1570: Building with Geometry, Drawing with Numbers


Publication History:

“1570: Building with Geometry, Drawing with Numbers,” in When is the Digital in Architecture, edited by Andrew Goodhouse, 33-44. Montréal: Centre Canadien d’Architecture, and Berlin: Sternberg Press, 2017 

The text posted here is an earlier, unedited draft and it may be different from the published version. Please only cite from copy in print

According to a commonplace of recent historiography, the Renaissance might have been the only period in architectural history when the rise of a new style was not related to technological change. The Gothic forms of the Middle Ages were abandoned and the old forms of classical antiquity were brought back to life and reinterpreted, but no new machinery, new material or new building technique accompanied this revolutionary change in architectural forms. True as this may be, one might argue that some technological change did nonetheless accompany the rise of Renaissance classicism. These technological changes may have gone unnoticed because they did not pertain directly to building technologies. In the Renaissance, as now, new information technologies, instead of building technologies, were the agents of change. New information technologies brought about some new devices of design that in their turn revolutionized the process of building and changed architectural forms.

The Renaissance design process disrupted the traditional, medieval way of getting things built, but the early-modern way to manufacture or to reproduce the architectural forms of classical antiquity was also completely different from the method of the ancients. The same forms were obtained using two very different technologies of design. The modern way, invented in the Renaissance, remained a staple of Western architecture for the five centuries that followed; it is only now being replaced. This is, perhaps, one reason why we are more likely to be aware of the historical watershed that took place in the sixteenth century. We tend to recognize the beginning of a historical age only when we have a perception that the age may be coming to an end.

To better illustrate my point, let me mention a very simple component of the system of the architectural orders that was a bestseller, so to speak, in classical antiquity, as it was for generations of modern classicists from the fifteenth century to the twentieth century.

The Attic or Doric base of a column, as described by Vitruvius, is composed of six superimposed parts. The rules for establishing the proportions of the path of each part, as explained by Vitruvius and marginally edited for clarity by Leon Battista Alberti fifteen centuries later, read as follows: First, you take the diameter of the column and you divide it into two equal parts. You divide that segment into three equal parts. Take away the lower third, that is, the plinth. Next, take what remains, make a new unit of it and divide it into four equal parts. Take away the upper fourth—that gives the upper torus. Take what is left, make a new unit out of it and divide it into two equal parts. Take the lower half, that is, the lower torus. Divide what is left into seven identical parts and take away the upper seventh and the lower seventh: that gives the two fillets. Take what is left, and, fortunately, it is over because there is nothing else to be proportioned. And that’s the end of the process.

Alberti, here acting as an editor of Vitruvius, guides the reader through a five-step sequence of successive divisions (fig. 1). Each step, however, is formally identical to any other step in the sequence, and each reads as follows, as I just emphasized: Take a segment, divide it into a given number of equal parts, take away one of these parts, take what is left, assume it as a new unit, then go back to step one and rerun the program, as we would say today, this time five times.

This way of determining the proportions, and then the dimensions of an architectural part, has its charms, but it is not the way we would do it. Our way, which is the modern way, came into being by steps in the course of the sixteenth century. First, images of the Attic base were printed thanks to the then-new technology of printing, something which neither Vitruvius nor Alberti had access to. And of course printing depended on the new-found availability of paper in the West. Serlio printed the proportional drawings of the base, proportionally drawn to scale, and he added the names of the parts, which of course can always help (fig. 2). Then a bit later, in the sixteenth century, both Vignola and Palladio printed the same scaled drawing, but they added the proportional or modular measurements of all the parts. Vignola used a module divided into eighteen parts and Palladio used a sexagesimal partition, as we still do with minutes and seconds (fig. 3, fig. 4). Vignola and Palladio could not use the decimal point for the simple but determinant reason that it had not yet been invented. However, these differences apart, this is a language both visual and numerical that twentieth-century engineers would still understand and would still be fully conversant with. This would be especially true for engineers trained in the imperial system, which is much closer to Vignola’s and Palladio’s fractional universe than to the Napoleonic empire of decimals.

In short, what we have seen here are two ways to produce the same object. The first way, the Vitruvius/Alberti way, which is classical but also medieval, is based on text and geometry. The second way, which is the Palladio/Vignola way, which is modern and basically still the one we use, is based on drawings and measurements. The result may be the same, but the two processes are not. In the first case, each operation in the sequence is an elementary geometrical partition, which can be performed mechanically—perhaps I should say manually—with a straightedge and a pair of compasses, and without any need to perform any number-based operations. A pair of compasses can divide a given segment into a given number of equal parts, without any need to measure the segment or to use numbers to calculate the result.

