Geometric Quotes

That's what I love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you to another book, and another bit there will lead you onto a third book. It's geometrically progressive - all with no end in sight, and for no other reason than sheer enjoyment.

Philosophy [nature] is written in that great book which ever is before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.

That's what I love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you onto another book, and another bit there will lead you onto a third book. It's geometrically progressive—all with no end in sight, and for no other reason than sheer enjoyment.

She is angle. I am curve. Together, we are geometric sculpture, and we make perfect sense.

CIRCLES OF LIFE Everything Turns, Rotates, Spins, Circles, Loops, Pulsates, Resonates, And Repeats. Circles Of life, Born from Pulses Of light, Vibrate To Breathe, While Spiraling Outwards For Infinity Through The lens Of time, And into A sea Of stars And Lucid Dreams. Poetry by Suzy Kassem

Satan, on the contrary, is thin, ascetic and a fanatical devotee of logic. He reads Machiavelli, Ignatius of Loyola, Marx and Hegel; he is cold and unmerciful to mankind, out of a kind of mathematical mercifulness. He is damned always to do that which is most repugnant to him: to become a slaughterer, in order to abolish slaughtering, to sacrifice lambs so that no more lambs may be slaughtered, to whip people with knouts so that they may learn not to let themselves be whipped, to strip himself of every scruple in the name of a higher scrupulousness, and to challenge the hatred of mankind because of his love for it--an abstract and geometric love.

We seem to inhabit a universe made up of a small number of elements-particles-bits that swirl in chaotic clouds, occasionally clustering together in geometrically logical temporary configurations.

There is still a difference between something and nothing, but it is purely geometrical and there is nothing behind the geometry.

You existed. You existed now as a fractal. Definition: A fractal is generally a rough or fragmented geometric shape that can be broken into parts, each of which is (at least approximately) a reduced-size copy of the whole. Maybe I was a fractal. Maybe the photographer was a fractal. Maybe we were all fractals.

Miracles do happen. You must believe this. No matter what else you believe about life, you must believe in miracles. Because we are all, every one of us, living on a round rock that spins around and around at almost a quarter of a million miles per hour in an unthinkably vast blackness called space. There is nothing else like us for as far as our telescopic eyes can see. In a universe filled with spinning, barren rocks, frozen gas, ice, dust, and radiation, we live on a planet filled with soft, green leaves and salty oceans and honey made from bees, which themselves live within geometrically complex and perfect structures of their own architecture and creation. In our trees are birds whose songs are as complex and nuanced as Beethoven’s greatest sonatas. And despite the wild, endless spinning of our planet and its never-ending orbit around the sun–itself a star on fire–when we pour water into a glass, the water stays in the glass. All of these are miracles.

I should attempt to treat human vice and folly geometrically... the passions of hatred, anger, envy, and so on, considered in themselves, follow from the necessity and efficacy of nature... I shall, therefore, treat the nature and strength of the emotion in exactly the same manner, as though I were concerned with lines, planes, and solids.

It snowed right before Jack stopped talking to Hazel, fluffy white flakes big enough to show their crystal architecture, like perfect geometric poems.

I guess a sock is also a geometric shape—technically—but I don't know what you'd call it. A socktagon?

He wasn't smart enough to see it, said Jason Bourne. He couldn't think geometrically.

That’s what I love about reading: one tiny thing will interest you in a book, and that tiny thing will lead you onto another book, and another bit there will lead you onto a third book. It’s geometrically progressive—all with no end in sight, and for no other reason than sheer enjoyment.

The pause - that impressive silence, that eloquent silence, that geometrically progressive silence which often achieves a desired effect where no combination of words, howsoever felicitous, could accomplish it.

So all night long the storm roared on: The morning broke without a sun; In tiny spherule traced with lines Of Nature’s geometric signs, In starry flake, and pellicle, All day the hoary meteor fell; And, when the second morning shone, We looked upon a world unknown, On nothing we could call our own. Around the glistening wonder bent The blue walls of the firmament, No cloud above, no earth below,— A universe of sky and snow!

Algebra applies to the clouds, the radiance of the star benefits the rose--no thinker would dare to say that the perfume of the hawthorn is useless to the constellations. Who could ever calculate the path of a molecule? How do we know that the creations of worlds are not determined by falling grains of sand? Who can understand the reciprocal ebb and flow of the infinitely great and the infinitely small, the echoing of causes in the abyss of being and the avalanches of creation? A mite has value; the small is great, the great is small. All is balanced in necessity; frightening vision for the mind. There are marvelous relations between beings and things, in this inexhaustible whole, from sun to grub, there is no scorn, each needs the other. Light does not carry terrestrial perfumes into the azure depths without knowing what it does with them; night distributes the stellar essence to the sleeping plants. Every bird that flies has the thread of the infinite in its claw. Germination includes the hatching of a meteor and the tap of a swallow's beak breaking the egg, and it guides the birth of the earthworm, and the advent of Socrates. Where the telescope ends, the microscope begins. Which of the two has a greater view? Choose. A bit of mold is a pleiad of flowers; a nebula is an anthill of stars. The same promiscuity, and still more wonderful, between the things of the intellect and material things. Elements and principles are mingled, combined, espoused, multiplied one by another, to the point that the material world, and the moral world are brought into the same light. Phenomena are perpetually folded back on themselves. In the vast cosmic changes, universal life comes and goes in unknown quantities, rolling everything up in the invisible mystery of the emanations, using everything, losing no dream from any single sleep, sowing a microscopic animal here, crumbling a star there, oscillating and gyrating, making a force of light, and an element of thought, disseminated and indivisible dissolving all, that geometric point, the self; reducing everything to the soul-atom; making everything blossom into God; entangling from the highest to the lowest, all activities in the obscurity of a dizzying mechanism, linking the flight of an insect to the movement of the earth, subordinating--who knows, if only by the identity of the law--the evolutions of the comet in the firmament to the circling of the protozoa in the drop of water. A machine made of mind. Enormous gearing, whose first motor is the gnat, and whose last is the zodiac.

