Geometry Architecture Quotes

Thus nature provides a system for proportioning the growth of plants that satisfies the three canons of architecture. All modules are isotropic and they are related to the whole structure of the plant through self-similar spirals proportioned by the golden mean.

The human mind is only capable of absorbing a few things at a time. We see what is taking place in front of us in the here and now, and cannot envisage simultaneously a succession of processes, no matter how integrated and complementary. Our faculties of perception are consequently limited even as regards fairly simple phenomena. The fate of a single man can be rich with significance, that of a few hundred less so, but the history of thousands and millions of men does not mean anything at all, in any adequate sense of the word. The symmetriad is a million—a billion, rather—raised to the power of N: it is incomprehensible. We pass through vast halls, each with a capacity of ten Kronecker units, and creep like so many ants clinging to the folds of breathing vaults and craning to watch the flight of soaring girders, opalescent in the glare of searchlights, and elastic domes which criss-cross and balance each other unerringly, the perfection of a moment, since everything here passes and fades. The essence of this architecture is movement synchronized towards a precise objective. We observe a fraction of the process, like hearing the vibration of a single string in an orchestra of supergiants. We know, but cannot grasp, that above and below, beyond the limits of perception or imagination, thousands and millions of simultaneous transformations are at work, interlinked like a musical score by mathematical counterpoint. It has been described as a symphony in geometry, but we lack the ears to hear it.

...a paradox: architecture can only escape the flatness imposed by drawing through drawing.

In Egypt, construction is male geometry, a glorification of the visible. The first clarity of intelligible form appears in Egypt, the basis of Greek Apollonianism in art and thought. Egypt discovers four-square architecture, a rigid grid laid against mother nature's melting ovals. Social order becomes a visible aesthetic, countering nature's chthonian invisibilities. Pharaonic construction is the perfection of matter in art. Fascist political power, grandiose and self-divinising, creates the hierarchical, categorical superstructure of western mind.

Here is the essence of mankind's creative genius: not the edifices of civilization nor the bang-flash weapons which can end it, but the words which fertilize new concepts like spermatoza attacking an ovum. It might be argued that the Siamese-twin infants of word/idea are the only contribution the human species can, will, or should make to the reveling cosmos. (Yes, our DNA is unique, but so is a salamander's. Yes, we construct artifacts, but so have species ranging from beavers to the architecture ants... Yes, we weave real fabric things from the dreamstuff of mathematics, but the universe is hardwired with arithmetic. Scratch a circle and pi peeps out. Enter a new solar system and Tycho Brahe's formulae lie waiting under the black velvet cloak of space/time. But where has the universe hidden a word under its outer layer of biology, geometry, or insensate rock?)

Likeness is not identity; orthographic projection is not orthography; drawing is not writing and architecture does not speak.

We therefore find that the triangles and rectangles herein described, enclose a large majority of the temples and cathedrals of the Greek and Gothic masters, for we have seen that the rectangle of the Egyptian triangle is a perfect generative medium, its ratio of five in width to eight in length 'encouraging impressions of contrast between horizontal and vertical lines' or spaces; and the same practically may be said of the Pythagorean triangle

Architecture’s most kissable aspect is its surface. Space is hard to get a hold on. Structure has historically been inadequately pliant. Geometry—well, who really wants to kiss a square? Architecture also has more surface and more kinds of surface than anything else: outside, inside, soft and hard, there’s a surface for everyone. Finally, surfaces are where architecture gets close to turning into something else and therefore exactly where it becomes vulnerable and full of potential.

A whole problematic then develops: that of an architecture that is no longer built simply to be seen (as with the ostentation of palaces), or to observe the external space (cf. the geometry of fortresses), but to permit an internal, articulated and detailed control – to render visible those who are inside it; in more general terms, an architecture that would operate to transform individuals: to act on those it shelters, to provide a hold on their conduct, to carry the effects of power right to them, to make it possible to know them, to alter them. Stones can make people docile and knowable. The old simple schema of confinement and enclosure — thick walls, a heavy gate that prevents entering or leaving — began to be replaced by the calculation of openings, of filled and empty spaces, passages and transparencies.

