Geometry Circle Quotes

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.

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

We are, all four of us, blood relatives, and we speak a kind of esoteric, family language, a sort of semantic geometry in which the shortest distance between any two points is a fullish circle.

A circle has no end.

It isn't a circle--it is simply a long line--as in geometry, you know, one that reaches into infinity. And because we cannot see the end--we also cannot see how it changes. And it is very odd by those who see the changes--who dream, who will not give up--are called idealists...and those who see only the circle we call them the "realists"!

His way had therefore come full circle, or rather had taken the form of an ellipse or a spiral, following as ever no straight unbroken line, for the rectilinear belongs only to Geometry and not to Nature and Life.

Today, suddenly, I reached an absurd but unerring conclusion. In a moment of enlightenment, I realized that I'm nobody, absolutely nobody. When the lightning flashed, I saw that what I had thought to be a city was in fact a deserted plain and, in the same sinister light that revealed me to myself, there seemed to be no sky above it. I was robbed of any possibility of having existed before the world. If I was ever reincarnated, I must have done so without myself, without a self to reincarnate. I am the outskirts of some non-existent town, the long-winded prologue to an unwritten book. I'm nobody, nobody. I don't know how to feel or think or love. I'm a character in a novel as yet unwritten, hovering in the air and undone before I've even existed, amongst the dreams of someone who never quite managed to breathe life into me. I'm always thinking, always feeling, but my thoughts lack all reason, my emotions all feeling. I'm falling through a trapdoor, through infinite, infinitous space, in a directionless, empty fall. My soul is a black maelstrom, a great madness spinning about a vacuum, the swirling of a vast ocean around a hole in the void, and in the waters, more like whirlwinds than waters, float images of all I ever saw or heard in the world: houses, faces, books, boxes, snatches of music and fragments of voices, all caught up in a sinister, bottomless whirlpool. And I, I myself, am the centre that exists only because the geometry of the abyss demands it; I am the nothing around which all this spins, I exist so that it can spin, I am a centre that exists only because every circle has one. I, I myself, am the well in which the walls have fallen away to leave only viscous slime. I am the centre of everything surrounded by the great nothing. And it is as if hell itself were laughing within me but, instead of the human touch of diabolical laughter, there's the mad croak of the dead universe, the circling cadaver of physical space, the end of all worlds drifting blackly in the wind, misshapen, anachronistic, without the God who created it, without God himself who spins in the dark of darks, impossible, unique, everything. If only I could think! If only I could feel!

The geometry of judgment is a circle. Hate is a snake that turns to consume itself from the tail, a circle that diminishes to a point, then to nothing. Pride is such a snake, and envy, and greed. Love, however, is a hoop, a wheel, that rolls on forever. We are rescued by those whom we have rescued. The saved become the saviors of their saviors.

Perfect hexagonal tubes in a packed array. Bees are hard-wired to lay them down, but how does an insect know enough geometry to lay down a precise hexagon? It doesn't. It's programmed to chew up wax and spit it out while turning on its axis, and that generates a circle. Put a bunch of bees on the same surface, chewing side-by-side, and the circles abut against each other - deform each other into hexagons, which just happen to be more efficient for close packing anyway.

Why is geometry often described as ""cold" and ""dry?" One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

In terms of systems design; ovals, circles and hexagons are more efficient than rectangles, squares and straight lines — Thats something to consider when designing supply chain systems.

And I, I myself, am the centre that exists only because the geometry of the abyss demands it; I am the nothing around which all this spins, I exist so that it can spin, I am a centre that exists only because every circle has one. I, I myself, am the well in which the walls have fallen away to leave only viscous slime. I am the centre of everything surrounded by the great nothing.

It never ceases to amaze me how adaptable social geometry can be. Within a couple of days I went from being the centre of the circle to an indefinite point outside its circumference.

Most right-handed people end up walking wide counterclockwise circles, because most right-handed people have left legs fractionally shorter than their right legs. Basic biology and geometry. I avoided that particular peril by stepping to the right of every tenth tree I came to, whether I thought I needed to or not.

