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Padovan's book has a range of titles for chapters (16 in all covering nearly four hundred pages) that are roughly as expected. They appear to fall broadly into three groups as follows: - The titles in the first half, which range from "Unit and Multiplier", though "The proportions of the Parthenon" to "Euclid: the golden section and the five regular solids," show a firm historical background in perhaps mathematical theory.
- The middle portion covering "Vitruvius", "Gothic Proportions" through to "The golden section and the golden module" with the additional help that the subtitles to each section within the chapters, cover practicalities from the Romans to Le Corbusier.
- The final chapter "The house as frame for living and a discipline for thought" with subtitles to sections dealing with "The search for a starting point" and "What system of proportion do we need?" seem to be reaching a conclusion and asking pertinent questions, if not answering them.
The table of contents is followed by three pages showing the list of approximately 150 figures. This is perhaps slightly less than might be expected and on flicking through the book it is apparent that there are fewer than ought to be there, since many are quite basic or tables of numbers. A book on proportion cries out for communication in visual terms more than words; and so to me it seems that the proportion of the two is wrong, but then this is a book about philosophy as much as about proportion and there are many other themes which permeate the work as well. For example, there is one which Padovan calls "empathy and abstraction"; then there is the cosmological links to architecture and also his championing of the Hans Van der Laan and his plastic number. These and other minor ones are threaded through the text, surfacing obviously or poking their head above the surface when least expected. However, it is not always easy to manoeuvre through the book to follow these threads (if, for example, you have a particular interest in them) without reading through the whole text since they are so tightly bound in; for example, the word cosmology does not appear in the index. I have to confess at this stage that it has taken me a year to read this book. This was partly because of the braiding of the multiple themes, but also because I found I had to concentrate to work through the mathematics for the following reasons. My interest in proportion is in the geometry and how and why it is used in architecture. I am also more pragmatic than philosophical, so the latter approach has made the book harder to read. I was also put off by the first paragraph, in the preface, which I think is worth quoting since it may explain why I found it such a challenge:
He also found out that his fellows, when he did become an architecture student, were no better at mathematics than he was. Padovan goes on to say that he had to discover for himself that numbers and geometrical constructions are beautiful in themselves. I will return to this topic later. However, I think it explains why I believe this book could have be written two hundred years ago, apart from the work of two architects that stand out. Le Corbusier and Hans van der Laan feature in recent history and obviously could not have been written about then since they were not born, but in principle, the impression is one that either mathematics is not required or architects are still dwelling in the past. This is not the case, as seen by some of the papers in the Nexus books and Journals. I would be interested to know how much of this view of the world is prevalent in general. Unfortunately, it is not just this sense of the past that comes over, but that proportion is a Western exclusive privilege.
Padovan's focus on number, as a convenient unit, also seems to limit the extent to which three dimensional and geometric proportions are considered in the work. In working through the mathematics behind proportion, just as the golden section is reduced to the Fibonacci numbers, he reduces all other proportions to series and tables of numbers. I can sympathise with this, because you have to know and understand to get this far, but many people are more at home with the geometry, because they can feel and see it and relate to it with their bodies. The relationship between number and space is further complicated by representation. This can be illustrated by the figures which are meant to convey the notion of arithmetic and geometric progression, preceded here by the notion of counting. The beginning of counting: 1+ 1=2 Arithmetic Progression Geometric Progression Ask yourself where is the multiplication in these figures. It is there, but it takes time to decode. Now an arithmetic progression is addition, and a geometric one is multiplication and, yes, multiplication is a sophisticated form of addition, but to push the visual explanation of a geometric progression back to arithmetic is a step backwards from two dimensions to one. Arithmetic is one dimensional: addition on a line. Multiplication is two dimensional; concerned with area. You cannot deal with proportion in the spaces we live in if you reduce it to cardinal numbers because that infers one dimension. There is an issue which has not been addressed, even though I believe Van der Laan understood it. We do not have an equivalent to continue the series arithmetic = 1, multiplication = 2. What = 3 relates to volume? The reason there is none is because we live in a space of three dimensions and not outside it, but more than that, the geometric tools used were still two dimensional (and of degree two also) until very recently: the ones of the ancient Greeks. This meant that they were not able to duplicate the cube in the famous Delian problem. They did have other tools, but because the use of these tools was restricted they effectively limited their use by architects. We do have the tools to construct in three dimensions, namely CAD programs in the computer. How soon will the 20 years or so we have had them allow us to break away from the 2000 years of ruler and compasses? More importantly, what was the special genius of architects like Palladio and Vitruvius which allowed them to build in space. I am not convinced it was number. Right from the preface Padovan points out one of his themes (for which he borrows the terminology of Wihelm Worringer) that he calls the quarrel between "empathy" and "abstraction". Padovan defines empathy as that "being ourselves part of nature, we have a natural affinity with it and an innate ability to know and understand it". Abstraction, on the other hand is "the contrary tendency to regard nature as elusive and perhaps ultimately unfathomable, and science and art as abstractions, artificial constructions that we hold up against nature to in order in some sense to grasp it and command it." One potential problem with Padovan's "empathy and abstraction"
is that to discuss them in terms of proportion, however much
he may support his argument, is fraught with problems. I believe
we impose conceptual ideas (namely mathematics) on the world
because it is easy to do and because they are a language which
enables us to speak to our fellow human beings. It is surprisingly
easier to understand something if it is symmetrical and ordered.
