Geometric Shapes Shapes Quotes

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

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.

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

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.)

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.

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

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.

As far as I could tell, the quickest way to a geeky guy's heart usually involved geometric shapes.

Basic geometric shapes communicate universal qualities common to all cultures. Practical design integrates them appropriately.

Nature itself rests on an internal foundation of archetypal principles symbolized by numbers, shapes, and their arithmetic and geometric relationships.

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.

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.

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.

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.

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.

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.

Deep down, it's all baseball, no matter what kind of geometrical shape you play it with.

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.

Driftglass," I said. "You know all the Coca-Cola bottles and cut-crystal punch bowls and industrial silicon slag that goes into the sea?" I know the Coca-Cola bottles." They break, and the tide pulls the pieces back and forth over the sandy bottom, wearing the edges, changing their shape. Sometimes chemicals in the glass react with chemicals in the ocean to change the color. Sometimes veins work their way through in patterns like snowflakes, regular and geometric; others, irregular and angled like coral. When the pieces dry, they're milky. Put them in water and they become transparent again.

Life, using matter to express itself in bodily shape, first traces a geometrical pattern. From the lowest form in crystals, upwards to more complicated patterns in the higher organisations—there is always first this geometrical pattern as skeleton.

Surprisingly, the tuner of the Coupling Equation which puts back the dimensions of the GPG to their original values is the GPG’s apex solid angle itself! Even the tip of the pyramid has a saying in projecting and expressing the geometrical shape of a Cube.

So it's not an architectural masterpiece. When Da5id and Hiro and the other hackers wrote The Black Sun, they didn't have enough money to hire architects or designers, so they just went in for simple geometric shapes. The avatars milling around the entrance don't seem to care.

Every inch of the interior space, high and low, glittered with arrangements of star-like patterns, all interwoven into a series of larger geometric shapes. The soaring domed ceilings glimmered from high above, a mirage of infinity that seemed to reach the heavens. Two large windows were thrown open to grant entrée to the sun: sharp shafts of light penetrated the room, further illuminating constellation after constellation of shattered glow. Even the floors were covered in mirrored tiles, though the delicate work was protected by a series of rich, intricately woven rugs.

I etch a pattern of geometric shapes onto a stone. To the uninitiated, the shapes look mysterious and complex, but I know that when arranged correctly they will give the stone a special power, enabling it to respond to incantations in a language no human being has ever spoken. ...Yet my work involves no witchcraft. The stone is a wafer of silicon, and the incantations are software. The patterns etched on the chip and the programs that instruct the computer may look complicated and mysterious, but they are generated according to a few basic principles that are easily explained.

Sacred shape and colour are the building blocks of creation.  Your Chakras emanate Sacred Shapes – Light Language to build the energy in your Aura, which in turn, is the building block for what happens in your life.  Sacred Geometric shapes and colour are what call physical reality into existence. I

Normal persons deprived of sensation progress from having mild to severe hallucinations, starting out with what looks very much like form constants (geometric patterns, mosaics, lines, rows of dots) and building to more developed, dream-like juxtapositions of perceptions the longer they remain in isolation.

The villa that Spiro had found was shaped not unlike a brick and was a bright crushed-strawberry pink with green shutters. It crouched in a cathedral-like grove of olives that sloped down the hillside to the sea, and it was surrounded by a pocket-handkerchief-size garden, the flower-beds laid out with a geometrical accuracy so dear to the Victorians, and the whole thing guarded by a tall, thick hedge of fuchsias that rustled mysteriously with birds.

It appears then that the essence of chess is its abstract structure. Names and shapes of pieces, colors of squares, whether the “squares” are in fact square, even the physical existence of board and pieces, are all irrelevant. What is relevant is the number and geometric arrangement of the “squares”, the number of types of piece and the number of pieces of each type, the quantitative-geometric power of each piece, etc. Everything else is a visual aid or a fairy tale.

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.

Most humans, it seems, still put up fences around their acts and thoughts – even when these are piles of shit – for they have no other way of delimiting them. Contrast Paleolithic cave paintings, in which animals and magical markings are overlayed with no differentiation or sense of framing. But when some of us have worked in natural settings, say in a meadow, woods, or mountain range, our cultural training has been so deeply ingrained that we have simply carried a mental rectangle with us to drop around whatever we were doing. This made us feel at home. (Even aerial navigation is plotted geometrically, thus giving the air a "shape".)

A fungal computer may sound fantastical, but biocomputing is a fast-growing field. Adamatzky has spent years developing ways to use slime molds as sensors and computers. These prototype biocomputers use slime molds to solve a range of geometrical problems. The slime mold networks can be modified—for instance, by cutting a connection—to alter the set of “logical functions” implemented by the network. Adamatzky’s idea of a “fungal computer” is just an application of slime-mold computing to another type of network-based organism. As Adamatzky observes, the mycelial networks of some species of fungus are more convenient for computing than slime molds.

Like Picasso and Braque, Mondrian explored the influential ideas of Paul Cézanne, who greatly influenced the analytic Cubists with his idea that all natural forms can be reduced to three figural primitives: the cube, the cone, and the sphere (Loran 2006; Kandel 2014). Mondrian recognized the plastic elements in analytic Cubism, and he began to echo the Cubists’ use of geometric shapes and interlocking planes. He reduced a specific object, such as a tree, to a few lines and then connected those lines to the surrounding space (fig. 6.4), thus entangling the branches of the tree with its surroundings. Yet whereas Cubist works played with simple shapes in a complex arena of shattered space, Mondrian’s art became more reductionist. He distilled figures to their most elemental forms, eliminating altogether the sense of perspective.

Many years passed before I learned of other ways to access the healthy and limitless part of my mind that psychedelic drugs had opened in my youth. In 2001, deep into a Vipassana course, a few days into silence and ten hours a day of meditation, I found myself in a psychedelic state. My body had become nothing but light, I was one with the universe and anything I could imagine was possible. I was a rock in an Alaskan stream purified by the freezing water rushing over me as a massive beautiful brown bear lumbered by. I looked up to see an intricate geometric pattern of shapes in motion in the air above; changing and unfolding, the most beautiful vivid and sharp color combinations to make Josef Albers cry with joy. I realized a profound simplicity of purpose, my focus crystal clear, I saw the beauty in all, and was overwhelmed with love and gratitude for all the joy and pain in my life. In that moment, I learned that no drug was ever necessary for a mind-opening experience.

