Essay on Geometry

“Design is form-making in order 
Form emerges out of a system of construction 
Growth is a construction 
In order is creative force”

Louis Kahn, Order Is, 1960

We are looking at the project for Mr. Kís house and wonder: how big should this room be and what about the one next to it? How many steps should a house measure in one direction and how many in the other? What is the correct height for each of the rooms?

These are seemingly simple questions for people  who are familiar with architecture. All you need is a good eye,  a feeling for proportions and there you are; in a short time weíve drawn up a plan and a section, a well-designed plan, with the right measurements for everything.

Not far from Venice, along the Naviglio del Brenta, near Mira, stands a remarkable building, an imposing and compact volume tucked into the lush Veneto plain. It is the suburban residence designed between 1556 and 1559 by Andrea Palladio for Nicolò Foscari, owner of the famous Ca’ Foscari on Venice’s Grand Canal. Apart from the Ionic pediment that articulates the entrance façade, it looks like a stereometric block of unusually vertical proportions; a somewhat ungraceful-looking building. Unlike other villas designed by Palladio, the architecture is more austere; few concessions are made to pleasing the eye. Despite this, we are attracted by the building, even though we don’t initially know why.

“In order to construct well and distribute his efforts to advantage, to obtain solidity and utility in the work [the builder] has taken measures, he has adopted a unit of measurement, he has regulated his work, he has brought in order. [ … ] But in deciding the form of the enclosure, the form of the hut, the situation of the altar and its accessories, he has had by instinct recourse to right angles—axes, the square, the circle. For he could not create anything otherwise which would give him the feeling that he was creating. For all these things—axes, circles, right angles—are geometrical truths, and give results that our eye can measure and recognize; whereas otherwise there would be only chance, irregularity and capriciousness. Geometry is the language of man.”5

For Le Corbusier, making architecture meant organizing, first and foremost, giving measure to things, otherwise “it would be down to chance, anomaly, arbitrary.” This was precisely why he saw the act of measuring, the choice of a module by the person building it, as the origin of the architecture, enabling him to regulate his work. The Swiss architect Charles-Edouard Jeanneret6, maintained that there was nothing worse, in architecture, than to proceed in an arbitrary manner without a logic that made every decision and every operation necessary. Order is therefore one of the fundamental prerogatives of architecture, its moral imperative. 

Organizing means measuring, ordering all the ­elements according to a rule fixed a priori, following a logical principle that holds all the measurements together; yet it is impossible to organize without a geometric reference system, a system that will allow all the parts to be in relation to each other. Order is achieved through a rigorous and disciplined respect for geometry.

It is no coincidence that Le Corbusier attributes the “fatal birth of architecture,”7 to geometry, a crucial factor in ­architecture. Geometry is therefore a necessary premise for ­architecture to take shape, first and foremost through the ­design and then through the building process. The plan is the place where geometry finds its natural home, the place of order and measure. 

The plan is where the principles of the subdivision of the space, the principle of organization are expressed. Without a plan, says Le Corbusier, there is an intolerable sensation of “shapelessness, of poverty, of disorder, of willfulness.” 

The plan is therefore the actual drawing of the architecture; a type of drawing that is, by definition, abstract in character. Unlike other forms of representation, plans do not describe and do not represent. Unlike elevation drawing, which in some ways simulates the actual vision of a man standing up; or the perspective that simulates the three-dimen­sionality of the actual space; the plan has no analogical relationship with the actual vision other than in the architect’s imagination. Artists do not draw plans; decorators do not draw plans. 

It is true to say that every organizing principle that forms the basis for architecture is put together by means of the rigorous control operated in the plan. The plan determines the structure of the project, the principle of sensation. 

It is through the rigorous design of the plan that ­geometry expresses the essence of the architecture, its order.

We are in front of Piero della Francesca’s Flagellation of Christ. The scene is rigorously divided into two parts, with three ­figures in the foreground on the right, in an outdoor city setting, and the actual flagellation, which takes place further away, on the left, in an interior. The two columns in the foreground frame the scene; the central column acts as a watershed between the external and the internal worlds. 

“And this angle is such excellence that it can never change, and the best geometers call it by another name Angulum iustitie, the angle of justice, since without its knowledge it is not possible to know right from wrong in any of our operations, nor without it is it possible in any way to provide an exact measure.”10

As we said, an organizing principle must be established. 

The fifteenth-century mathematician and monk Luca Pacioli constitutes a point of reference in this regard, for ­architects, theorists, and historians who have attempted over the centuries to identify a definitive system of geometric relations for organizing the shapes of art and architecture rationally. Pacioli believed mathematical proportion to be the foundation, or universal principle, of beauty, valid and verifiable in all the manifestations of mankind, from art to architecture to music. 