Is everyone with one face called a Milo?" "Oh no," Milo replied; "some are called Henry or George or Robert or John or lots of other things." "How terribly confusing," he cried. "Everything here is called exactly what it is. The triangles are called triangles, the circles are called circles, and even the same numbers have the same name. Why, can you imagine what would happen if we named all the twos Henry or George or Robert or John or lots of other things? You'd have to say Robert plus John equals four, and if the four's name were Albert, things would be hopeless." "I never thought of it that way," Milo admitted. "Then I suggest you begin at once," admonished the Dodecahedron from his admonishing face, "for here in Digitopolis everything is quite precise.

The interwoven spheres and vines ran along the bottom. I'd done some research, and I'd found this motif everywhere. These overlapping circles were ancient, tracing back to Pythagorean geometry--geometry, a measure of the world. In more mystical terms, the shape had always evoked tghe place where world overlap: dreaming with waking, death with life, the visible with the unseen. [p. 362]

By degrees, the bitterness at my heart diffused itself to the circumference of the circle in which my life went its cheerless mechanical round.

We ask the poet to reassure us by giving us a geometry of living, in which all things add up and cohere, to tell us how things buttress once another, circle round and intermelt.

In terms of systems design, shapes are important. Rectangles are not common in nature. That's probably because from a systems design perspective, rectangles often degrade efficiency instead of contributing to efficiency. Yet humans have designed an entire supply chain system based on rectangles, squares and straight lines. If we want to be more efficient, we should replace those rectangles, squares and straight lines with ovals, circles and hexagons. And maybe some other nature inspired geometries.

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

Philosophy is written in this all-encompassing book that is constantly open to our eyes, that is the universe; but it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders in a dark labyrinth.

I realized I was far more interested in the expansion of concentric rings than the geometry of perfect circles.

Putting the law above man is a problem in politics which I liken to that of squaring the circle in geometry.

Whenever we take the focus off ourselves and move it outward, we benefit. Life's most fortunate ironies are that what's best for the long run is best now, and selflessness serves our interests far better than selfishness. The wider our circle of considerations, the more stable we make the world—and the better the prospects for human experience and for all we might wish. The core message of each successive widening: we are one. The geometry of the human voyage is not linear; it's those ripples whose circles expand to encompass self, other, community, Life, and time.

The Greek mathematician Archimedes drew a shape with ninety-six sides around a circle and used the geometry he knew to measure the perimeter of the ninety-six sides shape. Then he divided that by the diameter of the circle.

I don't know how to feel or think or love. I'm a character in a novel as yet unwritten, hovering in the air and undone before I've even existed, amongst the dreams of someone who never quite managed to breathe life into me. I'm always thinking, always feeling, but my thoughts lack all reason, my emotions all feeling. I'm falling through a trapdoor, through infinite, infinitous space, in a directionless, empty fall. My soul is a black maelstrom, a great madness spinning about a vacuum, the swirling of a vast ocean around a hole in the void, and in the waters, more like whirlwinds than waters, float images of all I ever saw or heard in the world: houses, faces, books, boxes, snatches of music and fragments of voices, all caught up in a sinister, bottomless whirlpool. And I, I myself, am the centre that exists only because the geometry of the abyss demands it; I am the nothing around which all this spins, I exist so that it can spin, I am a centre that exists only because every circle has one.

Euler’s Formula encapsulates the whole of existence. It contains 0, the number of the monad (ontological zero); the number e that determines exponentiation; the number i that determines the imaginary domain (time); the number 1 that determines the domain of counting numbers (and with 0 creates the binary system of computing), and real numbers (space); -1, the number of the negative domain (antimatter); and the number π that determines the world of the circle and geometry. Euler’s Formula is the unquestionable God Equation.

The golden ratio, as well as the Great Pyramid as an expression of it, is an important key to our universe containing the Earth and the Moon. ... The ratio between the Earth and the Moon is in fact the basis for the mathematical concept of 'squaring the circle' ...