But is it not already an insult to call chess anything so narrow as a game? Is it not also a science, an art, hovering between these categories like Muhammad's coffin between heaven and earth, a unique yoking of opposites, ancient and yet eternally new, mechanically constituted and yet an activity of the imagination alone, limited to a fixed geometric area but unlimited in its permutations, constantly evolving and yet sterile, a cogitation producing nothing, a mathematics calculating nothing, an art without an artwork, an architecture without substance and yet demonstrably more durable in its essence and actual form than all books and works, the only game that belongs to all peoples and all eras, while no one knows what god put it on earth to deaden boredom, sharpen the mind, and fortify the spirit?

To the untrained (human) eye, Thalassinia looks like an expanse of coral reefs and volcanic formations. There are no straight lines or geometric shapes to give away the fact that the structures are actually mermade. (Get it? Mermade. Like mermaid, but…oh, never mind.)

I know death hath ten thousand several doors For men to take their exits; and 'tis found They go on such strange geometrical hinges, You may open them both ways: any way, for heaven-sake

To him, a stilted geometric love of arrangement was “system,” and indefatigable and feverish interest in the pettiest facets of day-to-day bureaucracy was “industry”, indecision when right was “caution”, and blind stubbornness when wrong, “determination.

Because it is occasionally possible, just for brief moments, to find the words that will unlock the doors of all those many mansions inside the head and express something - perhaps not much, just something - of the crush of information that presses in on us from the way a crow flies over and the way a man walks and the look of a street and from what we did one day a dozen years ago. Words that will express something of the deep complexity that makes us precisely the way we are, from the momentary effect of the barometer to the force that created men distinct from trees. Something of the inaudible music that moves us along in our bodies from moment to moment like water in a river. Something of the spirit of the snowflake in the water of the river. Something of the duplicity and the relativity and the merely fleeting quality of all this. Something of the almighty importance of it and something of the utter meaninglessness. And when words can manage something of this, and manage it in a moment, of time, and in that same moment, make out of it all the vital signature of a human being - not of an atom, or of a geometrical diagram, or of a heap of lenses - but a human being, we call it poetry.

Bees... by virtue of a certain geometrical forethought, knew that the hexagon is greater than the square and the triangle and will hold more honey for the same expenditure of material.

Certainly, the terror of a deserted house swells in geometrical rather than arithmetical progression as houses multiply to form a city of stark desolation. The sight of such endless avenues of fishy-eyed vacancy and death, and the thought of such linked infinities of black, brooding compartments given over to cob-webs and memories and the conqueror worm, start up vestigial fears and aversions that not even the stoutest philosophy can disperse.

In the candle's flickering light, the library's thousands of books emerged from the shadows, and for a moment Nicholas could not help admiring them again. During free time he had almost never looked up from the pages he was reading, but now he saw the books anew, from without rather than from within, and was reminded of how beautiful they were simply as objects. The geometrical wonder of them all, each book on its own and all the books together, row upon row, the infinite patterns and possibilities they presented. They were truly lovely.

A circle is the only geometric shape defined by its centre. No chicken and egg about it, the centre came first, the circumference follows. The earth, by definition, has a centre. And only the fool that knows it can go wherever he pleases, knowing the centre will hold him down, stop him flying out of orbit. But when your sense of centre shifts, comes whizzing to the surface, the balance has gone. The balance has gone. The balance my baby has gone.

Inhabited space transcends geometrical space.

A house that has been experienced is not an inert box. Inhabited space transcends geometrical space.

I tore open one side and reached in. My hand closed around something cold and metallic. I knew before I even pulled it out what it was. It was a silver stake. "Oh God," I said. I rolled the stake around, running my finger over the engraved geometric pattern at its base. There was no question. One-of-a-kind. This was the stake I'd taken from the vault in Galina's house. The one I'd- "Why would someone send you a stake?"asked Lissa. I didn't answer and instead pulled out the envelope's next item:a small note card. There, in handwriting I knew all too well, was: 'You forgot another lesson:Never turn your back until you know your enemy is dead. Looks like we'll have to go over the lesson the next time I see you-which will be soon. Love, D.

Be honest: did you actually read [the above geometric proof]? Of course not. Who would want to? The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious by insisting that it be 'rigorously proved' ... is to say to a student 'Your feelings and ideas are suspect. You need to think and speak our way.

Buckminster Fuller explained to me once that because our world is constructed from geometric relations like the Golden Ratio or the Fibonacci Series, by thinking about geometry all the time, you could organize and harmonize your life with the structure of the world.

It was not a great presence but a great absence, a geometric ocean of darkness that seemed to swallow heaven itself.

All of Nature follows perfectly geometric laws. The Ancient Egyptian, Greek, Peruvian, Mayan, and Chinese cultures were well aware of this, as Phi—known as the Golden Ratio or Golden Mean—was used in the constructions of their sculptures and architecture.

The moth settled onto the curtain and sat still. It was an astonishing creature, with black and white wings patterned in geometric shapes, scarlet underwings, and a fat white body with black spots running down it like a snowman's coal buttons. No human eye had looked at this moth before; no one would see its friends. So much detail goes unnoticed in the world.

That’s a geometric figure, that bird, he don’t exist without an angle.

I quickly learned that reading is cumulative and proceeds by geometrical progression: each new reading builds upon whatever the reader has read before.

Geometric and psychedelic shapes, mosaics, and mandalas...There is a certain calm in the chaos that most folks don't see" -Claudia

There are not wanting, it is true, some promulgators of paradoxes who maintain that there is no necessary connection between geometrical and moral Irregularity.

Back when I was a regular mortal kid, I didn't know much about combat. I had some murky ideas that armies would line up, blow trumpets, and then march forward to kill one another in an orderly fashion. If I thought about Viking combat at all, I would envision some dude yelling, SHIELD WALL! and a bunch of hairy blond guys calmly forming ranks and merging their shields into some cool geometric pattern like a polyhedron or a Power Ranger Megazord.

In life you have to avoid three geometric figures - vicious circles, love triangles and square minds

They came to know the night skys well. Western eyes that read more geometric constructions than those names given by the ancients.

You do the right thing and then you do the next right thing. Over time it adds up, and the geometric potential of success is unleashed.