The scientific principles that man employs to obtain the foreknowledge of an eclipse, or of anything else relating to the motion of the heavenly bodies, are contained chiefly in that part of science which is called trigonometry, or the properties of a triangle, which, when applied to the study of the heavenly bodies, is called astronomy; when applied to direct the course of a ship on the ocean, it is called navigation; when applied to the construction of figures drawn by rule and compass, it is called geometry; when applied to the construction of plans or edifices, it is called architecture; when applied to the measurement of any portion of the surface of the earth, it is called land surveying. In fine, it is the soul of science; it is an eternal truth; it contains the mathematical demonstration of which man speaks, and the extent of its uses is unknown.

We pass through vast halls, each with a capacity of ten Kronecker units, and creep like so many ants clinging to the folds of breathing vaults and craning to watch the flight of soaring girders, opalescent in the glare of searchlights, and elastic domes which criss-cross and balance each other unerringly, the perfection of a moment, since everything here passes and fades. The essence of this architecture is movement synchronized towards a precise objective. We observe a fraction of the process, like hearing the vibration of a single string in an orchestra of supergiants. We know, but cannot grasp, that above and below, beyond the limits of perception or imagination, thousands and millions of simultaneous transformations are at work, interlinked like a musical score by mathematical counterpoint. It has been described as a symphony in geometry, but we lack the ears to hear it.

After Us, the Salamanders!, The Future belongs to the Newts, Newts Mean Cultural Revolution. Even if they don't have their own art (they explained) at least they are not burdened with idiotic ideals, dried up traditions and all the rigid and boring things taught in schools and given the name of poetry, music, architecture, philosophy and culture in any of its forms. The word culture is senile and it makes us sick. Human art has been with us for too long and is worn-out and if the newts have never fallen for it we will make a new art for them. We, the young, will blaze the path for a new world of salamandrism: we wish to be the first newts, we are the salamanders of tomorrow! And so the young poetic movement of salamandrism was born, triton - or tritone - music was composed and pelagic painting, inspired by the shape world of jellyfish, fish and corals, made its appearance. There were also the water regulating structures made by the newts themselves which were discovered as a new source of beauty and dignity. We've had enough of nature, the slogans went; bring on the smooth, concrete shores instead of the old and ragged cliffs! Romanticism is dead; the continents of the future will be outlined with clean straight lines and re-shaped into conic sections and rhombuses; the old geological must be replaced with a world of geometry. In short, there was once again a new trend that was to be the thing of the future, a new aesthetic sensation and new cultural manifestoes; anyone who failed to join in with the rise of salamandrism before it was too late felt bitterly that he had missed his time, and he would take his revenge by making calls for the purity of mankind, a return to the values of the people and nature and other reactionary slogans. A concert of tritone music was booed off the stage in Vienna, at the Salon des Indépendents in Paris a pelagic painting called Capriccio en Bleu was slashed by an unidentified perpetrator; salamandrism was simply victorious, and its rise was unstoppable.

It's not unreasonable [...] to suppose that some kind of cosmic "sky-ground" religion lay behind the alignments to the solstices and the equinoxes at Watson Brake and at the other early sites--a religion sufficiently robust to ensure the continuous successful transmission of a system of geometry, astronomy, and architecture over thousands of years. John Clark is in no doubt. 'The evidence,' he says, 'suggests very old and widely disseminated knowledge about how to build large sites. The building lore persisted remarkably intact for so long that I think we can, and must, assume that it was part of special knowledge tied to ritual practice.' Where did this special knowledge come from before it appeared at Watson Brake?

As I explain at some length in 'The Crystal Sun' this particular angle, which we can call the 'golden angle,' is the precise value of the acute angle of of a right-angled 'golden triangle' that embodies the golden mean proportion .... The Danish art historian Else Kielland established with conclusive and absolutely overwhelming evidence and analysis that this angle was the basis for all Egyptian art and architecture. She did this in her monumental work 'Geometry in Egyptian Art' ..... The King's Chamber inside the Great Pyramid embodies no fewer than eight occurrences of the golden angle, and the coffer in the chamber embodies yet more.

