Saturday, May 11, 2013

THE MEASURE OF THE GOLDEN MEAN




Proportion is one of Howard Robertson’s subjects in his now much maligned and many-times reprinted Principles of Architectural Composition, Architectural Press, 1924. Other topics include ‘the consideration of unity,’ ‘the composition of masses,’ and ‘the element of contrast.’ They are all matters like ‘proportion’ that hold an irrelevant and quaint antique quality about them today, having no role in our current theory, analysis and critique of architecture. Maybe we need to have another look at this publication to reassess its relevance, or to try to understand where we have departed from its scrutiny to get to where we are now? We should never assume that ‘moving on,’ forgetting, is ‘progress’ or that ‘progress’ is ‘moving on’ - away from; changing for the sake of change. We must know where and what we have come from if we are to gauge the sense in our present and the relevance of our future. One needs to know why we have changed. Is it merely style; fashion; perhaps language itself? Has it something to do with the thrust for the ever-different outcome: the demands of ‘new’ PhD that must be unique for recognition like many other things in our times that favour the bespoke?




Robertson promotes the Golden Mean as an example of good proportion, with illustrations included to prove the point. Howard Robertson is not the first or the last to highlight the importance of proportion in architecture. Researchers on buildings from the Greek temple to the mediaeval cathedral have all used illustrations to make their point about origins and relationships of forms. More recently Le Corbusier promoted his Modular as a guide to good proportion in architecture. His ‘standing man’ graphic with its raised arm has become the icon of his system, indeed, his work, that he claims is based on the proportions of man: just which particular one is not made clear.



The Golden Mean, also known as the Golden Ratio and Golden Section, is only one proportion that has been singled out for attention. Illustrations are published to show how it appears in nature and in architecture, as are some other relationships too. These illustrations usually take the form of line drawings on top of, say, an image of a flower, a drawing of an elevation of a temple, or a cross section of a cathedral to prove the point, apparently unequivocally.






The Golden Mean is precise. Expressed algebraically:

(a + b) ÷ a = a ÷ b = (def) φ

where the Greek letter phi (φ) represents the golden ratio. Its value is:

φ = (1 + √5) ÷ 2 = 1.6180339887 . . . .






What is of concern is that the proof that proportional systems have played a role in architecture is usually a lot less precise than the fact of the relationships themselves, be this as simple as a square or a triangle, ad quadratum and ad triangulum, or as complex as the Fibonacci series. In mathematics, the Fibonacci series, also known as the Fibonacci numbers or Fibonacci sequence, are the numbers in the following integer sequence:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . .

By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. The series is like a chain of dance steps.

In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the recurrence relation

Fn = Fn-1 + Fn-2

with seed values

F0 = 0,  F1 = 1





In spite of the neat precision in the various systems that could be merely a square relationship, equal/equal, there often exists a gap that requires a leap of faith or a closed eye for one to accept that these things are or have been involved in either or both nature and art, and architecture. The ‘proofs’ seem to be much less rigorous than the precise proportions themselves, so much so that one is left with the suspicion that the proposition of the involvement of a certain scheme in any given circumstance has been somewhat forced; that the ‘facts’ have been fabricated, ‘massaged,’ to fit the theory - perhaps being specifically selected, chosen to fit so as to prove the point.





Time and time again, the ‘proofs’ are presented as small drawings overlayed with heavier lines that have been established according to the mathematics of the proportion. Superficially the whole proposition appears a simple ‘lay down misere.’ A closer look at the evidence, and further thought, usually raise more and more puzzling questions. Exactly what is being compared here? What distance is being identified? What height? At what gauge? Why have these dimensions been selected for examination? With, say, the section of the cathedral being illustrated in a drawing reproduced as a 50mm tall image, the scaled thickness of the line illustrating the proportional scaffold can become a full-sized 100 - 150mm thick measure in the real cathedral. What is this thickness really identifying? What is it defining? In spite of this uncertainty, the graphic always appears authoritative.





This looseness only further muddies the question about the precise location of the limits and the identity of the distance being referred to in the calculation. The first concern is just why the choice of this particular point or location holds any relevance: what is this place? Then there is the uncertainty: precisely where is this place? If this theory is to be tested, the measurement has to be able to be repeated, verified. Is it the centre of the column? Which one? What is the centre? Is it the face of the wall? Where? What is this face? Is it the edge of the moulding? Where? What? Why? And if this selected point is being indicated with a measure, say, 100mm thick, the difficulty in identifying the precise point of reference is only being further complicated by the inherent inaccuracy in the instrument - well, in the line being used to identify the location graphically. Does one measure to and from the centre of this line or to one edge? Which one?