And in between the two, in between the sky and the sea, were all the winds. And there were all the nights and all the moons. To be a castaway is to be a point perpetually at the centre of a circle. However much things may appear to change—the sea may shift from whisper to rage, the sky might go from fresh blue to blinding white to darkest black—the geometry never changes. Your gaze is always a radius. The circumference is ever great. In fact, the circles multiply. To be a castaway is to be caught in a harrowing ballet of circles.

The Great Pyramid was a fractal resonator for the entire Earth. It is designed according to the proportions of the cosmic temple, the natural pattern that blends the two fundamental principles of creation. The pyramid has golden ratio, pi, the base of natural logarithms, the precise length of the year and the dimensions of the Earth built into its geometry. It demonstrates.... As John Michell has pointed out in his wonderful little book, City of Revelation, 'Above all, the Great Pyramid is a monument to the art of 'squaring the circle''.

The goal of drawing a smooth border through the fractal of interpenetrating ethnic groups is an unsolvable geometry problem, and living with existing borders is now considered better than endless attempts to square the circle, with its invitations to ethnic cleansing and irredentist conquest.

Free will is an idealization of human beings that makes the ethics game playable. Euclidean geometry requires idealizations like infinite straight lines and perfect circles, and its deductions are sound and useful even though the world does not really have infinite straight lines or perfect circles. The world is close enough to the idealization that the theorems can usefully be applied. Similarly, ethical theory requires idealizations like free, sentient, rational, equivalent agents whose behavior is uncaused, and its conclusions can be sound and useful even though the world, as seen by science, does not really have uncaused events. As long as there is no outright coercion or gross malfunction of reasoning, the world is close enough to the idealization of free will that moral theory can meaningfully be applied to it.

Right from the beginning, I was convinced that Avenir is the better Futura,” said a confident Adrian Frutiger in a recent interview looking back at his 1988 creation. In some respects, his declaration was more than mere boasting — coming 60 years after Futura, Avenir remedied many of the compromises that Renner made in his quest for geometry. Frutiger abandoned pure circles and strictly even stroke weight for “corrected” curves and a bit of contrast.

And amid all this confusion I, what’s truly I, am the centre that exists only in the geometry of the abyss: I’m the nothing around which everything spins, existing only so that it can spin, being a centre only because every circle has one. I, what’s truly I, am a well without walls but with the walls’ viscosity, the centre of everything with nothing around it. — Fernando Pessoa, The Book of Disquiet. (Penguin Classics; New Ed edition, December 31, 2002) Originally published 1982.

Our choices are us. Envisioning choices as sea blue circles places a marker on our decisions: every word, each coffee, every button, and choices in between. The blue circles upon our daily choices are reflection points and moments of consideration. This is a choice is the breath before you look fully at loved ones or strangers and the breath before you change and reframe a pattern so as to not misplace a moment. The breath before you say I love you to you. Again and again. This is a choice.

The diameter divides into the circumference, you know. It ought to be three times. You'd think so, wouldn't you? But does it? No. Three point one four one and lots of other figures. There's no end to the buggers. Do you know how pissed off that makes me?" "I expect it makes you extremely pissed off," said Teppic politely. "Right. It tells me that the Creator used the wrong kind of circles. It's not even a proper number! I mean, three point five, you could respect. Or three point three. That'd look *right*." He stared morosely at the pie.

A philosopher/mathematician named Bertrand Russell who lived and died in the same century as Gass once wrote: “Language serves not only to express thought but to make possible thoughts which could not exist without it.” Here is the essence of mankind’s creative genius: not the edifices of civilization nor the bang-flash weapons which can end it, but the words which fertilize new concepts like spermatozoa attacking an ovum. It might be argued that the Siamese-twin infants of word/idea are the only contribution the human species can, will, or should make to the raveling cosmos. (Yes, our DNA is unique but so is a salamander’s. Yes, we construct artifacts but so have species ranging from beavers to the architect ants whose crenellated towers are visible right now off the port bow. Yes, we weave real-fabric things from the dreamstuff of mathematics, but the universe is hardwired with arithmetic. Scratch a circle and π peeps out. Enter a new solar system and Tycho Brahe’s formulae lie waiting under the black velvet cloak of space/time. But where has the universe hidden a word under its outer layer of biology, geometry, or insensate rock?)

Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as in a super-sensible intelligible world. Geometry deals with exact circles, but no sensible object is exactly circular; however carefully we may use our compasses, there will be some imperfections and irregularities. This suggests the view that all exact reasoning applies to ideal as opposed to sensible objects; it is natural to go further, and to argue that thought is nobler than sense, and the objects of thought more real than those of sense-perception. Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic. Rationalistic as opposed to apocalyptic religion has been, ever since Pythagoras, and notably ever since Plato, very completely dominated by mathematics and mathematical method.

❝ Outside and inside form a dialectic of division, the obvious geometry of which blinds us as soon as we bring it into play in metaphorical domains. It has the sharpness of the dialectics of ‘yes’ and 'no,’ which decides everything. Unless one is careful, it is made into a basis of images that govern all thoughts of positive and negative. Logicians draw circles that overlap or exclude each other, and all their rules immediately become clear. Philosophers, when confronted with outside and inside, think in terms of being and non-being. Thus profound metaphysics is rooted in an implitcit geometry which– whether we will or no–confers spatiality upon thought; if a metaphysician could not draw, what would he think? Open and closed, for him, are thoughts. They are metaphors that he attaches to everything, even his systems. In a lecture given by Jean Hyppolite on the subtle structure of denegation (which is quite different from the simple structure of negation) Hyppolite spoke of “a first myth of outside and inside.” And he added: “you feel the full significance of this myth of outside and inside in alienation, which is founded on these two term. Beyond what is expressed in their formal opposition lie alienation and hostility between the two.” And so, simple geometrical opposition becomes tinged with agressivity. Formal opposition is incapable of remaining calm.

One solution might be to liken the path of the light beam through a changing gravitational field to that of a line drawn on a sphere or on a surface that is warped. In such cases, the shortest line between two points is curved, a geodesic like a great arc or a great circle route on our globe. Perhaps the bending of light meant that the fabric of space, through which the light beam traveled, was curved by gravity. The shortest path through a region of space that is curved by gravity might seem quite different from the straight lines of Euclidean geometry. There was another clue that a new form of geometry might be needed. It became apparent to Einstein when he considered the case of a rotating disk. As a disk whirled around, its circumference would be contracted in the direction of its motion when observed from the reference frame of a person not rotating with it. The diameter of the circle, however, would not undergo any contraction. Thus, the ratio of the disk’s circumference to its diameter would no longer be given by pi. Euclidean geometry wouldn’t apply to such cases. Rotating motion is a form of acceleration, because at every moment a point on the rim is undergoing a change in direction, which means that its velocity (a combination of speed and direction) is undergoing a change. Because non-Euclidean geometry would be necessary to describe this type of acceleration, according to the equivalence principle, it would be needed for gravitation as well.

Gone were the landscape paintings of the past, the pictures of ancient Greek heroes, the portraits of women in their silks and feathers. Painters began to reduce everything to simple squares and circles, the intersection of triangles. They were thrilled by geometry. They talked about achieving weightlessness, of painting pictures that were no longer mired in the world. They wanted to leave the earth behind.

Eleven is important as the first number that allows us to begin to comprehend the measure of a circle. This is because, for practical purposes, a circle measuring seven across will measure eleven halfway around.

The ancients were obsessed with measures, and the number eleven is central in their metrological scheme. Shown opposite is the extraordinary fact that the size of the Moon relates to the size of the Earth as does three to eleven. What this means is that if we draw down the Moon to the Earth, as shown, then a heavenly circle through the moon will have a circumference equal to the perimeter of a square around the Earth. This is called 'squaring the circle'. Quite how the old druids worked this out we may never know, but they clearly did, for the Moon and the Earth are best measured in miles, as shown. A double rainbow also magically squares the circle.