Then she understood that what she needed was the motion to a purpose, no matter how small or in what form, the sense of an activity going step by step to some chosen end across a span of time. The work of cooking a meal was like a closed circle, completed and gone, leading nowhere. But the work of building a path was a living sum, so that no day was left to die behind her, but each day contained all those that preceded it, each day acquired its immortality on every succeeding tomorrow. A circle, she thought, is the movement proper to physical nature, they say that there's nothing but circular motion in the inanimate universe around us, but the straight line is the badge of man, the straight line of a geometrical abstraction that makes roads, rails and bridges, the straight line that cuts the curving aimlessness of nature by a purposeful motion from a start to an end. The cooking of meals, she thought, is like the feeding of coal to an engine for the sake of a great run, but what would be the imbecile torture of coaling an engine that had no run to make? It is not proper for man's life to be a circle, she thought, or a string of circles dropping off like zeros behind him--man's life must be a straight line of motion from goal to farther goal, each leading to the next and to a single growing sum, like a journey down the track of a railroad, from station to station to--oh, stop it!

what we do with shame on an individual level has potentially geometric consequences for any of the social systems we occupy, be that our family, place of employment, church or larger community. It

Creation takes place in bottlenecks . . . A creator who isn’t grabbed around the throat by a set of impossibilities is no creator. A creator’s someone who creates their own impossibilities, and thereby creates possibilities . . . it’s by banging your head on the wall that you find a way through. You have to work on the wall, because without a set of impossibilities, you won’t have the line of flight, the exit that is creation, the power of falsity that is truth. Your writing has to be liquid or gaseous simply because normal perception and opinion are solid, geometric … You have to open up words, break things open, to free earth’s vectors.

Everything is composed of small particles of itself and they are flying around in concentric circles and arcs and segments and innumerable other geometrical figures too numerous to mention collectively, never standing still or resting but spinning away and darting hither and thither and back again, all the time on the go. These diminutive gentlemen are called atoms. Do you follow me intelligently?

Yet there was no doubt that Theodore Roosevelt was peculiarly qualified to be President of all the people. Few, if any Americans could match the breadth of his intellect and the strength of his character. A random survey of his achievements might show him mastering German, French, and the contrasted dialects of Harvard and Dakota Territory; assembling fossil skeletons with paleontological skill; fighting for an amateur boxing championship; transcribing birdsong into a private system of phonetics; chasing boat thieves with a star on his breast and Tolstoy in his pocket; founding a finance club, a stockmen's association, and a hunting-conservation society; reading some twenty thousand books and writing fifteen of his own; climbing the Matterhorn; promulgating a flying machine; and becoming a world authority on North American game mammals. If the sum of all these facets of experience added up to more than a geometric whole - implying excess construction somewhere, planes piling upon planes - then only he, presumably, could view the polygon entire.

The true splendor of science is not so much that it names and classifies, records and predicts, but that it observes and desires to know the facts, whatever they may turn out to be. However much it may confuse facts with conventions, and reality with arbitrary divisions, in this openness and sincerity of mind it bears some resemblance to religion, understood in its other and deeper sense. The greater the scientist, the more he is impressed with his ignorance of reality, and the more he realizes that his laws and labels, descriptions and definitions, are the products of his own thought. They help him to use the world for purposes of his own devising rather than to understand and explain it. The more he analyzes the universe into infinitesimals, the more things he finds to classify, and the more he perceives the relativity of all classification. What he does not know seems to increase in geometric progression to what he knows. Steadily he approaches the point where what is unknown is not a mere blank space in a web of words but a window in the mind, a window whose name is not ignorance but wonder.

extraordinary success is sequential, not simultaneous. What starts out linear becomes geometric. You do the right thing and then you do the next right thing. Over time it adds up, and the geometric potential of success is unleashed. The domino effect applies to the big picture, like your work or your business, and it applies to the smallest moment in each day when you’re trying to decide what to do next.

Mathematical solutions are selected by the subliminal self on the basis of “mathematical beauty,” of the harmony of numbers and forms, of geometric elegance.

The months of the year, from January up to June, are a geometric progression in the abundance of distractions.

The sea-lentils tied to giant serpentine string beans, sea-liquor brine, sea-lyme grass, sea-moss, sea-cucumbers. He never knew the sea had such a lavish garden—sea-plumes, sea-grapes, sea-lungs. […] The sky put on its own evanescent spectacles, a pivoting stage, fugitive curtains, decors for ballets, floating icebergs, unrolled bolts of chiffon, gold and pearl necklaces, marabous of oyster white, scarves of Indian saris, flying feathers, shorn lambs, geometric architecture in snows and cotton. His theater was the clouds, where no spectacle repeated itself.

The old concept of chronological, orderly, symmetrical development of character died when it was discovered that the unconscious motivations are entirely at odds with fabricated conventions. Human beings do not grow in perfect symmetry. They oscillate, expand, contract, backtrack, arrest themselves, retrogress, mobilize, atrophy in part, proceed erratically according to experience and traumas. Some aspects of the personality mature, others do not. Some live in the past, some in the present. Some people are futuristic characters, some are cubistic, some are hard-edged, some geometric, some abstract, some impressionistic, some surrealistic!

Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.

Homesickness is a great teacher. It taught me, during an endless rainy fall, that I came from the arid lands, and like where I came from. I was used to dry clarity and sharpness in the air. I was used to horizons that either lifted into jagged ranges or rimmed the geometrical circle of the flat world. I was used to seeing a long way. I was used to earth colors--tan, rusty red, toned white--and the endless green of Iowa offended me. I was used to a sun that came up over mountains and went down behind other mountains. I missed the color and smell of sagebrush, and the sight of bare ground.

In One Dimensions, did not a moving Point produce a Line with two terminal points? In two Dimensions, did not a moving Line produce a Square wit four terminal points? In Three Dimensions, did not a moving Square produce - did not the eyes of mine behold it - that blessed being, a Cube, with eight terminal points? And in Four Dimensions, shall not a moving Cube - alas, for Analogy, and alas for the Progress of Truth if it be not so - shall not, I say the motion of a divine Cube result in a still more divine organization with sixteen terminal points? Behold the infallible confirmation of the Series, 2, 4, 8, 16: is not this a Geometrical Progression? Is not this - if I might qupte my Lord's own words - "Strictly according to Analogy"? Again, was I not taught by my Lord that as in a Line there are two bonding points, and in a Square there are four bounding Lines, so in a Cube there must be six bounding Squares? Behold once more the confirming Series: 2, 4, 6: is not this an Arithmetical Progression? And consequently does it not of necessity follow that the more divine offspring of the divine Cube in the Land of Four Dimensions, must have eight bounding Cubes: and is not this also, as my Lord has taught me to believe, "strictly according to analogy"?