Of course, all of this can be overcome today with the use of lasers to give what we are told are exact figures that can identify certain GPS-defined locations and the distances between these points with as many decimal places as the Golden Mean might choose to be expressed in. Such measurements can then become the subject of mathematical calculations to prove the point being made. Alas, even with this modern accuracy and the precise manipulations of the measurements in computed calculations - maybe because of this precision? - when the answers do not come out to the exact figure required for the ultimate evidence, the text usually blandly accepts, almost carelessly, nonchalantly, the ‘near enough’ figure as real proof of the facts. The graphics are added as additional ‘proof,’ as if seeing aided belief by distracting from the slight of hand - well, of calculation.





Why are we so ungracious to our ancestors? We seem happy to believe that, with their crude technologies, these ‘near enough’ figures can be assumed to be the verification of our preferences, when the proof of the accuracy and precision of our predecessors can be seen in their workmanship right before our eyes. Consider the setting out of a cathedral; e.g., the establishment of the levels for the footings and column bases, complete with their intricate complexities, all arrayed in the mayhem and mud: see http://voussoirs.blogspot.com.au/2012/05/gothic-thought.html  Consider the walls, the ribs, the vaults - how they form one organic element when fabricated from so many various parts by different workers over long time spans. This is not just a ‘near enough’ attitude - “She’ll be right. We’ve just got a few rough tools; bare feet: forgive us.” The proposition seems fanciful. Just cast the eye up any clustered column and gaze at the unity achieved, as Howard Robertson might suggest. Given this reality, why do we appear happy with such an ad hoc approach to proportions and proofs? Is it really a matter of conceit, of our favoured understandings and prejudices being confirmed, whatever? After all, we believe that we have the superior technologies; perhaps superior minds too?





Measurements are indeed tricky things, as Benoit B. Mandelbrot pointed out in his research into fractal geometry. He showed how the length of a coastline varies with the length of instrument being used to measure it: the shorter the tool, the longer the coastline. As the ruler being used to measure the coastline gets shorter, more intricate profiles can be accommodated, with less being averaged out of the length, so the coast gets‘longer.’ Given this enigmatic relationship, why are ‘proofs’ of architectural proportions left so vague without explanation, qualification or conditions? Why are we scared of questions? Why is the analysis always left as a near enough proposition as being good enough for a proof when, in, say, the study of proportions of the mediaeval cathedral, the mathematics is singularly precise: a square is a square in ad quadratum design, and a triangle is a triangle in the ad triangulum strategy. Do we assume that the mathematical inspiration was a rough guide only? If it was so important as a concept and as a beginning, surely it would not have been so carelessly accommodated? Given that the complexity of the rose window could be fabricated and assembled in its elevated location, why do we assume the medieval builders were satisfied with such extreme tolerances in their setting out of their buildings? We still seem to harbour the idea of the rude and crude ‘dark ages’ when considering the past, happy to believe that we live at the forefront of enlightenment - the outcome of progress. Maybe every era does?





The same can be said of the Greek temples too; and Egyptian temples and pyramids. Just what is being proved with these generalisations? Just what is the point - the precise points being identified and discussed to make the point that a certain proportional system has been instrumental in the implementation of the initial concept? Oversimplification does not seem to be good enough. It belittles the original makers.





This attitude and approach is clearly seen in Arnold Pacey’s book Medieval Architectural Drawing, (Tempus Publishing, Gloucestershire, 2007). He repeatedly makes assessments and draws parallels from circumstances that are explained as being almost a match - close enough, it seems, to allow one to say that such and such a strategy was indeed so. This seems strange. Is ‘close enough’ near enough to establish a fact?




p.73  Figure 3.6 Schematic cross-sections of churches, comparing the tall proportions achieved with ad quadratum design in a vaulted building (left), and the lower proportions with ad triangulum design (shown for a building with a timber roof structure.).



p.73-74

It is also possible that the tall proportions of the section of Westminster Abbey were based on a square of side equal to the overall width of the building. Towards the west end of the nave, where the structure is not complicated by the presence of the cloister, the overall width of the building, including its buttresses, is 31.7m (104ft). The internal height of the nave, to the crown of the vault, is 30.8m (101ft). The near equality of height to width tends to confirm that the design was conceived as being contained within a square. The ridge of the roof, though, rises significantly higher.