I felt as though the temple curtain had been drawn aside without warning and I, a goggle-eyed stranger somehow mistaken for an initiate, had been ushered into the sanctuary to witness the mystery of mysteries. I saw a phantasmagoria, a living tapestry of forms jeweled in minute detail. They danced together like guests at a rowdy wedding. They changed their shapes. Within themselves they juggled geometrical shards like the fragments in a kaleidoscope. They sent forth extensions of themselves like the flares of suns. Yet all their activity was obviously interrelated; each being's actions were in step with its neighbors'. They were like bees swarming: They obviously recognised each other and were communicating avidly, but it was impossible to know what they were saying. They enacted a pageant whose beauty awed me. As the lights came back on, the auditorium seemed dull and unreal.I'd been watching various kinds of ordinary cells going about their daily business, as seen through a microscope and recorded by the latest time-lapse movie techniques. The filmmaker frankly admitted that neither he nor anyone else knew just what the cells were doing, or how and why they were doing it. We biologists, especially during our formative years in school, spent most of our time dissecting dead animals and studying preparations of dead cells stained to make their structures more easily visible—"painted tombstones," as someone once called them. Of course, we all knew that life was more a process than a structure, but we tended to forget this, because a structure was so much easier to study. This film reminded me how far our static concepts still were from the actual business of living. As I thought how any one of those scintillating cells potentially could become a whole speckled frog or a person, I grew surer than ever that my work so far had disclosed only a few aspects of a process-control system as varied and widespread as life itself, of which we'd been ignorant until then.

Given that our brains were shaped by natural selection, it could hardly be otherwise. Natural selection is driven by the competition among genes to be represented in the next generation. Reproduction leads to a geometric increase in descendants, and on a finite planet not every organism alive in one generation can have descendants several generations hence. Therefore organisms reproduce, to some extent, at one another’s expense. If one organism eats a fish, that fish is no longer available to be eaten by another organism. If one organism mates with a second one, it denies an opportunity at parenthood to a third. Everyone alive today is a descendant of millions of generations of ancestors who lived under these constraints but reproduced nonetheless. That means that all people today owe their existence to having winners as ancestors, and everyone today is designed, at least in some circumstances, to compete. That does not mean that people (or any other animals) house an aggressive urge that must be discharged, an unconscious death wish, a rapacious sex drive, a territorial imperative, a thirst for blood, or the other ruthless instincts that are often mistakenly equated with Darwinism.

The panel delivery truck drew up before the front of the “Amsterdam Apartments” on 126th Street between Madison and Fifth Avenues. Words on its sides, barely discernible in the dim street light, read: LUNATIC LYNDON … I DELIVER AND INSTALL TELEVISION SETS ANY TIME OF DAY OR NIGHT ANY PLACE. Two uniformed delivery men alighted and stood on the sidewalk to examine an address book in the light of a torch. Dark faces were highlighted for a moment like masks on display and went out with the light. They looked up and down the street. No one was in sight. Houses were vague geometrical patterns of black against the lighter blackness of the sky. Crosstown streets were always dark. Above them, in the black squares of windows, crescent-shaped whites of eyes and quarter moons of yellow teeth bloomed like Halloween pumpkins. Suddenly voices bubbled in the night. “Lookin’ for somebody?” The driver looked up. “Amsterdam Apartments.” “These is they.” Without replying, the driver and his helper began unloading a wooden box. Stenciled on its side were the words: Acme Television “Satellite” A.406. “What that number?” someone asked. “Fo-o-six,” Sharp-eyes replied. “I’m gonna play it in the night house if I ain’t too late.” “What ya’ll got there, baby?” “Television set,” the driver replied shortly. “Who dat getting a television this time of night?” The delivery man didn’t reply. A man’s voice ventured, “Maybe it’s that bird liver on the third storey got all them mens.” A woman said scornfully, “Bird liver! If she bird liver I’se fish and eggs and I got a daughter old enough to has mens.” “… or not!” a male voice boomed. “What she got ’ill get television sets when you jealous old hags is fighting over mops and pails.” “Listen to the loverboy! When yo’ love come down last?” “Bet loverboy ain’t got none, bird liver or what.” “Ain’t gonna get none either. She don’t burn no coal.” “Not in dis life, next life maybe.” “You people make me sick,” a woman said from a group on the sidewalk that had just arrived. “We looking for the dead man and you talking ’bout tricks.” The two delivery men were silently struggling with the big television box but the new arrivals got in their way. “Will you ladies kindly move your asses and look for dead men sommers else,” the driver said. His voice sounded mean. “ ’Scuse me,” the lady said. “You ain’t got him, is you?” “Does I look like I’m carrying a dead man ’round in my pocket?” “Dead man! What dead man? What you folks playing?” a man called down interestedly. “Skin?” “Georgia skin? Where?” “Ain’t nobody playing no skin,” the lady said with disgust. “He’s one of us.” “Who?” “The dead man, that’s who.” “One of usses? Where he at?” “Where he at? He dead, that’s where he at.” “Let me get some green down on dead man’s row.” “Ain’t you the mother’s gonna play fo-o-six?” “Thass all you niggers thinks about,” the disgusted lady said. “Womens and hits!” “What else is they?” “Where yo’ pride? The white cops done killed one of usses and thass all you can think about.” “Killed ’im where?” “We don’t know where. Why you think we’s looking?” “You sho’ is a one-tracked woman. I help you look, just don’t call me nigger is all.

In learning general relativity, and then in teaching it to classes at Berkeley and MIT, I became dissatisfied with what seemed to be the usual approach to the subject. I found that in most textbooks geometric ideas were given a starring role, so that a student...would come away with an impression that this had something to do with the fact that space-time is a Riemannian [curved] manifold. Of course, this was Einstein's point of view, and his preeminent genius necessarily shapes our understanding of the theory he created. However, I believe that the geometrical approach has driven a wedge between general relativity and [Quantum Field Theory]. As long as it could be hoped, as Einstein did hope, that matter would eventually be understood in geometrical terms, it made sense to give Riemannian geometry a primary role in describing the theory of gravitation. But now the passage of time has taught us not to expect that the strong, weak, and electromagnetic interactions can be understood in geometrical terms, and too great an emphasis on geometry can only obscuret he deep connections between gravitation and the rest of physics...[My] book sets out the theory of gravitation according to what I think is its inner logic as a branch of physics, and not according to its historical development. It is certainly a historical fact that when Albert Einstein was working out general relativity, there was at hand a preexisting mathematical formalism, that of Riemannian geometry, that he could and did take over whole. However, this historical fact does not mean that the essence of general relativity necessarily consists in the application of Riemannian geometry to physical space and time. In my view, it is much more useful to regard general relativity above all as a theory of gravitation, whose connection with geometry arises from the peculiar empirical properties of gravitation.