The second way obliges the user to read the measurements with proportions in the drawing and to multiply these numbers by one or more other numbers in order to determine the final dimensions of the object. This second method, which presupposes—in fact requires—numeracy and the use of Hindu-Arabic numerals to perform the four basic operations of arithmetic, was a relatively new discipline in Europe in the fifteenth and sixteenth centuries. It was known as algorism, from the Latinized name of its inventor, Al-Khwarizmi, a ninth-century scientist from Baghdad, a city which still exists—or still existed two hours ago.

The old geometrical method had some advantages. It did not require the use of numbers—a decisive advantage at a time when most people did not know how to use numbers, and modern, Hindu-Arabic numbers did not exist. Roman numbers are not good for calculating. A sequence of geometrical instructions is a narration, a recital of sorts. It can be recited aloud, unfolding in real time: the time that is necessary to perform the operations that are described. And then as now, one remembers a story more easily than a list of telephone numbers. Geometry is the daughter of orality, and a good friend of memory. On the contrary, the new number-based instructions of Vignola and Palladio are difficult to memorize, and they are better recorded and transmitted in writing. They are even better recorded and transmitted in print. And mechanical reproduction reduces the risk of mistakes that would inevitably occur when copyists must transcribe pages and pages of apparently meaningless numbers.  Print made this transmission reliable.

The geometrical way, however, featured another even more crucial advantage. A geometrical construction, such as the division of a segment into two equal parts, is an entirely mechanical and analog operation that can be performed regardless of scale or size. With a small pair of compasses, it can be carried out at the scale of a drawing on paper, provided that you do have paper. With a larger pair of compasses, you can perform the very same operation, but at the scale of the building—or at any other scale, for that matter. I tried to bring along a small pair of compasses because I wanted to make an on-site demonstration, but that was foolish of me. I could not carry them on the plane; they were detected by a metal detector. I had to explain to the customs security officer what these things are. I said, “Well, I need them to argue, as I am trying to argue, that for centuries this was a weapon of mass construction.” In retrospect this was not a wise thing to say—security officers are not keen on learning the history of architecture. But anyway, my point is:_ geometrical constructions are a tool for building as well as a tool for drawing. In a geometrical environment the making of scaled project drawings may sometimes be unnecessary. Geometry can generate the real thing at real size on the real site without the need to go through the laborious mediation of a preliminary small-scale drawing on paper.

In contrast, small-scale proportional project drawings, with or without the addition of number-based, or digital, measurements, are separated both physically and ideologically from the materiality of building—again, thanks to paper. Project drawings exist and reside on paper. Such paper prefigurations of future buildings must at some later point be translated into real-size, full-scale, three-dimensional objects. This translation of drawings into buildings is an operation of proportional enlargement, also known in French as homothétie. Scaled project drawings must be enlarged by a factor 10, 50 or 100, or 96 in the imperial system, in order to be converted into stone. But—and this is the snag—this translation or proportional enlargement is not always an easy matter. In some cases, a three-dimensional model might help, but in most cases, the iron law of transference from two-dimensional drawings to three-dimensional objects applies—we can only measure what we can draw, and we can only build what we can measure in a drawing. In short, if you cannot draw it, you cannot measure it, and if you cannot measure it, you cannot build it.

It follows that within this logic, the forms that we can build are determined by the power or the potency of the mathematical language at our disposal. If this language is basic algorism, or the arithmetic of the four operations, as it was for centuries, we can better measure, and hence build, segments of straight lines that are all parallel or perpendicular to one another or that intersect at fixed angles on the same plane or on parallel planes. Such limits lead to objects that are grid-like, repetitive and discrete, as numbers are. In contrast, geometry can construct lines and surfaces that are continuous and bending, and curves that might be difficult, or even impossible, to measure. This is because geometry does not need to measure lines—lines are simply laid out mechanically. They are made on-site, full-size, using compasses, ropes, nails, chalk, chisels and all kinds of mechanical tricks.

Builders in classical antiquity constructed sophisticated curved surfaces and continuous lines that a twentieth-century engineer would have struggled to describe with numbers, such as the barely perceptible rise toward the centre of the platform, or stylobate, of a Greek temple, the spirals of the Ionic volute (fig. 5) or the entasis of the shaft of the column. The curved, continuous line of the entasis of the column was cut in stone onsite, full-size and without any need to measure it, which was just as good, because if classical builders had had to o measure it, they could not have done it.. Using a similar but more advanced geometrical construction, medieval stereotomy built complex curving surfaces that, up to twenty or thirty years ago, would have been almost impossible to draw and measure with numbers. Geometry is about continuous lines and surfaces. Numbers are discrete entities, and classical geometry neither needed nor used them. Indeed, some classical thinkers and scientists had little affection for numbers, and in the classical age, many practical issues that we now solve with numbers were solved with geometry.

