211,99 €</div></div></div><div class="field field-name-field-review-pages field-type-number-integer field-label-inline clearfix"><div class="field-label">Pages: </div><div class="field-items"><div class="field-item even">1465</div></div></div><span class="vocabulary field field-name-field-review-class field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/imu/mathematics-science-and-technology" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">Mathematics in Science and Technology</a></li></ul></span><div class="field field-name-field-review-website field-type-text field-label-hidden"><div class="field-items"><div class="field-item even"><a href="http://www.springer.com/gp/book/9783319055411" title="Link to web page">http://www.springer.com/gp/book/9783319055411</a></div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/00-general" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00 General</a></li></ul></span><span class="vocabulary field field-name-field-review-msc-full field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc-full/00a67" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">00a67</a></li></ul></span>Tue, 23 Jun 2015 15:04:32 +0000Adhemar Bultheel46274 at https://euro-math-soc.euhttps://euro-math-soc.eu/review/architecture-and-mathematics-antiquity-future#comments