The general tendency is to identify the secret correspondence of nature, in its cosmological and physical data, with the mathematical principle, with a number. In other words, the secret of the organization of shapes by means of relations is regarded as supernatural. First and foremost, the golden ratio, or divine proportion11, is the number that best expresses the unique characteristics of the Entity: “… just as in divinis one selfsame substance exists among the three persons so one selfsame portion of this kind always finds itself suitable among the three terms.”12 

Therefore the geometry, the mathematics, and the proportional system that can be established through prime numbers provide an ideal aesthetic for the different artistic representations, founded on the very structure of created things, like images of the Creator.

“Friedrich Nietzsche put it beautifully: ‘one must have chaos in one, to give birth to a dancing star’; if this is true we can expect the sum of our life and our life’s work to engender the birth of the biggest and most beautiful star: the requisite chaos has been able to prosper in us as never before.”16

There could be no better affirmation than Tessenow’s at the beginning of the chapter entitled “Die Ordnung—Orde” in his small book Hausbau und Dergleichen, to describe the state of architecture today: a chaotic situation generated by a huge ­proliferation of styles, vocabularies, and different ways of perceiving the discipline, dictated by architecture and architects’ need for originality at all costs; the constant quest for amazement and “invention.” 

This situation appears to be marked by a progressive loss of the sense and meaning of the project and a shift from disciplined, plan-based work to a profession in which image is paramount. A situation that, as Grassi suggests, has confined architects to “an increasingly limited and more wretched profession, because they are expected to show off, a profession that increasingly requires them to work as ‘artists’, let’s say, as ‘creatives’ as we say these days, in other words, to a profession ­reduced more and more to a job for ‘decorators.’”17 

Given this situation, it would seem wise to reflect once more on architecture as a discipline, to rediscover the foundations, the invariants, the aims; architecture as a transmissible art can be built on a logical basis18, an objective one; to think once more in terms of architecture “as a work that belongs to all and comes from all, as a work conceived to give order and shape to a city and ‘embody,’ as it were, the foundations and the aspirations of public life.”19 

These days it’s not a matter of choosing one style over another; of preferring “straight or crooked architecture, intelligent or stupid, rough or refined architecture.”20 It’s a matter of showing how architecture can still encapsulate collective values; how ­architecture can continue to give shape and identity to cities; or how it can be constantly renewed without losing its identity. 

We need to start thinking again about the foundations of the discipline, working out what to keep and what to let go; what is indispensible and what is superfluous; exploring the essence of architecture to establish whether or not there are ­invariables that are a structural part of architecture itself. We realize that geometry is one of these. Or rather, the need for an organizing system based on one or more ­geometric rules. This is a timeless principle impervious to fashion and the moment, rooted in the history of the discipline of ­architecture. 

Let us consider, for example, the simple rhythm of the series of arcades on the façade of the Coliseum or the ­colonnade of Palazzo Farnese, or the rhythm of the columns surrounding the Parthenon. Nothing could be simpler or more magnificent, the repetition of a single rhythm again and again, a single element that means nothing on its own, but when repeated several times becomes order, measure. 

Rhythm—or repetition—is therefore one of the ­fundamental elements of architecture because it coincides with the primeval act of measuring, of taking possession of a place; but it is also the entity that every human being perceives in the same way because it is what regulates the life of every organism, like the heartbeat; or the rhythm that measures the passing of time, which is at the heart of every musical composition. 

This means we have no option but to agree with ­Tessenow when he says: “Meanwhile it would be important if, as regards craftsmanship [meaning the work of architects], we could start by conquering the uncertainty that we feel when faced with order, repetition and uniformity, in themselves; from this point of view they represent the best and the most effective work tools. Good craftsmanship always fears origi­nality, but not what is customary or ­repetition, which always carries its own explanation.”21

“In any case, without further ado, Beauty can be defined as the harmony and concord of all parts achieved in such a manner that nothing could be added or taken away or altered except for the worse.” 

Leon Battista Alberti, De re aedificatoria, c. 1450

These days we might perhaps be accused of presumption or ­ingenuousness if we want to say any more about beauty. The categories of beauty—the philosophers might not like this—have now largely fallen out of favor with intellectuals. They seem to have fallen into disuse, starting with the epistemological rupture linked to the advent of the concept of the sublime22; which is the point at which the appraisal of beauty ceased to be a quest for objectivity in favor of the indeterminate, fascinating and murky field of the subjective. 

Everything seems possible when it comes to aesthetic judgment these days. Everything is potentially beautiful, or beauty no longer exists, which amounts to the same thing.

We then go into the Church of S. Lorenzo in Florence. The white-and-grey floor inside is fascinating; long courses of pietra serena mark out the space into large rectangular sections. These clear lines, which look as though they are drawn on a sheet of paper, define a perfect geometric matrix.