In other buildings, especially those intended to have a less towering profile, another option was to use the dimension of an equilateral triangle as a guide to design. In such a triangle, all the angles are 60 degrees, and all three sides are the same length. However the height of this triangle is less than the length of its base, and for an equilateral triangle, the height measures 86.6 per cent of the length of one side. This means that the elevation and section of a building designed ad triangulum will end up to be only 86.6 per cent as high as a building of the same width designed ad quadratum (Figure 3.6).



As an example, the internal width of the nave at Selby Abbey (West Riding of Yorkshire) is 17.7m (just over 58 ft). The height of the nave is 15.54m (51 ft), which is about 87 per cent of the width - close enough to the theoretical ratio for Eric Fernie to conclude that ‘Selby can justifiably be described as a design ad triangulum’.



Why is one asked to accept the measurement of one part of the cathedral only? What happens elsewhere? Why choose to measure the overall width and the internal height? What is the width and height of a cathedral? From what and where, to what and where? Why? Why is the ‘near equality’ assumed to mean anything? Why ignore the added height of the roof? Why do we get a description of an equilateral triangle, as if it were a scientific revelation? Why measure the internal width for ad triangulum? Why calculate the percentage to a decimal point and then accept a ‘close enough’ figure - and additional 0.4%? Why assume a cathedral began with the ambition of having a ‘less towering profile’? While measurements ‘tend to confirm’ the use of the square, ‘in other buildings . . another option was to use . . . an equilateral triangle.’ Why is there no doubt or uncertainty expressed on the use of the triangle?





p.102

The backs of the stalls at All Saints are formed of a series of planks set in a frame on which are carpenters’ assembly marks in the form of Roman numerals. On the northern set of stalls, I to V appear from left to right on vertical members of the frame, while the panels have numbers VI to X, running this time from right to left. Some of the planks at the left-hand end have been replaced, but two panels at the right-hand end show the system of numbering clearly.



The lower panels, behind the actual seats, are numbered differently, however. On one is what appears to be the numeral IV (which probably means six). On the next plank is a mark consisting of a circle with a smaller circle inside it (Figure 4.8). Small marks enclosed by these circles may indicate a Roman numeral, but are too indistinct for one to be sure. Some other planks are similarly marked to show where individual pieces should butt against one another, so the circles appear to be a form of assembly mark showing how the three planks forming the lower part of panel VI should fit together.



Why on earth should ‘IV’ ‘probably mean six’ when it seems to mean four? Why should an indecipherable mark be assumed to be a ‘Roman numeral’? Why assume the marks have anything to do with assembly? Pacey does explain the ‘IV’ idea later in his text: because of possible confusion with ‘IV’ and ‘VI’ when seen upside down, the mark for four was ‘IIII.’ Maybe, at least this theory accommodates the foibles of ordinary perception, reading, circumstance and co-incidence: see also http://voussoirs.blogspot.com.au/2013/03/on-copying-drawing.html

It is interesting that Pacey involves the whims of real life experience when it suits. He seems reluctant to entertain any variation to his proportional analysis other than to accept that things might be ‘near enough’ to be proof.






p. 93

Figure 4.3 Typical cross-section of a medieval aisled barn showing how, in principle, its proportion might be related to the height of an equilateral triangle (as in ad triangulum design) and how the proportions may also be presented by intersecting circles.



p.92-93

Moreover, just as churches were often planned using the perch or rod of 16½ft (5.03m) as a unit of length, so too were the Cressing Temple barns. The older barn, the Barley Barn, was originally 3 perches wide and 9 perches long. By contrast, the Wheat Barn at Cressing Temple is precisely 2½ perches in total width and its height corresponds to the height of an equilateral triangle with sides of that length.