This is life seen by life. I may not have meaning but it is the same lack of meaning that the pulsing vein has. I want to write to you like someone learning. I deepen the words as if I were painting, more than an object, its shadow. I don’t want to ask why, you can always ask why and always get no answer—could I manage to surrender to the expectant silence that follows a question without an answer? Though I sense that some place or time the great answer for me does exist. And then I shall know how to paint and write, after the strange but intimate answer. Listen to me, listen to the silence. What I say to you is never what I say to you but something else instead. It captures the thing that escapes me and yet I live from it and am above a shining darkness. One instant athematic theme unfurls without a plan but geometric like the successive shapes in a kaleidoscope. I slowly enter my gift to myself, splendor ripped open by the final song that seems to be the first. I enter the writing slowly as I once entered painting. It is a world tangled up in creepers, syllables, woodbine, colors and words—threshold of an ancestral cavern that is the womb of the world and from it I shall be born. And if I often paint caves that is because they are my plunge into the earth, dark but haloed with brightness, and I, blood of nature— extravagant and dangerous caves, talisman of the Earth, where stalactites, fossils and rocks come together, and where the animals mad by their own malign nature seek refuge. The caves are my hell. Forever dreaming cave with its fogs, memory or longing? eerie, eerie, esoteric greenish with the slime of time. All is weighted with sleep when I paint a cave or write to you about it—from outside it comes the clatter of dozens of wild horses stamping with dry hoofs the darkness, and from the friction of the hoofs the rejoicing is freed in sparks: here I am, I and the cave, in the very time that will rot us. I want to put into words but without description the existence of the cave that some time ago I painted—and I don’t know how. Only by repeating its sweet horror, cavern of terror and wonders, place of afflicted souls, winter and hell, unpredictable substratum of the evil that is inside an earth that is not fertile. I call the cave by its name and it begins to live with its miasma. I then fear myself who knows how to paint the horror, I, creature of echoing caverns that I am, and I suffocate because I am word and also its echo.

Then he went up to the window. His heart began pounding excitedly when he turned back the yellow linen of the curtain. An enchantingly beautiful spectacle was revealed before him — although today he immediately noticed that there was something strange in the entire aspect of this extensive and excellently arranged Garden. Precisely what amazed him he was still unable to say right away, and he began to examine the Garden attentively. What was there so unpleasant in its beauty? Why was the Youth's heart trembling so painfully? Was it that everything in the enchanted Garden was too exact. All the paths were laid out geometrically, and all were of the same width, and all were covered with precisely the same amount of yellow sand; the plants were all arranged with exaggerated orderliness; the trees were trimmed in the form of spheres, cones and cylinders; the flowers were arranged according to the various shades so that their composition was pleasing to the eye, but for some reason or other this wounded the soul. But giving it careful thought, what was there unpleasant in that orderliness which merely bore witness to the careful attention which someone paid to the Garden? Of course there was no reason for this to cause the strange apprehension which oppressed the Youth. But it was in something else as yet incomprehensible to the Youth. One thing was for certain, though, that this Garden did not resemble any other garden which the Youth had happened to see in his time. Here he saw giant flowers of an almost too brilliant color — at times it seemed that many-colored fires were burning in the midst of the luxuriant greenery — brown and black stalks of creeping growths, thick like tropical serpents; leaves of a strange shape and immeasurable size, whose greeness seemed to be unnaturally brilliant. Heady and languid fragrances wafted through the window in gentle waves, breaths of vanilla, frankincense and bitter almond, sweet and bitter, ecstatic and sad, like some joyous funereal mysterium. The Youth felt the tender yet lively touches of the gentle wind. But in the Garden it seemed as if the wind had no strength and lay exhausted on the tranquil green grass and in the shadows beneath the bushes of the strange growths. And because the trees and grass of the strange Garden were breathlessly quiet and could not hear the softly blowing wind above them and did not reply to it, they seemed to be inanimate. And thus they were deceitful, evil and hostile to man. ("The Poison Garden")

In 1931, amid that incredible transformation, a brilliant young Russian psychologist named Alexander Luria recognized a fleeting “natural experiment,” unique in the history of the world. He wondered if changing citizens’ work might also change their minds. When Luria arrived, the most remote villages had not yet been touched by the warp-speed restructuring of traditional society. Those villages gave him a control group. He learned the local language and brought fellow psychologists to engage villagers in relaxed social situations—teahouses or pastures—and discuss questions or tasks designed to discern their habits of mind. Some were very simple: present skeins of wool or silk in an array of hues and ask participants to describe them. The collective farmers and farm leaders, as well as the female students, easily picked out blue, red, and yellow, sometimes with variations, like dark blue or light yellow. The most remote villagers, who were still “premodern,” gave more diversified descriptions: cotton in bloom, decayed teeth, a lot of water, sky, pistachio. Then they were asked to sort the skeins into groups. The collective farmers, and young people with even a little formal education, did so easily, naturally forming color groups. Even when they did not know the name of a particular color, they had little trouble putting together darker and lighter shades of the same one. The remote villagers, on the other hand, refused, even those whose work was embroidery. “It can’t be done,” they said, or, “None of them are the same, you can’t put them together.” When prodded vigorously, and only if they were allowed to make many small groups, some relented and created sets that were apparently random. A few others appeared to sort the skeins according to color saturation, without regard to the color. Geometric shapes followed suit. The greater the dose of modernity, the more likely an individual grasped the abstract concept of “shapes” and made groups of triangles, rectangles, and circles, even if they had no formal education and did not know the shapes’ names. The remote villagers, meanwhile, saw nothing alike in a square drawn with solid lines and the same exact square drawn with dotted lines. To Alieva, a twenty-six-year-old remote villager, the solid-line square was obviously a map, and the dotted-line square was a watch. “How can a map and a watch be put together?” she asked, incredulous. Khamid, a twenty-four-year-old remote villager, insisted that filled and unfilled circles could not go together because one was a coin and the other a moon.