Now nature may use symmetry, but it may be that we see it as
a form of simplification and not see the full picture. In trying
to piece together the world in our consciousness our brains do
fantastic "computations" and rely both on past experience
and guesswork about which we are not aware. I believe good architects
are intuitive because they use this sense and compromise with
abstract (rather than abstraction) explanations afterwards to
both physically create and communicate the ideas to others. For
example, Le Corbusier is said to have drawn up plans and then
superimposed Modulor measurements on them [Evans 1995]. Secondly,
the Greeks knew about perceptual illusions and made columns taper
towards the top in order that their buildings look "correct".
So to talk about the proportions of a Greek temple is either
over simplifying or massaging the data. Now there is a difference
between the intuition example of Le Corbusier and Padovan's empathy.
Padovan describes Le Corbusier's system, but not his method.
Is he trapped by empathy? I find it difficult to have faith in
any discussion of the "proportions" of the Parthenon
in his description taken in isolation. It is like our struggling
to make sense of the world through science and art and a pointless
"abstraction". What is important is to find evidence
from working papers because that is the only way we can know
method and intent. All the diagrams in the book are redrawn as
far as I can tell since they are all in the same style. Having
been trained as a scientist, this leads me to be deeply suspicious
on two counts. I know for the most part there is no documentary
evidence available, and if there is, it is very easy to adjust
the data to meet the point you are trying to make.
The historical aspects and the links to cosmology seem well
covered in perhaps a fairly standard way. I would praise the
scholarly and not the "sacred geometry" point of view
but I am hesitant to be too drawn to this. Surprisingly, there
is no mention in this book until a brief note towards the end
in a comparison of Ruskin's Victorian ideas of decoration, contrasted
with Le Corbusier's. Padovan infers that Ruskin's decoration
is vulgar ("a sort of tattooing or make up applied to the
surface of the building"), but I am not sure what to make
of Le Corbusier's approach of "a mask covering it up".
This is a great omission since decoration plays a great part
in some types of architecture and defines proportion in other
ways than he considers; one only has to think of Greek columns
to start the flow. Although the book has the sub-title "Science Philosophy Architecture", I can find no science in it. There is quite a bit of mathematics, but mathematics is not science. Mathematics starts from a set of premises, or axioms, and deduces something from there. Philosophically, there is also the question as to whether mathematics is discovered or is the invention of the human mind. I do not recall this being discussed in what is a very philosophical book. But it is vital to any discussion on proportion. Moreover, the science of proportion is also vitally important. It is no good building a model of paper and then scaling it up in stone without knowing that the science and engineering properties of the materials. Without this knowledge, disaster can strike. Buildings have fallen down before the practical science was understood. This brings us back to the practical understanding, and as an example I will take the very simple one of Roriczer's Geometria Deutsch (c1486) which displayed the secrets of the masons and which Dürer used in his books. Padovan discusses the method for construction of a pentagon which is famously in Roriczer's book (and copied by Durer). He notes it is an approximate method; which it is, although quite accurate. He seems to think that Roriczer and the masons would have used Euclid's method. What he fails to realise is the practical difficulties in which the two methods differ. Roriczer's method is a so called "rusty compass" construction. Once you have set the side of the pentagon, the compass setting does not have to be altered. This is of tremendous importance in practice. Euclid's method is not simpler when you take practicality into account. This practical aspect is important in other ways. Padovan mainly discusses systems which are Euclidean, that is constructable with the ruler and compasses and which can only lead to quadratic equations and not cubic or higher. This means that certain regular polygons such as heptagons and enneagons (9 sides) cannot be constructed with these tools, although there are quite accurate approximate methods and the Greeks, particularly Archimedes, had ways of using other tools for accurate construction. When such items are seen by pure mathematicians they will puzzle over how "it cannot be done, but they did it" [Hancox 1997]. But we must be aware of the practical aspects, what was done rather than what is theoretically possible following a mathematically logical deduction. There is also a strong case for ignoring this artificial barrier and using modern tools like CAD programs to explore new proportions that were not easily possible in the past. [1]
Padovan says that Van der Laan takes great pains to stress that there is no mysticism associated with the plastic number, but even so Padovan seems to be promoting it and Van der Laan as the successor to Le Corbusier and the Modulor. He makes the pertinent point that despite the "success" of the Modulor, it did not take off as an idea. We know it was just one aspect of the golden section movement which began with Fechner and perhaps reached its peak through Jay Hambidge. Surprisingly, I found no mention of the practical use Van der Laan made of it the plastic number. Padovan goes through Van der Laan's pebble sorting hypothesis which smacks of Fechner and his golden section experiments. That is not to say I do not believe it has potential, I am more afraid of harm being done before it has been weaned properly. I would also say at this point that when I did look at Padovan's book on Van der Laan's work, small though it is in practice, I was not immediately drawn to the photographic results. I believe I would have a different view were I to see the buildings, but books are a poor way to experience architecture. The title of the final chapter is, "The