These conclusions about the use of triangles in design arise entirely from modern measurements of the existing buildings, not from any medieval records or drawings showing how carpenters worked. Hence, although the proportions of the Cressing Temple barns are fully consistent with the use of ad triangulum design, it is open to Laurie Smith to argue that the proportions of the buildings were derived by drawing circles rather than triangles. Since equilateral triangles can be fitted neatly within circles, Smith’s theory about how circles were used leads to the same results as Gibson’s triangles. Figure 4.3, which is more basic than either authors’ diagrams, shows how a notional aisled barn can be proportioned by these geometrical figures. Smith claims more generally that there is ‘considerable evidence for circle-based design in the medieval period.’ This accords with the experience mentioned in previous chapters that craftsmen frequently drew circles (rather than squares or triangles) on walls and tracing floors.




. . . and so on. This last statement seems to ignore the fact that a compass can easily be used to create a square and a triangle. Why assume anything? If the matter is so inconclusive in this context, why is it apparently promoted as being relevant in other situations? Is this all just a guessing game of maybe? To what end?*





The point is that there is much assumed, much guessed and much read into what has been chosen to be measured and assumed. One has to ask if the measurements have been selected only because they are close to fitting the theory. What units did the medieval builders use? Why does geometry matter at all? Has it got something to do with Masonic mysticism? We might like things to be so, but we do not have to generalise to prove our expectations. Are we only cheating ourselves?





Until architects become as rigorous, as open and as co-operative as scientists - well, as scientists should be - architects will always languish in the world of pseudo-science and pseudo-art, mocked by others who work with much more precision, questioning and testing. Architects will continue to be seen as lazy, indulgent dilettantes working in a ‘closed shop,’ interested only in themselves and their own ideas, irrespective of client needs and budgets, if guessing is seen to be good enough. Grasping for status in things vague and hazy will always be seen as being dodgy, doubtful.





While our analyses and assessments might initially all be based on feeling and intuitions, the outcomes of these perceptions gain their rigour in their development and testing: c.f. the Periodic Table. Are we afraid of the facts? What equivalent is there in architecture? The Golden Section? Is it all merely a hopeful ideal? Perhaps we should never even attempt to rationalise or explain beauty with any precision, or without it. While our intrigues with other minds and the management of making wonder will always stir us, we should remember that there are no simple or rational roads to or rules for beauty.


Why do we never acknowledge that our history in architecture is predominantly the history of religion and its ‘expression’ - (it is not the best word to use, but it crudely states the idea)? It may be an anathema to us today to recognise this, because it will highlight a marked difference in our lives and our interests. Measuring is fraught with failures. Remember Mandelbrot’s coast: nature. The beauty of fractals lies in the rigour of the indefinite definite - or is it the definite indefinite? - with its infinite self-similarity. Has this any relevance for us today apart from the strange shapes that can be made with such mathematics - consider Storey Hall in Melbourne and the pentagonal geometry of Roger Penrose. Our computers allow us to do nearly anything.




Still, near enough may indeed be good enough. It might have to be because we have no other place for perfection but in a vision, perhaps of paradise. Maybe we need to concentrate more on personal integrity and commitment, honesty and humility, as we grapple with our past and move to our futures. This might modify our grand desires for personal proofs and maintain our contentment, seeking satisfaction in the revelation of beauty in the ordinary rather than the extraordinary.





How is this possible? How can materials and forms be manipulated to create meaning? The Golden Mean, like other proportional systems, might indeed become quite average when translated from its mystical numerical identity into form. It may be only our desire, perhaps a pure whim that gives its numerical magic meaning that we seek to make some sense of. We may have to look elsewhere, in ourselves, in our experience, for substance. We have a strange habit of locating meaning everywhere except in ourselves, in our own being. Is the avoidance of this understanding part of our great effort, our great desire to measure? Do we rationalise matters to distract ourselves, to allay our doubts, to avoid our inner silences? Try measuring ‘I am’: good architecture does, with a definitive lack of certainty; with, dare one say it?, love.




"Design is not making beauty, beauty emerges from selection, affinities, integration, love."  Louis Kahn


*P.S.
Matters become only more enigmatic:
p.137

The north range of the building once had a window. It was later reconstructed as a conventional transomed window, but the dimensions of what remains show that the original oriel was 3.36m wide. The splayed lines in the drawing imply two options for the window, one 2.58m wide and the other 3.48m wide. That tends to confirm that the second option on the wall was a full-size representation of what was constructed.
Arnold Pacey appears to have allowed for some extreme tolerances in this assumption.

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