In North America the evidence is that hunter-gatherers bounced back quite successfully within less than a millennium of the onset of the Younger Dryas, and thereafter there is a thin but fairly continuous archaeological record. What is mysterious is not so much the early appearance of mound-building in this new age--perhaps as early as 8,000 years ago, as we've seen--or the sophistication of sites such as Watson Brake 5,500 years ago, nor even their obvious astronomical and geometrical connections to later vast earthworks such as Moundville and Cahokia, but that in this early monumental architecture of the New World memes of geometry, astronomy, and solar alignments consistently appear that are also found in the early monumental architecture of the Old World at iconic sites such as Stonehenge and the Great Pyramid of Giza. A tremendous leap forward in agricultural know-how, coupled with the sudden uptake of eerily distinctive spiritual ideas concerning the afterlife journey of the soul, also often accompanies the architectural memes. It's therefore hard to avoid the impression that some kind of 'package' is involved here.

An architectural curiosity. And a flawed comparison, as the geometry of this chamber is quite different. The great curse of the artistic education, or what you call the liberal arts, is the understanding of the world through metaphor. Understanding things for what they resemble, rather than what they are in themselves. I think that is what draws you to the battle against conspiracy. Am I correct?” “I guess. I call it false narrativization.” Rostam laughed, and the sound echoed back around them “Also it brings in listeners.

Aldus Barnes, a structural engineer by training and member of the Advanced Geometry Unit (AGU) at Arup, has formed many successful collaborations and earned a prominent place for himself in architecture by adopting the language and skills of architects. "Talk in terms of texture and density, instead of torsion and shear. That way they don't think you are just another nerd," Barnes advises the young members of his team.

How did nature manage to evolve such complicated architecture? Mandelbrot's point is that the complications exist only in the context of traditional Euclidean geometry. As fractals, branching structures can be described with transparent simplicity, with just a few bits of information. Perhaps the simplest transformations that gave rise to the shapes devised by Koch, Peano, and Sierpinski have their analogue in the coded instructions of an organism's genes. DNA surely cannot specify the vast number of bronchi, bronchioles, and alveoli or the particular spatial structure of the resulting tree, but it can specify a repeating process of bifurcation and development. Such processes suit nature's purposes. When E.I. Dupont de Nemours & Company and the United States Army finally began to produce a synthetic match for goose down, it was by finally realizing that the phenomenal air-trapping ability of the natural product came from the fractal nodes and branches of down's key protein, keratin. Mandelbrot glided matter-of-factly from pulmonary and vascular trees to real botanical trees, trees that need to capture sun and resist wind, with fractal branches and fractal leaves. And theoretical biologists began to speculate that fractal scaling was not just common but universal in morphogenesis. They argued that understanding how such patterns were encoded and processed had become a major challenge to biology.

Cutting stone to the optimum basic size for building secure, aesthetically pleasing combinations was tantamount to establishing the smallest prime integer without which there would be no arithmetic or geometry. Working by trial and error with units, Greek mathematical thinking arrived at the idea of beautiful proportions, which in turn formed the basis of all the arts, not just architecture. Masonry as a method led to principles that eventually came to govern the whole of western art.

Despite the emphasis that the Greeks placed on theory, they had little interest in applying it to any kind of technology that would have changed their views of the manageability of the future. When Archimedes invented the lever, he claimed that he could move the earth if only he could find a place to stand. But apparently he gave no thought to changing it. The daily life of the Greeks, and their standard of living, were much the same as the way that their forebears had subsisted for thousands of years. They hunted, fished, grew crops, bore children, and used architectural techniques that were only variations on themes developed much earlier in the Tigris-Euphrates valley and in Egypt. Genuflection before the winds was the only form of risk management that caught their attention: their poets and dramatists sing repeatedly of their dependence on the winds, and beloved children were sacrificed to appease the winds. Most important, the Greeks lacked a numbering system that would have enabled them to calculate instead of just recording the results of their activities.9 I do not mean to suggest that the Greeks gave no thought to the nature of probability. The ancient Greek word εικος (eikos), which meant plausible or probable, had the same sense as the modern concept of probability: “to be expected with some degree of certainty.” Socrates defines εικος as “likeness to truth.”10 Socrates’ definition reveals a subtle point of great importance. Likeness to truth is not the same thing as truth. Truth to the Greeks was only what could be proved by logic and axioms. Their insistence on proof set truth in direct contrast to empirical experimentation. For example, in Phaedo, Simmias points out to Socrates that “the proposition that the soul is in harmony has not been demonstrated at all but rests only on probability.” Aristotle complains about philosophers who, “. . . while they speak plausibly, . . . do not speak what is true.” Elsewhere, Socrates anticipates Aristotle when he declares that a “mathematician who argues from probabilities in geometry is not worth an ace.”11 For another thousand years, thinking about games and playing them remained separate activities.