Leon Battista Alberti, referring to Vitruvius, delivers an extremely effective definition of beauty. It consists of the harmony ­between all the parts of a building, with a logical and rational correspondence between the parts and the whole, so that each obeys a precise law. Nothing can be added or taken away from this correspondence without destroying the harmony of the whole. 

Beauty, according to Alberti, therefore, is primarily tied to the concept of harmony or, to be more precise, of Concinnitas,23 described by Alberti as the equilibrium and order born of a respect for the mathematical law of proportionality. Harmony is achieved by respecting an objective proportional system that eschews individual whim and is based on “exact and constant criteria.” 

Throughout the Renaissance, the proportional ­system thought to have been triggered directly by the immutable laws governing the known universe was the Pythagorean ­Harmonic Scale, in which the discovery—by Pythagoras—of the existence of a perfect relationship between some fractions of the first whole numbers of the decimal scale (1, 2, 3, 4 …)  and the different tones of the musical scale, rendered the ­system universally valid and revelatory of celestial harmonies. In other words, if a string of a determinate length vibrates, it emits a sound; if this string is cut in half—ratio 1:2—the sound will be the same but an octave higher. If the string is cut by two-thirds—ratio 2:3—the sound will be a fifth higher; and if cut by three-quarters, it will sound like a fourth, and so on. 

By translating music into geometry, the Renaissance architects adopted the harmonic proportions as the foundation for every geometric system. The elementary ratios of 1:1, 1:2, 2:3, and 3:4—and this also holds true for more complex ratios—became the essential ratios to which the proportions of all the buildings should conform. This great Renaissance lesson on proportions is just as valid today. It is a lesson that places the image of idealized beauty at the heart of the project and, especially, the need for a geometrical structure as its foundation. 

Going back to Alberti, we realize that the “exact and constant criteria” are precisely the definition of this geometrical structure, capable of ordering an entire project regardless of personal preferences. This is one of the central themes of Alberti’s work, the focus on defining a canon, rules that will define the foundations of the discipline. The whole of De re aedificatoria is a systematic attempt to make architecture transmissible and knowable, an attempt to get to the root of the problem in order to build the foundations of an art that constantly repeats itself, even as it evolves. 

Despising what we do not understand is a defect typical of the ignorant, according to Alberti. It denies that architecture is an art founded on knowledge. It denies the fact that in order to build competently, the rules must be deduced from a study of the past.24

We now come to Santa Maria delle Carceri in Prato. This is a small but very powerful building, its interior measuring just 26 × 26 meters. The plan is very simple and is based on a Greek cross; it is one of the earliest Renaissance examples of a Greek cross plan. The four, rather short, arms meet in the central space, its geometry based on the figures of the circle and the square.

The proportional ratios could not be more obvious. The whole thing is based on the ratio of 1:1 or 1:2. The depth of the arms is equal to half their width—1:2—while the height of the back walls, up to the base of the vault equals the width—1:1. The height of the vault above the arms, in the key, is one and a half times their width, while the impost of the dome over the central space is equal to twice the side of the basic square. A system of molded pietra serena pilasters and horizontal strips mark out the geometric order of the space on the white sur­faces, acting as a sort of guide for the eye, enabling us to identify the essential elements of the space.

“Beauty will result from the form and correspondence of the whole, with respect to the several parts, of the parts with regard to each ­other, and of these again to the whole; that the structure may appear an entire and compleat body, wherein each member agrees with the other, and all necessary to compose what you intend to form.”27

Palladio was an extremely practical architect; in The Four Books on Architecture, his theses are illustrated with few words but a great many drawings. Palladio used drawing to try and describe with the greatest possible accuracy principles, rules, and proportions for good architecture. His practical sense appears to reflect his faith in the Aristotelian doctrine of experience; and his faithfulness to the ancient prototypes reveals a certain familiarity with the doctrine of mimesis. Thus imitating classical buildings, those that have achieved the peak of perfection, does not mean passively replicating previous models; on the contrary, it means attempting to keep to the authority of the classical rules—the deep spirit of the consummate masters— in order to generate or perhaps regenerate architecture. 

Palladio wrote, “And altho’ variety and things new may please everyone, yet they ought not to be done contrary to the precepts of art, and contrary to that which reason dictates; whence one sees that altho’ the antients did vary, yet they never departed from the universal and necessary rules of art, as shall be seen in my books of antiquities.”28 

The villas designed by Palladio as a whole are a demonstration of all this. They exemplify great rigor and equally great in­ventiveness, demonstrating that, through the rigorous and systematic application of the same rules of a general nature, a huge variety of formal results can be achieved. In the rigor of the layouts, in the almost obsessive repetition of some of his themes and in his quest for a preordained formal order, Palladio has more in common with a contemporary artist than with a Renaissance man. 