Elephanta caves, Mumbai-- I entered a world made of shadows and sudden brightness. The play of the light, the vastness of the space and its irregular form, the figures carved on the walls: all of it gave the place a sacred character, sacred in the deepest meaning of the word. In the shadows were the powerful reliefs and statues, many of them mutilated by the fanaticism of the Portuguese and the Muslims, but all of them majestic, solid, made of a solar material. Corporeal beauty, turned into living stone. Divinities of the earth, sexual incarnations of the most abstract thought, gods that were simultaneously intellectual and carnal, terrible and peaceful. ............................................................................ Gothic architecture is the music turned to stone; one could say that Hindu architecture is sculpted dance. The Absolute, the principle in whose matrix all contradictions dissolve (Brahma), is “neither this nor this nor this.” It is the way in which the great temples at Ellora, Ajanta, Karli, and other sites were built, carved out of mountains. In Islamic architecture, nothing is sculptural—exactly the opposite of the Hindu. The Red Fort, on the bank of the wide Jamuna River, is as powerful as a fort and as graceful as a palace. It is difficult to think of another tower that combines the height, solidity, and slender elegance of the Qutab Minar. The reddish stone, contrasting with the transparency of the air and the blue of the sky, gives the monument a vertical dynamism, like a huge rocket aimed at the stars. The mausoleum is like a poem made not of words but of trees, pools, avenues of sand and flowers: strict meters that cross and recross in angles that are obvious but no less surprising rhymes. Everything has been transformed into a construction made of cubes, hemispheres, and arcs: the universe reduced to its essential geometric elements. The abolition of time turned into space, space turned into a collection of shapes that are simultaneously solid and light, creations of another space, made of air. There is nothing terrifying in these tombs: they give the sensation of infinity and pacify the soul. The simplicity and harmony of their forms satisfy one of the most profound necessities of the spirit: the longing for order, the love of proportion. At the same time they arouse our fantasies. These monuments and gardens incite us to dream and to fly. They are magic carpets. Compare Ellora with the Taj Mahal, or the frescoes of Ajanta with Mughal miniatures. These are not distinct artistic styles, but rather two different visions of the world.

listened to the rhythm of the waves and imagined the crystal clear blue waters cleansing her of any lower energies, any remnants of shadows. She pictured the sun as God’s golden light filling her from within. As she did every morning and night, she cleared, grounded, and protected herself, calling on the archangels and ascended masters to help her along the way. Her mother taught her long ago that shielding herself psychically and spiritually was every bit as important as protecting herself physically. She took in a deep, cleansing breath, then another, and another. On each inhale, she imagined herself breathing in the essence of the Universe—Divine white light and perfect love. With each exhale, she imagined herself eliminating a gray mist that represented any negativity or fear. She invoked Archangel Michael to cut any cords of fear. She asked Archangel Michael, Archangel Raguel, Archangel Jophiel, Archangel Haniel, and the Ascended Masters El Morya and Lady Nada to turn on the spiritual vacuum cleaner and vacuum away any lower energies or remnants of shadows that might be in her field. She imagined the pure waters of a beautiful waterfall washing away any lingering energies. Then she called upon Archangel Metatron to use his geometric shapes to clear and open all of her chakras.

The light travelers felt the familiar tingling and vibrational pull at the top of their heads, and the great rush of cool air pressed in on them. The familiar green fluorescent geometric shapes and symbols collided with their vibrating bodies. As the third spiral drew them up, it dissolved into very fine hair-like ribbons of light that seemed to meld into all things, all time, all space. The teens realized they had access to all knowledge—past, present, future. Using her intent Drew, asked to see the inside of the South Portal at Aramu Muru. In the next second she saw a clash of light and dark, distorted inhuman faces, hellacious other-worldly images.

The trajectory curves produced by the ball thrown into the air or the orbital curves of the planets orbiting the sun were of great interest to mathematicians. Treating algebraic systems was developed by medieval Islam scholars. Descartes showed how to use the algebraic term (x, y) to describe a geometric shape, showing what is known as Cartesian coordinates and how they were drawn using x, y and graphs. A straight line graph has characteristics that are easy to calculate. 카톡【AKR331】텔레【RDH705】라인【SPR331】위커【SPR705】 저희는 7가지 철칙을 바탕으로 거래를 합니다. 고객들과 지키지못할약속은 하지않습니다 1.정품보장 2.총알배송 3.투명한 가격 4.편한 상담 5.끝내주는 서비스 6.고객님 정보 보호 7.깔끔한 거래 포폴,에토미,수면제 팔아요 The known formula from the Babylonian times was able to calculate the area under the straight line. This slope (the rate of change represented by the slope of the straight line) is the value of the y coordinate divided by the change of the associated x coordinate. However, these values ​​are more difficult to calculate in the curve. Before Newton, mathematicians realized that one way to do this was to calculate an approximation. Calculate the curve as continuous straight lines, and the area under the curve as continuous squares and triangles. Using more or less rectangles and triangles, you can get a more accurate approximation, but this is still only an approximation. Newton began challenging this problem before he reached Ulussof. In February 1665 he was still in the third year of college. He knew that the French mathematician Fermat and his mentor Bera both explained the formula for a particular curve. He began to wonder if they could be generalized to all curves. "I got a hint about this method from how to draw Fermat's tangents and generalized it," he later said. The key to this problem was his ability to use infinite water. Newton realized this. Instead of adding to infinity, the sum associated with an infinite series is similar to a finite set of goals or limits. And we could use this to find the curve as a rectangle. Effective using infinite numbers and giving small squares to the area under the curve. This is 'integral'.

Studies have shown that the time to complete even simple tasks, such as sorting geometric shapes, significantly degrades when multitasking. Of

There are only a limited number of sporadic groups, and one of them does indeed have the geometric interpretion with the highest number of dimensions. It's the Monster Group, and the shape it corresponds to can exist only in 196,883 dimensions. This boggles my mind. As you travel up past hundreds of thousands of dimensions, with only a few predictable infinite families of shapes to keep you company, suddenly, out of the blurred monotony, a shape flashes into existence for a single dimensional space. It wasn't there in 196,882D and has gone again by 196,884D. In that one tiny window, a shape beyond any human comprehension exists. It is a real mathematical object, as much as a triangle or a cube. The title of Griess's 1982 paper gives the Monster its other, more affectionate name: the Friendly Giant. We will never be able to picture the Friendly Giant, but we know it exists.

In his ... 'Geometrical peculiarities of the Pyramids', Ballard shows the relationship between the equal area theory and the golden number. After checking Herodotus' statement via dimensions Ballard concludes: 'I have therefore the authority of Herodotus to support the theory which I shall subsequently set forth, that this pyramid was the exponent of lines divided in mean and extreme ratio.