Most [Shetland patterns] are fairly small and simple, as textile patterns go. Almost all are symmetrical, with eight smaller parts. As a result, most are geometric rather than representational. ... One last feature, less easy to define but easy enough to recognise ... is the liking for little motifs and for a pattern to be 'finished'.

The association of multiplication with vector rotation was one of the geometric interpretation's most important elements because it decisively connected the imaginaries with rotary motion. As we'll see, that was a big deal.

I review all I know, but can synthesize no meaning. When I doze, the Fact, the certain accomplished calamity, wakes me roughly like a brutal nurse. I see it crouching inflexibly in a corner of the ceiling. It comes down in geometrical diagonal like lightning. It says, I remain, I AM, I shall never cease to be: your memory will grow a deathly glaze: you will forget, you will fade out, but I cannot be undone. Thus every quarter hour it puts the taste of death in my mouth, and shows me, but not gently, how I go whoring after oblivion.

People conceptualize conditioning in different ways," he said. "Some think it's a ladder straight up. Others see plateaus, blockages, ceilings. I see it as a geometric spiraling upward, with each spin of the circle taking you a different distance upward. Some spins may even take you downward, just gathering momentum for the next upswing. Sometimes you will work your fanny off and see very little gain; other times you will amaze yourself and not really know why.

I remember a little later wondering why things always turn out to be diametrically opposed to what you expect them to be. It's no good even trying to predict what this opposite will be because it always fools you and turns out to be the opposite of that, if you see what I mean. If you think this is geometrically impossible all I can say is that you don't know my life

Another means of silently lessening the inequality of property is to exempt all from taxation below a certain point, and to tax the higher portions or property in geometrical progression as they rise. Whenever there are in any country uncultivated lands and unemployed poor, it is clear that the laws of property have been so far extended as to violate natural right. The earth is given as a common stock for man to labor and live on. Letter to James Madison, October 28, 1785

A tenth-century copy of Archimedes’s Method of Mechanical Theorems. In it, Archimedes had ingeniously applied mechanical laws, such as the law of the lever, to find the volume and area of geometric shapes. Two thousand years before Newton, he had come tantalizingly close to deriving calculus. However, in the thirteenth century this work was scraped off and overwritten with a prayer book.

Darkness. The door into the neighboring room is not quite shut. A strip of light stretches through the crack in the door across the ceiling. People are walking about by lamplight. Something has happened. The strip moves faster and faster and the dark walls move further and further apart, into infinity. This room is London and there are thousands of doors. The lamps dart about and the strips dart across the ceiling. And perhaps it is all delirium... Something had happened. The black sky above London burst into fragments: white triangles, squares and lines - the silent geometric delirium of searchlights. The blinded elephant buses rushed somewhere headlong with their lights extinguished. The distinct patter along the asphalt of belated couples, like a feverish pulse, died away. Everywhere doors slammed and lights were put out. And the city lay deserted, hollow, geometric, swept clean by a sudden plague: silent domes, pyramids, circles, arches, towers, battlements.

In the pentagram, the Pythagoreans found all proportions well-known in antiquity: arithmetic, geometric, harmonic, and also the well-known golden proportion, or the golden ratio. ... Probably owing to the perfect form and the wealth of mathematical forms, the pentagram was chosen by the Pythagoreans as their secret symbol and a symbol of health. - Alexander Voloshinov [As quoted in Stakhov]

gold light burned faintly. From his cosy window seat, Mario was tracing a frost-flower on the windowpane with an unsure finger. Were its perfectly-rendered geometric patterns a product of nature, or were they an artefact of metaphysics? Was the frost-flower to the Masters what a work of Art was to him? Did the Masters of Strings truly control every aspect of reality? The fractal flower slowly melted under Mario’s fingertip. “No work of chance here,” he bitterly thought. “This was by design.

Outside the moon had come out. It was full, a disk of bright silver. I saw a large, dramatic spider web on my back porch that must have been made while I was in the house with my mind in turmoil; the spider was just finishing the outer circle of it. The moon illuminated the strands of the big taut web so that it seemed to be made of pure light. It was dazzling, geometric and mysterious, and it calmed me just to stop and look at it, at the elaboration and power of life that could make such a design.

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth. —Galileo Galilei, The Assayer, 1623

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.

Every unique thing in nature is related to the whole, and partakes of the perfection of the whole. Each particle is a microcosm, and faithfully renders the likeness of the world. In geometric harmony of the cosmos there are ways that resemble, there are universal patterns, from blood vessels, to winter trees or to a river delta, from nautilus shell to spiral galaxy, from neurons in the brain to the cosmic web. A whole universe of connections is in your mind – a universe within a universe – and one capable of reaching out to the other that gave rise to it. Billions of neurons touching billions of stars – surely spiritual.

I have little interest in illustration, which lacks a kind of transcendental quality. It is too literal. I find typography more straightforward, conceptual, and appealing, with its strict geometric vocabulary. There is a bridge between typographic design and fine art, especially since typography possesses a complex subtlety. The idea, the method, and the honesty in expression are central to a designer who works with type.

When Leonardo da Vinci wanted to create a whole new style of painting, one that was more lifelike and emotional, he engaged in an obsessive study of details. He spent endless hours experimenting with forms of light hitting various geometrical solids, to test how light could alter the appearance of objects. He devoted hundreds of pages in his notebooks to exploring the various gradations of shadows in every possible combination. He gave this same attention to the folds of a gown, the patterns in hair, the various minute changes in the expression of a human face. When we look at his work we are not consciously aware of these efforts on his part, but we feel how much more alive and realistic his paintings are, as if he had captured reality.

Emerson said that a library is a magic chamber in which there are many enchanted spirits. They wake when we call them. When the book lies unopened, it is literally, geometrically, a volume, a thing among things. When we open it, when the book surrenders itself to its reader, the aesthetic event occurs. And even for the same reader the same book changes, for the change; we are the river of Heraclitus, who said that the man of yesterday is not the man of today, who will not be the man of tomorrow. We change incessantly, and each reading of a book, each rereading, each memory of that rereading, reinvents the text. The text too is the changing river of Heraclitus.