Let us take a look at these rules. The first and most obvious is the inclusion of a great hall, the main one, positioned in the center of the building, in line with the axis of the main entrance. The hall, the real centerpiece of the compo­sition, could be interpreted as the generator of the surrounding spaces; sometimes by means of an operation involving an ­addition and sometimes by means of an operation involving the subdivision of a larger geometric figure. 

The second rule is the presence of alternating narrow and wide bays approaching the entrance. The narrow bays, alongside the great hall, usually contain a pair of staircases, symmetrically arranged in line with the main axis. 

The third rule is the presence of a transverse axis, perpendicular to the entrance axis, which does not necessarily also become the axis of symmetry, but which determines the development of the cruciform plan. 

Let us take a 400-year leap ahead to Berlin in 1962, the year in which Mies van der Rohe embarked on the design for the Neue Nationalgalerie. The design is the upshot of lengthy reflection over the problem of covering large rooms with a steel roof, with which he had already had to contend in previous situations30. The technical issue was resolved by moving the load-bearing components to the exterior of the building, thus completely freeing up the internal space. 

“The way is essential and determines the conclusive or concluded character of the work. Formation determines form and is therefore the greater of the two.

Thus form may never be regarded as solution, result, end, but should be regarded as genesis, growth, essence. Form as phenomenon is a dangerous chimera. 

Form as movement, as action is a good thing, active form is good. Form as rest, as end, is bad. 

Passive, finished form is bad. Formation is good. Form is bad; form is the end, death. Formation is movement, act. Formation is life.”31

Paul Klee, The Thinking Eye

There is an even more slippery and delicate issue that concerns beauty: how to establish a relationship between geometry and essence in art. The word essence has such broad and complex philosophical implications that they are way beyond the scope of this essay. Yet it seems to be the appropriate word for ­describing a behavior, the way many artists behave in front of their own work, a way of understanding art as a tool for ­exploring reality and the real. Let us see why.

Let us imagine we are standing in front of Paul Klee’s painting In the Current Six Thresholds but we don’t know its title. At first glance it would appear to be composed of a series of horizontal colored lines and spaces, some of them interrupted by just a few vertical lines, not many. The painting is ­totally abstract. All we can say at first glance concerns the ­relationships between the colors and proportions between the various horizontal stripes. There is a slight progression from left to right, making the lines and horizontal spaces closer and narrower on the left, while they widen and thin out ­towards the right. The chiaroscuro alternation of the horizontal spaces follows the same progression and is more marked on the left side of the painting, becoming nuanced on the right. The change of rhythm occurs at the point at which the horizontal scansion meets the vertical lines; the horizontal rhythm doubles precisely with every vertical “leap.” 

That’s it. As lovers of abstract art, our soul is satisfied. It is a perfect construction of “pure figurative ratios”: lines and patches of color are distributed on the canvas without any pretension towards depiction. Through a process of analytical decomposition, the painter has critically isolated the painting tools and has created the object, the aim of his work32. At a certain point, we happen to find out the name of the painting. There is nothing abstract about its title; it’s called In the Current Six Thresholds.

Let us take a step back in time for a moment, to the period of the Renaissance, in which geometry—as we have seen—evolved from tool to objective, transcending its own mandate to become the narration of a concept.

“Weary of the mute sterility of irregular volumes, I proceeded to study regular volumes. What I first noted was their regularity, their symmetry and their variety; and I perceived that that was what ­constituted their shape and their form. What is more, I realized that regularity alone had given man a clear conception of the shape of ­volumes, and so he gave them a definition which, as we shall see, ­resulted not only from their regularity and symmetry but also from their variety.”35

What is so fascinating about Boullée’s architecture? Firstly, its austerity and rigor. Producing an enormous number of theoretical projects, free of the compromises that the professional practice of architecture necessarily implies, gave Boullée the freedom to put together and clarify a concept of architecture over time, which has become increasingly solid, and in which, as we shall see, form and ideal tend to coincide. 

This concept of architecture is fascinating also because of its didactic relevance. All Boullée’s theoretical work was geared to a rational systematizing of knowledge in architecture. As Aldo Rossi says: “Boullée is a rationalist architect in the sense that, having built a logical system of architecture, he continually attempts to verify its principles through his various ­projects; and the rationality of the design consists of sticking to this system.”36 In this sense, Boullée shares the ambition and the patience of the Renaissance treatise-writers; he tries to found or refound the discipline on the definition and characteristics of a series of fundamental types. Types that coincide with the main public buildings and institutions of the city, such as the Monument, the Basilicas, the Theatre, the Sovereign’s Palace, the Law Courts, the National Palace, the Municipal Palace, the Circus, the Public Library, Funerary Monuments, and City Gates. He strove for a logical dimension for architecture, even though, as we have said, on the threshold of this attempt at systemization, a subjective and inevitably autobiographical ­datum/element would always creep in.37