Plato argued that the material world of visible things was but a shadow of the true reality of eternal forms. He proceeds to explain the nether world of eternal blueprints most completely in the case of the elements of matter: earth, air, fire, and water. These he represents by geometrical solids: the earth by a cube, water by an icosahedron, air by an octahedron, and fire by a tetrahedron. His position is that ultimately the elements are just these solid geometrical shapes not simply that they possess geometrical shapes as one of their properties. The transmutation of elements one into the other is then explained by the merger and dissolution of triangles. This strictly mathematical description characterizes Plato's discussion of many other physical problems, For him, mathematics is a pointer to the ultimate reality of the world of forms that overshadows the visible world of sense data. The better we can grasp it, the closer we can come to true knowledge. Thus, for Plato, mathematics is more fundamental, truer, closer to the eternal forms of which the visible world is an imperfect reflection, than the objects of physical science. Because the world is mathematical at its deepest level, all visible phenomena will have mathematical aspects and be describable by mathematics to a greater or lesser extent, determined by their closeness to their underlying forms.

contrast, the right hemisphere doesn’t march in the single-file formation of A-B-C-D-E. Its special talent is the ability to interpret things simultaneously. This side of our brains is “specialized in seeing many things at once: in seeing all the parts of a geometric shape and grasping its form, or in seeing all the elements of a situation and understanding what they mean.”9 This makes the right hemisphere particularly useful in interpreting faces. And it confers on human beings a comparative advantage over computers.

I etch a pattern of geometric shapes onto a stone. To the uninitiated, the shapes look mysterious and complex, but I know that when arranged correctly they will give the stone a special power, enabling it to respond to incantations in a language no human being has ever spoken. ...Yet my work involves no witchcraft. The stone is a wafer of silicon, and the incantations are software. The patterns etched on the chip and the programs that instruct the computer may look complicated and mysterious, but they are generated according to a few basic principles that are easily explained.

We must listen with all our hearts when time speaks to us. These are not just a row of huge stones laid out and arranged in a specific and tidy geometric shape, but it is trying to tell us its beautiful and interesting story. The story of determination, will, and strength, is the story of civilization in a nutshell. Time speaks to us on behalf of those who preceded us".

Innovation itself can be viewed as a stack of interconnected elements. Consider a chair, for instance, which combines wood or other solid materials with a specific geometric shape designed for comfortable seating, primarily suited for humans. Adding cushions to the chair further enhances its innovation by increasing comfort.

Don Juan drew a diagram in the dust, a geometrical shape with eight points. The eight points, organized from small to large, were called reason, talking, dreaming, seeing, feeling, will, the known, and the unknown.

Western Texas was just such a project: a grandiose scheme, germinated in secret, and unlikely to bear fruit for years. As laid out in private correspondence with Adolf Douai and other co-conspirators in Texas, the plan called for the "immigration of one or two thousand staunch and steadfast northern men, supporters of Freedom." These infiltrators should come quietly and in small groups at first, forming a "nucleus" in alliance with free- state Germans. Thereafter, migrants from the North and Europe would "pour in," aided by new railroad lines. Olmsted kept refining and expanding on this plan, long after his return from Texas. It became, in effect, a dry run for his career as a landscape architect, including blueprints for a string of planned communities across the frontier of the Cotton Kingdom. "I have a private grand political hobby which I must display to you," he disclosed to a Northern ally, in a letter filled with geometric shapes, lines, and arrows. The sketch was nothing less than a sweeping design for winning what Olmsted called the "war between the power of Slavery and of Freedom on this continent.

Honestly? I have no idea,’ I laugh. ‘It seems to involve lots of geometric shapes made up of tiny dots of paint. It’s really not my kind of thing, but it makes him happy, and he’ll be even happier if he actually sells one at this exhibition. Now, are you cooking tonight or am I?’ ‘How is the new flat?’ Thomas’s mother asks me. It’s the first night of the exhibition and I’m carefully pouring prosecco into the glasses she’s borrowed from one of the local supermarkets.

Look for geometric shapes in your environment. Pay attention to the patterns in objects, but also their influence in the abstract. How does it feel when people are in pairs, lines, circles, or triangles? When are bridges in effect? Try to get in touch with the archetypical energies of different shapes in different circumstances.

must reason your way through the problem. Using line only, draw one simple geometric shape, such as a square, triangle or circle. Without overlapping or intersecting, draw a different shape. Now, draw another. Choose your favorite. Make the other 2 like your favorite. Enlarge one of the shapes. Reduce one of them. Make one shape touch one edge of the page. Make the other

String theorists have found special pairs of geometrical shapes for space that have completely different features when each is probed by unwrapped strings. They also have completely different features when each is probed by wrapped strings. But-and this is the punch line-when probed both ways, with wrapped and unwrapped strings, the shapes become indistinguishable. what the unwrapped strings see on one space, the wrapped strings see on the other, and vice versa, rendering identical the collective picture gleaned from the full physics of string theory.

He discovered its self-accumulating, self-reproducing nature and gave the spiral a motto (perhaps the only one associated with a geometric shape): Eadem mutato resurgo—Although changed, I arise again the same.

The difference in inclination between the two upper shafts in the Great Pyramid of Giza was coherently engineered with its dimensions and not separately therefrom. In other words, the engineering application of the shafts is an indivisible element from the whole geometrical shape of the pyramid itself.

Joining the world of shapes to the world of numbers in this way represented a break with the past. New geometries always begin when someone changes a fundamental rule. Suppose space can be curved instead of flat, a geometer says, and the result is a weird curved parody of Euclid that provides precisely the right framework for the general theory of relativity. Suppose space can have four dimensions, or five, or six. Suppose the number expressing dimension can be a fraction. Suppose shapes can be twisted, stretched, knotted. Or, now, suppose shapes are defined, not by solving an equation once, but by iterating it in a feedback loop. Julia, Fatou, Hubbard, Barnsley, Mandelbrot-these mathematicians changed the rules about how to make geometrical shapes. The Euclidean and Cartesian methods of turning equations into curves are familiar to anyone who has studied high school geometry or found a point on a map using two coordinates. Standard geometry takes an equation and asks for the set of numbers that satisfy it. The solutions to an equation like x^2 + y^2 = 1, then, form a shape, in this case a circle. Other simple equations produce other pictures, the ellipses, parabolas, and hyperbolas of conic sections or even the more complicated shapes produced by differential equations in phase space. But when a geometer iterates an equation instead of solving it, the equation becomes a process instead of a description, dynamic instead of static. When a number goes into the equation, a new number comes out; the new number goes in, and so on, points hopping from place to place. A point is plotted not when it satisfies the equation but when it produces a certain kind of behavior. One behavior might be a steady state. Another might be a convergence to a periodic repetition of states. Another might be an out-of-control race to infinity.

The floor is a fifteenth-century revival of medieval Cosmatesque mosaic style. The Cosmati family developed their unmistakable technique in Rome in the twelfth and thirteenth centuries. This decorating style was a fantasy of geometric shapes and swirls in cut pieces of colored glass and marble (much of which was “recycled” from pagan Roman palaces and temples). Stunning examples of authentic Cosmati floors and decorations can be found in some of the oldest and most beautiful churches, basilicas, and cloisters in Rome and southern Italy.