The effect is both domestic and wild, equal parts geometric and chaotic. It's the visual signature of small, diversified farms that creates the picture-postcard landscape here, along with its celebrated gastronomic one. Couldn't Americans learn to love landscapes like these around our cities, treasuring them not just gastronomically but aesthetically, instead of giving everything over to suburban development? Can we only love agriculture on postcards?

Now in my eleven years of conventional life I had learned many things and one of them is what it means to be convicted of rape--I do not mean the man who did it, I mean the woman to whom it was done. Rape is one of the Christian mysteries, it creates a luminous and beautiful tableau in people's minds; and as I listened furtively to what nobody would allow me to hear straight out, I slowly came to understand that I was face to face with one of those feminine disasters, like pregnancy, like disease, like weakness; she was not only the victim of the act but in some strange way its perpetrator; somehow she had attracted the lightening that struck her out of a clear sky. A diabolical chance--which was not chance--had revealed her to all of us as she truly was, in her secret inadequacy, in that wretched guiltiness which she had kept hidden for seventeen years but which now finally manifested in front of everybody. Her secret guilt was this: She was Cunt. She had "lost" something. Now the other party to the incident had manifested his essential nature, too; he was Prick--but being Prick is not a bad thing. In fact, he had "gotten away with" something (possibly what she had "lost"). And there I was at eleven years of age: She was out late at night. She was in the wrong part of town. Her skirt was too short and that provoked him. She liked having her eye blacked and her head banged against the sidewalk. I understood this perfectly. (I reflected thus in my dream, in my state of being a pair of eyes in a small wooden box stuck forever on a grey, geometric plane--or so I thought.) I too had been guilty of what had been done to me, when I came home from the playground in tears because I had been beaten up by bigger children who were bullies. I was dirty. I was crying. I demanded comfort. I was being inconvenient. I did not disappear into thin air.

Appliqué is very popular here. To my eye it has a facile look about it, as if the maker has not thought hard but simply cut out whatever shape has taken her fancy and sewn it on to a bit of cloth. Piecing together patchwork, on the other hand, requires more consideration and more accuracy; that is why I like it, though some say it is too cold and geometrical.

I was sitting on the seashore, half listening to a friend arguing violently about something which merely bored me. Unconsciously to myself, I looked at a film of sand I had picked up on my hand, when I suddenly saw the exquisite beauty of every little grain of it; instead of being dull, I saw that each particle was made up on a perfect geometrical pattern, with sharp angles, from each of which a brilliant shaft of light was reflected, while each tiny crystal shone like a rainbow…. The rays crossed and recrossed, making exquisite patterns of such beauty that they left me breathless…. Then, suddenly, my consciousness was lighted up from within and I saw in a vivid way how the whole universe was made up of particles of material which, no matter how dull and lifeless they might seem, were nevertheless filled with this intense and vital beauty. For a second or two the whole world appeared as a blaze of glory. When it died down, it left me with something I have never forgotten and which constantly reminds me of the beauty locked up in every minute speck of material around us.

I exist, I think, I feel pain. Is all this as certain as a geometric truth? Yes. Why? It is because these truths are proved by the same principle that a thing cannot be and not be at the same thime. I cannot at the same time exist and not exist, feel and not feel. A triangle cannot at the same time have and not have 180 degrees, which is the sum of two right angles.

Back when I was a regular mortal kid, I didn’t know much about combat. I had some murky ideas that armies would line up, blow trumpets, and then march forward to kill one another in an orderly fashion. If I thought about Viking combat at all, I would envision some dude yelling SHIELD WALL! and a bunch of hairy blond guys calmly forming ranks and merging their shields into some cool geometric pattern like a polyhedron or a Power Ranger Megazord. Actual battle was nothing like that. At least, not any version I’d ever been in. It was more like a cross between interpretive dance, lucha libre wrestling, and a daytime talk show fight.

Boldwood looked at her—not slily, critically, or understandingly, but blankly at gaze, in the way a reaper looks up at a passing train—as something foreign to his element, and but dimly understood. To Boldwood women had been remote phenomena rather than necessary complements—comets of such uncertain aspect, movement, and permanence, that whether their orbits were as geometrical, unchangeable, and as subject to laws as his own, or as absolutely erratic as they superficially appeared, he had not deemed it his duty to consider.   He

Frederick consistently used his central position to concentrate against one fraction of the enemy, and he always employed tactics of indirect approach. Thereby he gained many victories. But his tactical indirect approach was geometrical rather than psychological-unprepared by the subtler forms of surprise favoured by Scipio-and for all their executive skill, these manoeuvres were narrow. The opponent might be unable to meet the following blow, owing to the inflexibility of his mind or his formations, but the blow itself did not fall unexpectedly.

The memory of human blood manifests now as a kind of visceral reaction to seeing people's veins and their necks. The skin on a neck appears to me as different from the skin anywhere else on a body. It seems as thin and consumable as rice paper wrapped around a sweet. It is too blank compared with skin everywhere else, as though it is asking to have marks made on it, like very expensive calligraphy paper, or cold-pressed Fabriano. Often, I wonder whether the urge I have to make art is the same as the urge to consume and destroy the blankness of a human neck. While at art college, I read that the best paper used by artists in the seventeenth century was made from the skins of lamb fetuses. This skin was soft and absorbent, and had an even texture right across its surface. For a long time, the process of creating art has been linked to the killing of living things. My dad, even, used fine silk stretched across wooden frames in his own work as a painter. Once, when we still had some of his pieces, I looked at the odd geometric shapes he created on a huge sheet and thought about all the silkworms who had had their cocoons torn open before they were able to become moths.

I am conscious that an equal division of property is impracticable. But the consequences of this enormous inequality [in Europe] producing so much misery to the bulk of mankind, legislators cannot invent too many devices for subdividing property,...[One] means of silently lessening the inequality of property is to exempt all from taxation below a certain point, and to tax the higher portions of property in geometrical progression as they rise.