When one circle is drawn over another like this so that they pass through each others' centers, then an important almond shape, the vesica piscis, literally 'fish's bladder' is formed. It is one of the first things that circles can do. Christ is often depicted inside a vesica.

The Golden Ratio defines the squaring of a circle. Stated in mathematical terms, this says: Given a square of known perimeter, create a circle of equal circumference. According to some, in ancient Egypt, this mathematical mystery was encoded in the measurements of the Great Pyramid of Giza.

Within the Unity of the gathered and the awakened visualizing and connecting as a sacred circle there can become manifest a quantum Divine 'Buddha field' as a geometry of light transference telepathy and amplification. It is a clear method for Universal Mind awakening.

The Scientific Worldview meant to make things as clear as possible, since that is the purpose of any manifesto. And indeed, as a summary of the Vienna Circle, the text is still unsurpassed. In just a few pages it neatly described the group’s historical background and its highest mission—a collective crusade against all metaphysical and theological doctrines. Only the results of experimentation and logical analysis were admitted—nothing else. The manifesto listed the names of the Circle’s members, and the problems to be tackled: namely the foundations of mathematics, physics, geometry, biology, psychology, and the social sciences. An encyclopedia could not have done better.

With Ghirlandaio and Fillipino Luppi dead, and Botticelli in a permanent state of depression, Raphael found an affluent audience starved for works of the highest quality. Florentine mercantile society fell in love with his potrayals of the Madonna and Child and the Holy Family-and with him, personally, for his gentle character. The provincialism of his master Perugino had heretofore kept Raphael’s genius under wraps. Leonardo taught him the power of unified, lucid compositions based on geometry, particularly the triangle and the circle. During Michelangelo’s absence, Raphael’s company was sought by everyone, including Michelangelo’s valued friends Taddeo Taddei and Agnolo Doni. In fact, he was such a frequent guest at Taddei’s home, where he would have had plenty of opportunities to study Michelangelo’s tondo, that Raphael gave his patron two paintings as thanks for his many kindnesses, and painted the Madonna deil Cardellino as a wedding gift for his friend Lorenzo Nasi, Taddei’s cousin. In 1505, the Carrera year, Raphael painted the portraits of Doni and his wife, Maddalena. The out-of-towner whom Michelangelo had dismissed as a mere nuisance had grown up.

Circles are the simplest curves in geometry. Yet, surprisingly, measuring them—quantifying their properties with numbers—transcends geometry.

Ive’s design team had obsessed over the rounded corners of the phone and become advocates of Bézier curves, a concept from computer modeling used to eliminate the transition breaks between straight and curved surfaces. The Bézier geometry gave the iPhone rounded corners that arched like a sculpture. A standard rounded corner consists of a single-radius arch or a quarter circle, whereas their curves were mapped through a dozen points, creating a more gradual and natural transition. Meanwhile, Forstall used a standard three-point curve in the corners of iPhone apps. Each time Ive opened his iPhone, he could see the difference between the phone’s carefully crafted corners and the software’s clunky corners. He was powerless to change those features because Jobs excluded him from software design meetings. He could only look at them and fume.

Like numbers, human experience has abstract and practical sides, feeling and reason, and in each of us one or the other tends to dominate. Belief does not persuade a scientist, and science does not persuade a believer. Too ardent an embrace of reason leads to irrational thinking, and too ardent an embrace of feeling leads to madness. William James says that religious mysticism is only half of the possible mysticisms, the others are forms of insanity. These are the states in which mystical convictions circle back on a person and pessimistically invert notions of divinity into notions of evil.

In geometry, as in nature, the circle is the archetypal shape of wholeness and inclusion. It is an effective shape for nonprofits or community-focused efforts.

context. Take 5 heel-to-toe steps in each of the four directions from a center point. From one “corner” to the next will be 7 heel-to-toe steps, and when you connect those four corners with straight lines, the resulting square will measure 28 footsteps around the edges.1 (See Figure 3-1.) Whether used as a square or curved outward into a circle, 28 reminds us of the Moon phases, a worthy underpinning to any magical space.