But in the seventeenth century numbers took over. Differential calculus empowered numbers to describe continuity, and through analytic geometry, curves could be written down as algebraic equations. This is in fact what we mostly still do, as for most of us an ellipse is an x-y function, not a concoction to be obtained mechanically with a rope, a stencil and two nails, which is what Serlio could have done. And it is also well known, as Greg Lynn has been reiterating for years, that architects did not start to use calculus as a tool to create forms—as a device of design—until some ten or perhaps fifteen years ago. This was when computers first made differential calculus available to the masses, so to speak— not so much calculus, as the possibility of visualizing continuous functions generated by algebraic equations. And as we all know, this brought formal continuity back to the centre stage of architecture after an exile of almost five centuries.

I am abridging the story and simplifying here a bit: continuity of form did not completely disappear during the five centuries of the dominion of the number. Let us just think of the survival of traditional stereotomy well into the seventeenth century, and occasionally beyond. Or let us think of Antoni Gaudí, Erich Mendelsohn or even the later work of Le Corbusier. But in each of these cases there is some explanation. During the age of architectural numeracy, non-measurable forms could still be built following the traditional geometrical approach, or by using the modern number-based method in disguise—that is, cheating. We must keep these exceptions in mind. Yet what follows from all the above is a challenging and at times exciting historical paradox.

If all or even only some of the above is true, we must come to the conclusion that one of the main consequences of the digital revolution in architecture is the revival of geometry as a tool for design. As with most revivals, this is not exactly a revival: some more recent developments in geometry are now also involved, and what is being brought back to life is geometry translated first into a new, number-based format by seventeenth-century calculus, and second, into a new machine-readable format by twentieth-century electronic computing. This new geometrical tool for design is managed by machines, and the objects that we can produce using computer-based geometry are machine made rather than handmade. We can now mass-produce what used to be artisanal pièces uniques—a marginal point in the economy of this paper, but a major point in the global economy of the present, as this is one reason why we must use the new technologies and make the most of them.

But geometry is still geometry, regardless of the machines that process it—compasses or computers. Not only is geometry about continuity of form, it is also, as it always was, a process that is mostly indifferent to scale. The separation between design and building site, an estrangement that started with the rise of architectural numeracy and the availability of paper in the Renaissance, is now being epistemically challenged by file-to-factory technologies, whereby the same software manages computer-generated images as well as the three-dimensional manufacturing of an object. In time, the gap between design and production, which started in the sixteenth century, will most likely be reduced by the logic of the new digital tools. These tools are new, automatically calculated in a three-dimensional space of x-y-z coordinates. Although endless two-dimensional images of an object can be printed out at will, the source and matrix of all of these variable manifestations make a virtual substitute for the object itself (fig. 6). All parts and each point of this digital archetype can be automatically drawn, measured and built. The iron law of transference from drawing to building—if you cannot measure it, you cannot build it—has ceased to be. In a digital environment, if you have a drawing you already have all of its measurements. Or to be precise, you don’t have them; your computer has them.

It is a commonplace of the digital revolution that the new digital environment is in many ways the print environment in reverse. As many have pointed out, the new digital environment is closer to the age of the manuscript as it existed before the age of print, than it is to the age of print that is now coming to an end. An assessment of the first ten years of the digital revolution in architecture would appear to reinforce and to corroborate this assumption. Numeracy can exert its influence over architectural design only when numbers and drawings can be printed together. It is the alliance of Arabic numerals and printed images that brought about the rise of architectural numeracy and changed the course of architectural history in the sixteenth century. Now, as it seems, the new digital tools are bringing architectural design back to an Edenic state of pure geometry, which is where architecture lived and thrived for centuries before that paradise was lost, as it fell under the dominion of numeracy and of print.

But if this is so, and this is my conclusion, then Thomas Aquinas—a very unlikely name in this environment—and, right on the eve of a revolution of print and numeracy, Leon Battista Alberti could probably understand the present digital environment and the principles of contemporary digital design much better than Walter Benjamin or Mies van der Rohe could or would, to mention only two of the most eminent advocates of art and design in the age of identical mechanical reproduction. Aquinas and Alberti lived, as we do, in a universe of variable media. For them, the fixity of print and measured drawings had yet to come. For us, the fixity of print and measured drawings has already gone. And it is certainly one of the most significant paradoxes that marks the latest stage in the evolution of number-based computing that, thanks to computers, we can now mostly forget about numbers and when necessary manipulate intersecting curves and bending surfaces regardless of scale and measurability, just as our ancestors did in the time of compasses. Computers are just as good, and to be honest, in many ways, I think they could even be better.


When is the Digital in Architecture


“1570: Building with Geometry, Drawing with Numbers,” in When is the Digital in Architecture, edited by Andrew Goodhouse, 33-44. Montréal: Centre Canadien d’Architecture, and Berlin: Sternberg Press, 2017