Elizebeth raced to stay on top of the shifting codes, the proliferation of patterns. Her worksheets grew weird, beautiful. She filled the grid squares with letters and numbers that made different geometric shapes when you stepped back and looked at the worksheet from a distance. Some of the shapes were parallelograms, some looked like stairs, other like labyrinths. She pulled mischievious letters from the sky and sorted them on the page. The invisible world was all out of whack, misaligned, and she had this set of tricks to knock it back into order.

FDM (Fused Deposition Modeling) printers There is no hard and fast classification of the FDM 3D printers Cartesian 3D Printers: These are the most common type, operating on a straightforward Cartesian coordinate system with linear rails guiding movement along the X, Y, and Z axes. They are recognized for their simplicity and reliability. Delta 3D Printers: Delta printers employ a triangular configuration of three arms attached to moving carriages at the printer's apex. The print head hangs from these carriages, executing precise movements to craft the intended object. Delta printers excel in speed and consistency, particularly in producing tall items. CoreXY 3D Printers: CoreXY printers utilize a distinctive belt-driven mechanism to maneuver the print head across the X and Y axes. This design separates the print head's motion from that of the build platform, resulting in swifter and more accurate prints. Enthusiasts favor CoreXY printers for their speed and precision. Polar 3D Printers: Polar printers feature a circular build platform and a print head that moves both radially and vertically. This configuration facilitates continuous rotation of the print bed, enabling the creation of objects with intricate geometric shapes. Polar printers are commonly employed for crafting artistic and sculptural pieces. SCARA 3D Printers: SCARA (Selective Compliance Articulated Robot Arm) printers utilize a robotic arm mechanism to navigate the print head in a two-dimensional plane. This design offers rapid and precise movement, making SCARA printers ideal for producing small, intricate objects with exceptional accuracy. Each variant of FDM 3D printer has its own strengths and is tailored to diverse applications, spanning from hobbyist endeavors to industrial-scale manufacturing.

Viktor Frankl used the metaphor of geometric dimensions to illustrate challenges in perception and understanding. Just as a three-dimensional cylinder projected onto a two-dimensional plane can appear as different shapes depending on the angle, our perspectives are limited by the "conceptual dimensions" we inhabit. Focusing on one framework or worldview casts blind spots on issues outside its purview. Like the cylinder, reality contains more complexity than any single viewpoint can capture. What appears contradictory from a limited vantage point may be reconciled from a broader perspective. Self has this broad perspective. Frankl suggested cultivating multi-dimensional awareness (Self's awareness) to overcome biases and grasp truth more wholly. Though we cannot transcend our situatedness (parts and ego), we can seek to understand the diverse dimensions that comprise the fullness of reality. Awareness of our frames allows us to interpret experiences with more wisdom and nuance.

The enthralling elegance of mathematics defines the circular motion of the stars, the angles of observation, the tallies of our economics, the shapes of our tools, the counting of our assets, and the creation of delicious food, compelling art, and captivating music. The span of our evolution provides a humbling realization that the brain has been mathematically evolving for thousands of years before any evidence we may currently have. Concepts and ideas have continually built upon previous mathematical truths, which humans have frequently proved, tested, and retested to establish new foundational truths. These scientific truths linger in the mathematical cognizance of the birds that count their hatchlings, the bees that geometrically shape their homes, the dogs that count their treats, and the seals that understand the flow of the oceans currents.

Four oxygen atoms surround a single zirconium atom in such a way as to make a geometric shape known as a tetrahedron, a four-sided pyramid, which is the strongest geometric shape possible.

The scenery that opened before me was composed of shades of black and white, and of trees woven together in lines along the boundaries between the fields. In places where the grass had not been cut, the snow had failed to blanket the fields in a uniform plane of white. Blades of grass were poking through its cover; from a distance it looked as if a large hand had begun to sketch an abstract pattern, by practicing some short strokes, fine and subtle. I could see the beautiful geometric shapes of fields, strips and rectangles, each with a different texture, each with its own shade, sloping at different angles toward the rapid winter Dusk. And our houses, all seven, were scattered here like a part of nature, as if they had sprung up with the field boundaries, and so had the stream and little bridge across it—it all seemed carefully designed and positioned, perhaps by the very same hand that had been sketching.

Renormalization had entered physics in the 1940s as a part of quantum theory that made it possible to calculate interactions of electrons and photons. A problem with such calculations, as with the calculations Kadanoff and Wilson worried about, was that some items seemed to require treatment as infinite quantities, a messy and unpleasant business. Renormalizing the system, in ways devised by Richard Feynman, Julian Schwinger, Freeman Dyson, and other physicists, got rid of the infinities. Only much later, in the 1960s, did Wilson dig down to the underlying basis for renormalization’s success. Like Kadanoff, he thought about scaling principles. Certain quantities, such as the mass of a particle, had always been considered fixed—as the mass of any object in everyday experience is fixed. The renormalization shortcut succeeded by acting as though a quantity like mass were not fixed at all. Such quantities seemed to float up or down depending on the scale from which they were viewed. It seemed absurd. Yet it was an exact analogue of what Benoit Mandelbrot was realizing about geometrical shapes and the coastline of England. Their length could not be measured independent of scale. There was a kind of relativity in which the position of the observer, near or far, on the beach or in a satellite, affected the measurement. As Mandelbrot, too, had seen, the variation across scales was not arbitrary; it followed rules. Variability in the standard measures of mass or length meant that a different sort of quantity was remaining fixed. In the case of fractals, it was the fractional dimension—a constant that could be calculated and used as a tool for further calculations. Allowing mass to vary depending on scale meant that mathematicians could recognize similarity across scales.

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

Studies have shown that multitasking degrades the performance of completing even simple tasks, such as sorting geometric shapes.3 The serious impact of this phenomenon was shown by Harvard researchers Dr. Steven Wheelwright and Dr. Kim Clark. They wrote about the problem of spreading engineers’ attention over too many projects simultaneously. As the number of projects went up, the time spent on productive tasks (e.g., problem-solving, interpreting data) went down by more than half, from 70% or more of their time to about 30%. The increased nonproductive activities included status meetings (communicating and coordinating across teams), switching costs (time required to reestablish context from one project to another), and so forth.