The machine aesthetic bought something familiar to modern experience, but remote from architecture, to bear on modern architecture.

…the extent of Ronchamp’s resemblance to other things is exceptionally broad. Nor is it only an issue of extent. More is at stake. The nature of the building alters drastically when what it resembles cannot be traced back to Le Corbusier’s mind; it becomes less predictable and its meaning is less stable.

Passyunk Avenue (pronounced pashunk by the locals) cuts a rude swath across an otherwise orderly grid of streets in South Philadelphia. Except for Passyunk (and Moyamensing) Avenue, the neighborhood is composed of a uniform matrix of numbered and named streets—one big street followed by two little streets. Viewed on a map, they form ninety-degree angles and predictable intersections. Passyunk Avenue, or simply Passyunk, is the great disruptor of this comforting geometry. Irregular and meandering, its slashing path intersects with the more obedient byways. Together they form a unique gridwork of inconvenient crossings and odd angles. The cumulative result is one of strangely shaped buildings. Their pointy corners puncture curious cells of dead space—the spaces between. While born of necessity, the resulting architecture created by these acute angles also manages to be strangely beautiful, an exotic visage in a sea of pretty faces. If you’ve ever seen the famous photo of Sophia Loren giving the side-eye to Jayne Mansfield, that’s Passyunk—South Philly’s middle finger to white bread Center City.

Varro produced an influential encyclopaedia, Nine Books of Disciplines, in which he outlined nine arts: grammar, rhetoric, logic, arithmetic, geometry, astronomy, musical theory, medicine and architecture. Later writers omitted the last two arts.79 In Rome, by the end of the first century AD, education had been more or less standardised and the seven liberal arts identified. In turn, these would become the basis of medieval education,

The spread of concrete also spawned whole new types of architecture. One of its earliest apostles was the American architect Frank Lloyd Wright,56 who understood that concrete made possible entirely new forms. Take the inverted ziggurat of the Solomon R. Guggenheim Museum that Wright designed in New York. Wright created its fanciful geometry with “gun-placed concrete,” aka gunite, a form of the compound made with more sand and less gravel than ordinary concrete, which allows it to be sprayed from a nozzle57 directly onto a vertical surface. Try doing that with brick. Wright’s work paved, so to speak, the way for Walter Gropius’s Bauhaus School, Le Corbusier’s International school, and Richard Neutra’s modernist creations. From Modernism grew Brutalism, the stark, angular, proudly concrete-heavy style that became popular after World War II. Today that term is often applied more broadly to the generic mode that has come to define so much of the visual landscape of our cities—the bluntly utilitarian look of near-identical factories and warehouses, the quadrangular shapes of institutional buildings and cheap apartment blocks, the coldly functional sweep of highway overpasses.

For instance, Le Corbusier and Amedee Ozenfant proposed a theory of painting and architecture which would be based primarily on Platonic forms: cones, spheres, cylinders, cubes, etc. They argued that only these simple forms were universal, and that they would in fact set off "identical sensations" in "everyone on earth- a Frenchman, a Negro, a Laplander”. In essence they were arguing for a universal language of the emotions- Purisme which would cut through the Babel of contending, eclectic languages. The individual words of this language would be the psychophysical constants found by psychologists. A flat line would mean "repose," a blue color "sadness,'' a jagged, diagonal line "activity,'' and so on until the whole gamut of emotions” (82>83) had been built up. They argued, as Plato often did, that nature had constructed within us a fixed language based on efficiency, geometry and function; this language of the emotions was the most economical and pure one-hence Purisme.

It is often assumed that a symbol, while not necessarily representing a truth, represents a belief, and so speaks directly for a historical frame of mind.

As time passes, what was once an unpredictable step sideways becomes a definitive step forward for art.

It is a conservation of momentum: what is diminished by time is augmented by the pencil. As the idea fades, the agitation around it has to increase to keep it alive; and as the scope of the architectural profession also reduces, the designer’s visible impact on the object increases, as if to recall lost powers.

The truly haunting Other is not what lies outside the text but what lies outside the picturesque garden of the Western world.

Architecture led to the vanishing point which led to architecture.