He couldn’t understand how I could be so bright in my other classes and a total failure in his. I must not be trying hard enough, he told me. In truth, I didn’t care, and I didn’t want to care. Geometric shapes with their dangerously sharp angles seemed the mortal enemies of words, and formulas were written in a hostage-taking language that made me despair of ever freeing the words from their captors. The best I could do was to avoid the enemy and save myself.

With scientific discovery and invention proceeding, we are told, at the rate of geometric progression, a generally passive and culture-bound people cannot cope with the multiplying issues and problems. Unless individuals, groups, and nations can imagine, construct, and creatively revise new ways of relating to these complex changes, the lights will go out. Unless man can make new and original adaptations to his environment as rapidly as his science can change the environment, our culture will perish.

The road goes west out of the village, past open pine woods and gallberry flats. An eagle's nest is a ragged cluster of sticks in a tall tree, and one of the eagles is usually black and silver against the sky. The other perches near the nest, hunched and proud, like a griffon. There is no magic here except the eagles. Yet the four miles to the Creek are stirring, like the bleak, portentous beginning of a good tale. The road curves sharply, the vegetation thickens, and around the bend masses into dense hammock. The hammock breaks, is pushed back on either side of the road, and set down in its brooding heart is the orange grove. Any grove or any wood is a fine thing to see. But the magic here, strangely, is not apparent from the road. It is necessary to leave the impersonal highway, to step inside the rusty gate and close it behind. By this, an act of faith is committed, through which one accepts blindly the communion cup of beauty. One is now inside the grove, out of one world and in the mysterious heart of another. Enchantment lies in different things for each of us. For me, it is in this: to step out of the bright sunlight into the shade of orange trees; to walk under the arched canopy of their jadelike leaves; to see the long aisles of lichened trunks stretch ahead in a geometric rhythm; to feel the mystery of a seclusion that yet has shafts of light striking through it. This is the essence of an ancient and secret magic. It goes back, perhaps, to the fairy tales of childhood, to Hansel and Gretel, to Babes in the Wood, to Alice in Wonderland, to all half-luminous places that pleased the imagination as a child. It may go back still farther, to racial Druid memories, to an atavistic sense of safety and delight in an open forest. And after long years of spiritual homelessness, of nostalgia, here is that mystic loveliness of childhood again. Here is home. An old thread, long tangled, comes straight again.

The world’s oldest map, the Babylonian Map of the World, had a little circle bored through the center. Scholars explained that the hole had come from using a compass to trace the two outer rings of the map. Oghi was captivated more by that hole than by the geometric shapes engraved in the clay tablet, and had stared at it for a long time in the darkened exhibit room of the British Museum. That dark, narrow hole went as deep as the memory of an age that no one could ever return to. The only way to reach that lost age was through that hole, but the hole itself could never be reached.

Fractals are a kind of geometry, associated with a man named Mandelbrot. Unlike ordinary Euclidean geometry that everybody learns in school—squares and cubes and spheres—fractal geometry appears to describe real objects in the natural world. Mountains and clouds are fractal shapes. So fractals are probably related to reality. Somehow. “Well, Mandelbrot found a remarkable thing with his geometric tools. He found that things looked almost identical at different scales.” “At different scales?” Grant said. “For example,” Malcolm said, “a big mountain, seen from far away, has a certain rugged mountain shape. If you get closer, and examine a small peak of the big mountain, it will have the same mountain shape. In fact, you can go all the way down the scale to a tiny speck of rock, seen under a microscope—it will have the same basic fractal shape as the big mountain.

Whether we like it or not, if we are to pursue a career in science, eventually we have to learn the “language of nature”: mathematics. Without mathematics, we can only be passive observers to the dance of nature rather than active participants. As Einstein once said, “Pure mathematics is, in its way, the poetry of logical ideas.” Let me offer an analogy. One may love French civilization and literature, but to truly understand the French mind, one must learn the French language and how to conjugate French verbs. The same is true of science and mathematics. Galileo once wrote, “[The universe] cannot be read until we have learnt the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles, and other geometrical figures, without which means it is humanly impossible to understand a single word.

Standing in the courtyard with a glass eye; only half the world is intelligible. The stones are wet and mossy and in the crevices are black toads. A big door bars the entrance to the cellar; the steps are slippery and soiled with bat dung. The door bulges and sags, the hinges are falling off, but there is an enameled sign on it, in perfect condition, which says: “Be sure to close the door.” Why close the door? I can’t make it out. I look again at the sign but it is removed; in it’s place there is a pane of colored glass. I take out my artificial eye, spit on it and polish it with my handkerchief. A woman is sitting on a dais above an immense carven desk; she has a snake around her neck. The entire room is lined with books and strange fish swimming in colored globes; there are maps and charts on the wall, maps of Paris before the plague, maps of the antique world, of Knossos and Carthage, of Carthage before and after the salting. In the corner of the room I see an iron bedstead and on it a corpse is lying; the woman gets up wearily, removes the corpse from the bed and absent mindedly throws it out the window. She returns to the huge carven desk, takes a goldfish from the bowl and swallows it. Slowly the room begins to revolve and one by one the continents slide into the sea; only the woman is left, but her body is a mass of geography. I lean out the window and the Eiffle Tower is fizzing champagne; it is built entirely of numbers and shrouded in black lace. The sewers are gurgling furiously. There are nothing but roofs everywhere, laid out with execrable geometric cunning.

For me, madness was definitely not a condition of illness; I did not believe that I was ill. It was rather a country, opposed to Reality, where reigned an implacable light, blinding, leaving no place for shadow; an immense space without boundary, limitless, flat; a mineral, lunar country, cold as the wastes of the North Pole. In this stretching emptiness, all is unchangeable, immobile, congealed, crystallised. Objects are stage trappings, placed here and there, geometric cubes without meaning. People turn weirdly about, they make gestures, movements without sense; they are phantoms whirling on an infinite plain, crushed by the pitiless electric light. And I - I am lost in it, isolated, cold, stripped purposeless under the light.