Flower of life: A figure composed of evenly-spaced, overlapping circles creating a flower-like pattern. Images of the Platonic solids and other sacred geometrical figures can be discerned within its pattern. FIGURE 3.14 FLOWER OF LIFE The Platonic solids: Five three-dimensional solid shapes, each containing all congruent angles and sides. If circumscribed with a sphere, all vertices would touch the edge of that sphere. Linked by Plato to the four primary elements and heaven. FIGURE 3.15 PENTACHORON The applications of these shapes to music are important to sound healing theory. The ancients have always professed a belief in the “music of the spheres,” a vibrational ordering to the universe. Pythagorus is famous for interconnecting geometry and math to music. He determined that stopping a string halfway along its length created an octave; a ratio of three to two resulted in a fifth; and a ratio of four to three produced a fourth. These ratios were seen as forming harmonics that could restore a disharmonic body—or heal. Hans Jenny furthered this work through the study of cymatics, discussed later in this chapter, and the contemporary sound healer and author Jonathan Goldman considers the proportions of the body to relate to the golden mean, with ratios in relation to the major sixth (3:5) and the minor sixth (5:8).100 Geometry also seems to serve as an “interdimensional glue,” according to a relatively new theory called causal dynamical triangulation (CDT), which portrays the walls of time—and of the different dimensions—as triangulated. According to CDT, time-space is divided into tiny triangulated pieces, with the building block being a pentachoron. A pentachoron is made of five tetrahedral cells and a triangle combined with a tetrahedron. Each simple, triangulated piece is geometrically flat, but they are “glued together” to create curved time-spaces. This theory allows the transfer of energy from one dimension to another, but unlike many other time-space theories, this one makes certain that a cause precedes an event and also showcases the geometric nature of reality.101 The creation of geometry figures at macro- and microlevels can perhaps be explained by the notion called spin, first introduced in Chapter 1. Everything spins, the term spin describing the rotation of an object or particle around its own axis. Orbital spin references the spinning of an object around another object, such as the moon around the earth. Both types of spin are measured by angular momentum, a combination of mass, the distance from the center of travel, and speed. Spinning particles create forms where they “touch” in space.

A third measure, the convex hull, suggests putting a hypothetical rubber band around a district.5 The score is the percentage of the area surrounded by the hypothetical rubber band that is in the district. The measure would yield a high score for regular geometric shapes such as a square, rectangle, pentagon, and so on. However, a district in which portions have been cut away in order to avoid including certain populations would have lower scores. Each

No one tries to talk me out of a migraine aura. I never try to interpret the shimmering geometric shapes or figure out the scintillating stairways crawling in the corners of my vision. No matter how hard I stare, I’ll never see my friend’s eye. I just navigate by what I can see. I’m gentle with myself, and my friends care for me while I wait for it to go away. This same gentle patience is the treatment for OCD. I needed the patience to remember that OCD is a broken record, thoughts endlessly looping between the thalamus, cortex, and cingulate gyrus. The scratch that connected the record grooves was only deepened by researching, ruminating on, and then carefully avoiding things that scared me. I had to find a new way of knowing—so I could move on with the music.

The weaving, waving field of geometric shapes and lines folds and falls over me, or I fall into it. I am seeing small spherical globules of white light, like pearls, that are glistening, shining moist, and perfectly aligned and interconnected in complex three-dimensional webs, reminiscent of Buckminster Fuller’s dymaxion structures, yet always changing, unfolding and enfolding. These webs are what constitutes my body, clustering in certain areas to make organs like my eyes. They also constitute all other bodies and forms around me. Each individual is a kind of cluster in this infinite ever-changing molecular web. Each thought or feeling or experience is also a local cluster in this holographic matrix of all possibilities. A sun of pure white light radiates out from the center of the swirling, pearl-studded crystalline grid. It is too intensely bright for me to maintain the focus of attention, so gradually I lose awareness of it and emerge back out of the infinite oneness back into my body-form (RM).

Grid art has only gotten better over time. In a Times crossword from 2009 by Elizabeth Gorski, the black squares at the grid’s center formed a spiral, with THE SOLOMON R GUGGENHEIM / MUSEUM as answers spanning the top of the spiral, and—for the geometrically impaired—SPIRAL SHAPE across the bottom. Eight artworks hanging in the spiral-shaped Guggenheim museum appeared as clues, with each artist hung as an answer in the puzzle.

All the art experts, all the big galleries, if not maybe quite all of the humble folk who look at them, agree Jackson Pollock’s splatter paintings do indeed count as great art. And JP intended it to be art too. But what’s curious about most of the most radical artists of the post-Second World War period is that they came from nowhere to prominence with the support of . . . the CIA! Yes, the American secret services actively promoted (through books, funding schemes, newspapers and of course galleries) radical art as part of a labyrinthine strategy to undermine the Soviet Union. This was all part of a special strategy to win over intellectuals – including philosophers – described as ‘the battle for Picasso’s mind’ by one former CIA agent, Thomas Braden, in a television interview in the 1970s. Tom Braden was responsible for dispensing money under the heading Congress for Cultural Freedom. Naturally, most of the people he gave money to had no idea that the funds, and hence the artistic direction, actually came from the CIA. Intellectuals and great artists, after all, hate being told what to think. And what was the communist empire doing meanwhile? They were promoting, through galleries, public funding and so on, a very different kind of art supposedly reflecting communist political values. ‘Soviet realism’ was a kind of reaction to ‘Western Impressionism’ (all those dotty – pointilliste the art-experts call them – landscapes and swirling, subjective shapes) and ensured that people in the paintings looked like people, decent, hard-working types too, and what’s more were doing worthy things – like making tractors or (at least) looking inspirationally at the viewer. When Soviet art wasn’t figurative (as this sort of stuff is called), it was very logical and mathematical, full of precise geometrical shapes and carefully weighted blocks of colour.

Contrary to Piaget’s theory, it turns out that babies come into this world with an innate, non-verbal “number sense” and the ability to “guesstimate” the relative number of things. Newborns just two days old are, in fact, even capable of doing a kind of numbers matching game. Researchers found that when they played a specific number of syllables to newborns, the babies were able to match it to the correct number of geometric shapes. For example, when the newborns were played “tuuuuu tuuuu tuuuu tuuu,” they would look longer at a picture with four squares; when twelve syllables were played, they looked longer at a picture with twelve squares. Even more impressive, the ability of infants to link the number of sounds to the number of objects at six months of age often predicted their eventual math ability.

Plastic molds are cheaper, so they’re a reasonable choice if you’re just trying out candle making and don’t want to spend a whole lot of money at first. They also come in a lot of fun shapes like orbs, pyramids, and other geometrical solids. You won’t find such variety in metal molds.