Even were we to agree, we could notice that things do not develop by continuous extrusion through time like toothpaste squeezed from a tube.

The mystery of building is that many things conspire to stand as one.

Art needs to be imaginatively undressed in order to be appreciated, and so does the world.

…the opposite of man is woman. Language enable us to deny this, but makes affirmation easier.

Mathematics, after all, lives on unambiguous exactitude, whereas there are types of art that die of it.

A person alone in a room is like the soul in the body.

I do not wish integrity to be taken in some vague metaphorical sense; I mean it quite literally.

Imagination is not held within the mind, but it is potentially active in all the areas of transition from persons to objects or pictures. It operates, in other words, in the same zones as projection and its metaphors.

Design is action at a distance.

...architecture begins and ends in pictures…

I am exalted at the thought that I see through his eyes as well as mine.

Is Ronchamp reminiscent of the Ark left stranded on Ararat, or like ‘bits of broken china thrown on top of the hill’? Is the roof like a bird’s wing, or does the whole edifice look like a decoy? Is it an alighting dove or a sitting duck? Good taste bids us to suppress the latter in favour of the former, although the latter is as easy to see.

The Modular, good fable that it is, tells a story about a hidden regulatory agency, but it is not that agency itself.

The Primary Act. As they entered the cinema, Dr Nathan confided to Captain Webster, ‘Talbert has accepted in absolute terms the logic of the sexual union. For him all junctions, whether of our own soft biologies or the hard geometries of these walls and ceilings, are equivalent to one another. What Talbert is searching for is the primary act of intercourse, the first apposition of the dimensions of time and space. In the multiplied body of the film actress - one of the few valid landscapes of our age - he finds what seems to be a neutral ground. For the most part the phenomenology of the world is a nightmarish excrescence. Our bodies, for example, are for him monstrous extensions of puffy tissue he can barely tolerate. The inventory of the young woman is in reality a death kit.’ Webster watched the images of the young woman on the screen, sections of her body intercut with pieces of modern architecture. All these buildings. What did Talbert want to do - sodomize the Festival Hall? Pressure Points. Koester ran towards the road as the helicopter roared overhead, its fans churning up a storm of pine needles and cigarette cartons. He shouted at Catherine Austin, who was squatting on the nylon blanket, steering her body stocking around her waist. Two hundred yards beyond the pines was the perimeter fence. She followed Koester along the verge, the pressure of his hands and loins still marking her body. These zones formed an inventory as sterile as the items in Talbert’s kit. With a smile she watched Koester trip clumsily over a discarded tyre. This unattractive and obsessed young man - why had she made love to him? Perhaps, like Koester, she was merely a vector in Talbert’s dreams. Central Casting. Dr Nathan edged unsteadily along the catwalk, waiting until Webster had reached the next section. He looked down at the huge geometric structure that occupied the central lot of the studio, now serving as the labyrinth in an elegant film version of The Minotaur . In a sequel to Faustus and The Shrew , the film actress and her husband would play Ariadne and Theseus. In a remarkable way the structure resembled her body, an exact formalization of each curve and cleavage. Indeed, the technicians had already christened it ‘Elizabeth’. He steadied himself on the wooden rail as the helicopter appeared above the pines and sped towards them. So the Daedalus in this neural drama had at last arrived. An Unpleasant Orifice. Shielding his eyes, Webster pushed through the camera crew. He stared up at the young woman standing on the roof of the maze, helplessly trying to hide her naked body behind her slim hands. Eyeing her pleasantly, Webster debated whether to climb on to the structure, but the chances of breaking a leg and falling into some unpleasant orifice seemed too great. He stood back as a bearded young man with a tight mouth and eyes ran forwards. Meanwhile Talbert strolled in the centre of the maze, oblivious of the crowd below, calmly waiting to see if the young woman could break the code of this immense body. All too clearly there had been a serious piece of miscasting. ‘Alternate’ Death. The helicopter was burning briskly. As the fuel tank exploded, Dr Nathan stumbled across the cables. The aircraft had fallen on to the edge of the maze, crushing one of the cameras. A cascade of foam poured over the heads of the retreating technicians, boiling on the hot concrete around the helicopter. The body of the young woman lay beside the controls like a figure in a tableau sculpture, the foam forming a white fleece around her naked shoulders.