Now we can see what makes mathematics unique. Only in mathematics is there no significant correction-only extension. Once the Greeks had developed the deductive method, they were correct in what they did, correct for all time. Euclid was incomplete and his work has been extended enormously, but it has not had to be corrected. His theorems are, every one of them, valid to this day. Ptolemy may have developed an erroneous picture of the planetary system, but the system of trigonometry he worked out to help him with his calculations remains correct forever. Each great mathematician adds to what came previously, but nothing needs to be uprooted. Consequently, when we read a book like A History of Mathematics, we get the picture of a mounting structure, ever taller and broader and more beautiful and magnificent and with a foundation, moreover, that is as untainted and as functional now as it was when Thales worked out the first geometrical theorems nearly 26 centuries ago. Nothing pertaining to humanity becomes us so well as mathematics. There, and only there, do we touch the human mind at its peak.

He was dressed in odd and garish fashion and his clothes were embroidered with signs that had about them the geometric look of instructions, perhaps a game. He wore jewelry of jade and silver and his hair was long and blacker than his age would seem to warrant. He told the boy that although he was a huérfano still he must cease his wanderings and make for himself some place in the world because to wander in this way would become for him a passion and by this passion he would become estranged from men and so ultimately from himself. He said that the world could only be known as it existed in men's hearts. For while it seemed a place that contained men it was in reality a place contained within them and therefor to know it one must look there and come to know those hearts and to do this one must live with men and not simply pass among them. He said that while the huérfano might feel that he no longer belonged among men he must set this feeling aside for he contained within him a largeness of spirit which men could see and that men would wish to know him and that the world would need him even as he needed the world for they were one. Lastly he said that while this itself was a good thing like all good things it was also a danger.

And are we not guilty of offensive disparagement in calling chess a game? Is it not also a science and an art, hovering between those categories as Muhammad’s coffin hovered between heaven and earth, a unique link between pairs of opposites: ancient yet eternally new; mechanical in structure, yet made effective only by the imagination; limited to a geometrically fixed space, yet with unlimited combinations; constantly developing, yet sterile; thought that leads nowhere; mathematics calculating nothing; art without works of art; architecture without substance – but nonetheless shown to be more durable in its entity and existence than all books and works of art; the only game that belongs to all nations and all eras, although no one knows what god brought it down to earth to vanquish boredom, sharpen the senses and stretch the mind. Where does it begin and where does it end? Every child can learn its basic rules, every bungler can try his luck at it, yet within that immutable little square it is able to bring forth a particular species of masters who cannot be compared to anyone else, people with a gift solely designed for chess, geniuses in their specific field who unite vision, patience and technique in just the same proportions as do mathematicians, poets, musicians, but in different stratifications and combinations. In the old days of the enthusiasm for physiognomy, a physician like Gall might perhaps have dissected a chess champion’s brain to find out whether some particular twist or turn in the grey matter, a kind of chess muscle or chess bump, is more developed in such chess geniuses than in the skulls of other mortals. And how intrigued such a physiognomist would have been by the case of Czentovic, where that specific genius appeared in a setting of absolute intellectual lethargy, like a single vein of gold in a hundredweight of dull stone. In principle, I had always realized that such a unique, brilliant game must create its own matadors, but how difficult and indeed impossible it is to imagine the life of an intellectually active human being whose world is reduced entirely to the narrow one-way traffic between black and white, who seeks the triumphs of his life in the mere movement to and fro, forward and back of thirty-two chessmen, someone to whom a new opening, moving knight rather than pawn, is a great deed, and his little corner of immortality is tucked away in a book about chess – a human being, an intellectual human being who constantly bends the entire force of his mind on the ridiculous task of forcing a wooden king into the corner of a wooden board, and does it without going mad!

In two days they began to come upon bones and cast-off apparel. They saw halfburied skeletons of mules with the bones so white and polished they seemed incandescent even in that blazing heat and they saw panniers and packsaddles and the bones of men and they saw a mule entire, the dried and blackened carcass hard as iron. They rode on. The white noon saw them through the waste like a ghost army, so pale they were with dust, like shades of figures erased upon a board. The wolves loped paler yet and grouped and skittered and lifted their lean snouts on the air. At night the horses were fed by hand from sacks of meal and watered from buckets. There was no more sickness. The survivors lay quietly in that cratered void and watched the whitehot stars go rifling down the dark. Or slept with their alien hearts beating in the sand like pilgrims exhausted upon the face of the planet Anareta, clutched to a namelessness wheeling in the night. They moved on and the iron of the wagontires grew polished bright as chrome in the pumice. To the south the blue cordilleras stood footed in their paler image on the sand like reflections in a lake and there were no wolves now. They took to riding by night, silent jornadas save for the trundling of the wagons and the wheeze of the animals. Under the moonlight a strange party of elders with the white dust thick on their moustaches and their eyebrows. They moved on and the stars jostled and arced across the firmament and died beyond the inkblack mountains. They came to know the nightskies well. Western eyes that read more geometric constructions than those names given by the ancients. Tethered to the polestar they rode the Dipper round while Orion rose in the southwest like a great electric kite. The sand lay blue in the moonlight and the iron tires of the wagons rolled among the shapes of the riders in gleaming hoops that veered and wheeled woundedly and vaguely navigational like slender astrolabes and the polished shoes of the horses kept hasping up like a myriad of eyes winking across the desert floor. They watched storms out there so distant they could not be heard, the silent lightning flaring sheetwise and the thin black spine of the mountain chain fluttering and sucked away again in the dark. They saw wild horses racing on the plain, pounding their shadows down the night and leaving in the moonlight a vaporous dust like the palest stain of their passing.

At the time, I paid no heed to the emblem above the door of a compass crossed with a square; the library had been founded by Masons. There, in the quiet shadows, I read for hours from the books that the kind librarian allowed me to take from the shelves: fairy tales, adventure stories, adaptations of classics for children, and dictionaries of symbols. One day while browsing among the shelves I ran across a yellowed volume: Les Tarots by Eteilla. All my efforts to read it were in vain. The letters looked strange and the words were incomprehensible. I began to worry that I had forgotten how to read. When I communicated my anguish to the librarian, he began to laugh. “But how could you understand it; it’s written in French, my young friend! I can’t understand it either!” Oh, how I felt drawn to those mysterious pages! I flipped through them, seeing many numbers, sums, the frequent occurrence of the word Thot, some geometric shapes . . . but what fascinated me most was a rectangle inside which a princess, wearing a three-pointed crown and seated on a throne, was caressing a lion that was resting its head on her knees. The animal had an expression of profound intelligence combined with an extreme gentleness. Such a placid creature! I liked the image so much that I committed a transgression that I still have not repented: I tore out the page and brought it home to my room. Concealed beneath a floorboard, the card “STRENGTH” became my secret treasure. In the strength of my innocence, I fell in love with the princess.