Research conducted by Egyptian architect Dr. Ibrahim Karim over thirty years has demonstrated the amazing effects of geometrical shapes. One study, led by the Egyptian National Research Centre, showed that simple shapes could stop the replication of bacteria. Most frequently, he surrounded the subjects of his experiments with materials formed into various shapes, such as triangles, squares, or circles; he has also created an extensive index of thought-provoking shapes that integrate other shapes, such as spirals and lines, each of which promotes different changes, such as the healing of heart disease or the growth of new cells in the body.

Infants prefer to look at dots that move in biological patterns rather than random ones. They will look longer at geometrical shapes that seem to be self-propelled than ones that seem to move passively. Children also have a bias toward life in the way they learn: they can learn about animals faster than inanimate objects, and they hold on to the memories of what they learn longer. Our knowledge of life, in other words, arises long before we can tell ourselves what we know.

But to some people home was home, a complex of feeling far beyond rationality, a sort of grid or gravitational field in which the personality itself took its geometrical shape. While for others, a place was just a place, and the self free of all that, the same no matter where it was. One kind lived in the Einsteinian curved space of home, the other in the Newtonian absolute space of the free self.

Much of our intellectual development is the story of how we learn to sort impressions: self or environment, rocks, trees, clouds, books, cats. It is a story of how we learn to judge and recognize colors, numbers, shapes, and abstract concepts. When we learn a new category, our internal model of the world rotates - often slightly, occasionally more.... Some shifts are emotional: holding a newborn in your hands and understanding just what a rich and varied life will come to this tiny seed of an individual, looking into the eyes of an animal and recognizing a kinship despite having traveled very different evolutionary paths. Some shifts are abstract: learning the crystalline pure beauty of a geometric proof's logic. Fractal geometry also represents a shift, both emotional and abstract...

ARCHIMEDES PALIMPSEST, C. 10TH–13TH CENTURY 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.

Originally, Smale had hoped to explain all dynamical systems in terms of stretching and squeezing—with no folding, at least no folding that would drastically undermine a system’s stability. But folding turned out to be necessary, and folding allowed sharp changes in dynamical behavior. Smale’s horseshoe stood as the first of many new geometrical shapes that gave mathematicians and physicists a new intuition about the possibilities of motion. In some ways it was too artificial to be useful, still too much a creature of mathematical topology to appeal to physicists. But it served as a starting point.

Among other elements, the test had a vestigial examination in drawing, and Mandelbrot discovered a latent facility for copying the Venus de Milo. On the mathematical sections of the test—exercises in formal algebra and integrated analysis—he managed to hide his lack of training with the help of his geometrical intuition. He had realized that, given an analytic problem, he could almost always think of it in terms of some shape in his mind. Given a shape, he could find ways of transforming it, altering its symmetries, making it more harmonious. Often his transformations led directly to a solution of the analogous problem. In physics and chemistry, where he could not apply geometry, he got poor grades. But in mathematics, questions he could never have answered using proper techniques melted away in the face of his manipulations of shapes.

The vogue for geometrical architecture and painting came and went. Architects no longer care to build blockish skyscrapers like the Seagram Building in New York, once much hailed and copied. To Mandelbrot and his followers the reason is clear. Simple shapes are inhuman. They fail to resonate with the way nature organizes itself or with the way human perception sees the world. In the words of Gert Eilenberger, a German physicist who took up nonlinear science after specializing in superconductivity: “Why is it that the silhouette of a storm-bent leafless tree against an evening sky in winter is perceived as beautiful, but the corresponding silhouette of any multi-purpose university building is not, in spite of all efforts of the architect? The answer seems to me, even if somewhat speculative, to follow from the new insights into dynamical systems. Our feeling for beauty is inspired by the harmonious arrangement of order and disorder as it occurs in natural objects—in clouds, trees, mountain ranges, or snow crystals. The shapes of all these are dynamical processes jelled into physical forms, and particular combinations of order and disorder are typical for them.

Poem of the Phalanx (Perception as Visual Personal Art) Memories, shard, intersect and twitch, Create images anew as they inter and switch. Amid blackness eternal, the ground breaks the day And the shape which cuts the ground— Phalanx in time—reapers way. 5 Thoughts as geometric planes galley the night mind, Images thoughted, float raging ever by. Comets to the mind–bolt outta the black they mortise and fly– Disappear they do–into their midnighted cry. (Yea, evil is wrought from the want of the want of Love’s lost ought. 10 Goodness wrights of the abundance of Love in blood ’twas bought. —Live the moment within God’s Mind too, For once missed she will pass by you. But He alone shall remember thy days, For in His Heart He will hold thy ways. 15 (. . . Surmount untold; reproaching its summits hidden self face, Can’t make for a daydrop of lost opportunity and regret’s disgrace. Yes, eternities of regrets can never create The day’s bested instance that was forsaked). Fleets of illusion harbor and snag, 20 Bristled spears impale with emotive jags. Willish anvil beaten and enhammored in bers red embs, Guards the hellgates unhinged in forged remembered contems. (Aye, the anvil of will beaten and wrought Sentinels the gate ripped in forged oughts). 25 Phalanx of dreams penetrate they deep, Guard thy soul they do lest the enemy storms thy keep. They rancor and barb thyself under penalty of arms, And kill the dragons that would do thee most harm. Yea, in the Belly of the Beast thy wounds do feel pierced, 30 For Love Eternal must cut darkness as the Spirit is so fierce. The hour of shadows exalt—! ’Gainst the Christ in His plain splin‴try array– Yet curshed in a moment on that ill-fated day. The way of caution doth forbear to tread beyond the mire In those bleak hours when the ‘Powers that Be’ seek to solace thee in thy soulish desires. 35 Mercy travails deep upon the Fires of His Winds To heal by His cut; His own everlasting His– Is to die to Love Eternal with He, –as He now does and is . . . Hell for others, heaven for some, His work ’tis finished all given and in all thrust done. 40 As Love rejoices His shed blood run red for thee—, —Things Divined and precioius for you and for me forever in He (The spear that killed Him gave Him life –the enemy’s travesty). Phalanx comes, phalanx goes, Wither are thou—dost thousest know? 45 Are ye pierced through and through out within? Seek his face so life may begin Sharp keys to hell the warriors doth say, Yet unlock they the gate to heaven’s pathway. End

The columns of the nave resembled massive palm trees soaring forty-five meters high but rather than forming any kind of expected ceiling they branched into a kaleidoscopic array of twisting geometric forms. Nothing was flat. There were no right angles or even conventional arches. Everything was a dizzying array of soft and hard shapes and angles no less complicated than the stalks of a plant or the longitudinal section of a seashell.

Now I don’t have any dreams, or if I dream, I dream of dark geometric shapes drifting noiselessly across a pink, black, and purple pixelated colorscape. Screen-saver dreams.