house as frame for living and a discipline for thought". However, far from being a comforting conclusion, Padovan's conclusions are slightly disconcerting. For example, he asks the question "What system of proportion do we need?" Padovan keeps echoing that the whole must have a relation to the parts as if he is trying to convince himself. To me this is part of the definition of proportion, something which he brought out time and time again. It is also fundamental to the concept of fractals. Now this may just be the current fashion in geometry, but it can describe the geometry of a cloud or mountain and a plant in a way that the ancient geometry never will be able to get near. Perhaps he should be asking a different question "What can different proportions do for me?". There is not one proportion, but each proportion has a use and for a purpose. This would at least get away from the way of looking at the past as the golden age we cannot hope to reach again; it is like the summers of one's youth always being idyllic. If Vitruvius was a great architect, he is long dead and we are living 2000 years after his death and times have changed. The architect Richard Rogers said recently that "the fear of beauty is destroying our urban environment" [Rogers 2001] . This was in reply to another piece by Sir Stuart Lipton's the chairman of the Commission for Architecture and the Built Environment with a remit to improve the low standard of British Architecture. He had said that the improvements he was trying to seek "are not at the bottom about aesthetics." Richard Rogers replies that aesthetics is precisely what architecture is about. He berates British Architects for the acres of concrete blocks and glass boxes because they are building for "the quickest budget in the quickest time". Of course he blames the politicians, but my worry is that Richard Padovan may be typical of teachers our architects in the way he cannot reach a conclusion and give a lead after such an in depth study. If so, then we are in for a rough ride in the future.
It is becoming harder for anyone to be able to encompass more than a fraction of the knowledge in one's own discipline never mind to go across many. There is definitely a place for philosophy and the study of aesthetics but I believe there is a requirement for the specialities of architecture and mathematics to be talking to one another, and there needs to be such multidisciplinary modules in courses for architects. So I would like to return to Richard Padovan's quote at the beginning of the book, where he said he was bad at mathematics, and was not taught about the beauty in geometry and number. If he had been able to call on such help earlier rather than have to struggle, I feel he would have been able to get to grips with it much more and we could have had some answers. The appreciation of aesthetics in science and mathematics not just art belongs in school. The architect Jonathan Hale's book, His last quote is from Hans Van der Laan and why he became an architect. Van der Laan tells a story about his first arithmetic meeting where the teacher put an apple on his desk then added more until there were five. The teacher then divided them into two groups of two and three apples and "we were supposed to conclude that these together made five apples. But in order then to teach us that these 2 + 3 = 5, even without any apples, the apples were sliced up and divided among the boys… But all my life I have been unwilling to forget those apples, and that is why I a became an architect and not a mathematician." I am not sure of the precise point Van der Laan was making, nor Padovan in ending the book that way. Was he saying that the mathematics was taught badly, or that there is a mystery which can't be solved by mathematics, or that he needed to get to the root of understanding another way or to be creative through architecture? Have I missed the point of the book, or perhaps it is a philosopher's tease? I can counter it with another, by a quote from David Hilbert, one of the giants of mathematics at the turn of the twentieth century, who wrote a famous book called "Mathematics and the Imagination" with S Cohn-Vossen. He was in a group where someone said a mathematician they knew had become a novelist and there were exclamations of surprise as to how such a thing could happen. Hilbert said it was easy - the man lacked sufficient imagination to be a mathematician but had enough to be a novelist. Padovan's penultimate quote is much more direct and is one of Le Corbusier saying that man becomes an abstraction when he closes his eyes but that if he builds it is with his eyes open and that architecture is judged by "what eyes see, by the head that turns and the legs that walk". Will someone write a book about proportion that does this by opening our eyes, moving our heads and taking us on a walk. Richard Padovan has been brave to tackle such a subject and has made a balanced result in bringing a vast amount of detail together in one volume and on that score alone it is worth having on your bookshelf. I don't think it the last word on proportion by any means, since the questions that are not asked and those not answered leave much in the way of inspiration. But does it help me make sense of the role of proportion in the world? - I don't think it does. I can't quite place what is missing, but it might just be that I have experienced the beauty of numbers and geometry which I would have like to have been enriched by the experience of an architect. I need the walk that Le Corbusier recommends in the three dimensional world of architecture. I don't think a book can satisfy that need. It needs the dynamics of a video if the real life walk cannot be made. But it is a start that leaves me wanting that walk.
Hancox, Joy. 1997. Padovan, Richard. 1999. Padovan, Richard. 1994. Dom Hans Van der Laan:
Modern Primitive. Rogers, Richard. 2001. Steinbach, Peter. 1997. Golden Fields: a case
for the heptagon. Stewart, Ian. 1996. Tales of a neglected number.
Penguin
Dictionary of Curious and Interesting Geometry and has written his own book on modelling
geometrical surfaces called Sliceforms, some of which
are in the "Strange
Surfaces" exhibit
in the Science Museum in London.
Copyright ©2002 Kim Williams top of
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