The differences between religions are reflected very clearly in the different forms of sacred art: compared with Gothic art, above all in its “flamboyant” style, Islamic art is contemplative rather than volitive: it is “intellectual” and not “dramatic”, and it opposes the cold beauty of geometrical design to the mystical heroism of cathedrals. Islam is the perspective of “omnipresence” (“God is everywhere”), which coincides with that of “simultaneity” (“Truth has always been”); it aims at avoiding any “particularization” or “condensation”, any “unique fact” in time and space, although as a religion it necessarily includes an aspect of “unique fact”, without which it would be ineffective or even absurd. In other words Islam aims at what is “everywhere center”, and this is why, symbolically speaking, it replaces the cross with the cube or the woven fabric: it “decentralizes” and “universalizes” to the greatest possible extent, in the realm of art as in that of doctrine; it is opposed to any individualist mode and hence to any “personalist” mysticism. To express ourselves in geometrical terms, we could say that a point which seeks to be unique, and which thus becomes an absolute center, appears to Islam—in art as in theology—as a usurpation of the divine absoluteness and therefore as an “association” (shirk); there is only one single center, God, whence the prohibition against “centralizing” images, especially statues; even the Prophet, the human center of the tradition, has no right to a “Christic uniqueness” and is “decentralized” by the series of other Prophets; the same is true of Islam—or the Koran—which is similarly integrated in a universal “fabric” and a cosmic “rhythm”, having been preceded by other religions—or other “Books”—which it merely restores. The Kaaba, center of the Muslim world, becomes space as soon as one is inside the building: the ritual direction of prayer is then projected toward the four cardinal points. If Christianity is like a central fire, Islam on the contrary resembles a blanket of snow, at once unifying and leveling and having its center everywhere.

The ten-dimensional universe was a world of light. The entire universe was constructed upon energy exchange between photons. All particles and antiparticles were formed from photons, and their mutual annihilation resulted in yet more photons. Thus, everything occurred at the speed of light, which was infinite. Even more marvelously, because particles and antiparticles balanced each other out, the total energy level of the universe was zero. This was an incredible state of symmetry, and the foundation of the emergence of intelligence in the ten-dimensional universe.” Matter, life, sentience, civilization … in this Edenic universe, everything was part of the same whole. All matter possessed life, and all life possessed sentience, and all sentience existed in a state of harmonious civilization. Unlike the three-dimensional universe, in which lonely stars hung in the vast emptiness of space, the entire universe was a living being. All life was but a part of this one grand Life, and all intelligence but a component of the highest Intelligence. The dark forest state was an impossibility for this transcendent being of unified matter and spirit. Tianming felt awe at the root of his soul. This was the original form of the Master. The ten-dimensional universe was itself alive. Sophon offered more explanations. “The Master is not an individual like a human being; rather, it is the sum of an infinite number of self-aware consciousnesses. Every consciousness shares the awareness of every other consciousness as well as the presence of the universe itself, and yet each possesses an independent will. This is a state of existence simply unimaginable by humans: individual presence absolutely, seamlessly harmonized with the universe, akin to the geometric construction of the Spirit you saw.

Now to picture the mechanism of this process of construction and not merely its progressive extension, we must note that each level is characterized by a new co-ordination of the elements provided—already existing in the form of wholes, though of a lower order—by the processes of the previous level. The sensori-motor schema, the characteristic unit of the system of pre-symbolic intelligence, thus assimilates perceptual schemata and the schemata relating to learned action (these schemata of perception and habit being of the same lower order, since the first concerns the present state of the object and the second only elementary changes of state). The symbolic schema assimilates sensori-motor schemata with differentiation of function; imitative accommodation is extended into imaginal significants and assimilation determines the significates. The intuitive schema is both a co-ordination and a differentiation of imaginal schemata. The concrete operational schema is a grouping of intuitive schemata, which are promoted, by the very fact of their being grouped, to the rank of reversible operations. Finally, the formal schema is simply a system of second-degree operations, and therefore a grouping operating on concrete groupings. Each of the transitions from one of these levels to the next is therefore characterized both by a new co-ordination and by a differentiation of the systems constituting the unit of the preceding level. Now these successive differentiations, in their turn, throw light on the undifferentiated nature of the initial mechanisms, and thus we can conceive both of a genealogy of operational groupings as progressive differentiations, and of an explanation of the pre-operational levels as a failure to differentiate the processes involved. Thus, as we have seen (Chap. 4), sensori-motor intelligence arrives at a kind of empirical grouping of bodily movements, characterized psychologically by actions capable of reversals and detours, and geometrically by what Poincaré called the (experimental) group of displacement. But it goes without saying that, at this elementary level, which precedes all thought, we cannot regard this grouping as an operational system, since it is a system of responses actually effected; the fact is therefore that it is undifferentiated, the displacements in question being at the same time and in every case responses directed towards a goal serving some practical purpose. We might therefore say that at this level spatio-temporal, logico-arithmetical and practical (means and ends) groupings form a global whole and that, in the absence of differentiation, this complex system is incapable of constituting an operational mechanism. At the end of this period and at the beginning of representative thought, on the other hand, the appearance of the symbol makes possible the first form of differentiation: practical groupings (means and ends) on the one hand, and representation on the other. But this latter is still undifferentiated, logico-arithmetical operations not being distinguished from spatio-temporal operations. In fact, at the intuitive level there are no genuine classes or relations because both are still spatial collections as well as spatio-temporal relationships: hence their intuitive and pre-operational character. At 7–8 years, however, the appearance of operational groupings is characterized precisely by a clear differentiation between logico-arithmetical operations that have become independent (classes, relations and despatialized numbers) and spatio-temporal or infra-logical operations. Lastly, the level of formal operations marks a final differentiation between operations tied to real action and hypothetico-deductive operations concerning pure implications from propositions stated as postulates.