Al-Kind¯ı (ca. 800 – ca. 870), the first to develop philosophical thought as such in Arabic, was a polymath in the translated sciences and very much a product of his age. Like other scientists of his time, he gathered around him a wide circle of individuals capable of advising him on various issues and translating the relevant texts. He commissioned translations of scientific subjects and he himself wrote on all the sciences: astrology, astronomy, arithmetic, geometry, music, and medicine – he even has a treatise on swords.36 This broad and synoptic view of all sciences, along with the spirit of encyclopedism fostered by the translation movement for the half century before his time, led him to an overarching vision of the unity and interrelatedness of all knowledge. At the same time, and as a result of this view, he developed a unitary epistemological approach – namely, that of mathematics. His goal became to approach mathematical accuracy in his argumentation; influenced by both Ptolemy and Euclid, he held mathematical or geometrical proof to be of the highest order

Fractals are beautiful to look at when graphed and, thus, had been used in arts and the sciences.  It first appeared in art in the 19th century, at a painting of Mt. Fuji, which shows a great wave that threatens an open boat, wherein the dimension of the wave is the approximation of a circle’s diameter.  This is an example of a natural fractal, way before Felix Hausdorff first presented the theory of the fractal dimension in 1868.[viii]  They are used to incorporate nature into artistic elements and, thus, had been used to highlight pieces of visual arts.  It became well known when genetic programming entered the world in the 20th century, which optimized parameters of what is called “Mandelbrot sets” that are useful in generating certain biomorphs. 

And amid all this confusion I, what’s truly I, am the centre that exists only in the geometry of the abyss: I’m the nothing around which everything spins, existing only so that it can spin, being a centre only because every circle has one. I, what’s truly I, am a well without walls but with the walls’ viscosity, the centre of everything with nothing around it.

The thought turned him topsy-turvy. It seemed to summarize the whole worthless way of the world--if there was one. And versions of it began to flutter wildly through his head. You have to look round to see straight. Good enough. Useful. And the rough places plain. But all that's geometry. But it measures the earth. You have to go slow to catch up. Eat to get thin? no, but fast to grow fat, that was a fine one. Then lose to win? fail to succeed? Risky. Stop to begin. The form made noiseless music--lumly lum lum or lum-lee-lee lum--like fill to empty, every physical extreme. Die to live was a bit old hat. But default to repay. And lie to be honest. He liked the ring of that. Flack! I'm white in order to be black. Sin first and saint later. Cruel to be kind, of course, and the hurts in the hurter--that's what they say--a lot of blap. That's my name, my nomination: Saint Later. Now then: humble to be proud; poor to be rich. Enslave to make free? That moved naturally. Also multiply to subtract. Dee dee dee. Young Saint Later. A list of them, as old as Pythagoras had. Even engenders odd. How would that be? Eight is five and three. There were no middle-aged saints--they were all old men or babies. Ah, god--the wise fool. The simpleton sublime. Babe in the woods, roach in the pudding, prince in the pauper, enchanted beauty in the toad. This was the wisdom of the folk and the philosopher alike--the disorder of the lyre, or the drawn-out bow of that sane madman, the holy Heraclitus. The poet Zeno. The logician Keats. Discovery after discovery: the more the mice eat, the fatter the cats. There were tears and laughter, for instance--how they shook and ran together into one gay grief. Dumb eloquence, swift still waters, shallow deeps. Let's see: impenitent remorse, careless anxiety, heedless worry, tense repose. So true of tigers. Then there was the friendly enmity of sun and snow, and the sweet disharmony of every union, the greasy mate of cock and cunt, the cosmic poles, war that's peace, the stumble that's an everlasting poise and balance, spring and fall, love, strife, health, disease, and the cold duplicity of Number One and all its warm divisions. The sameness that's in difference. The limit that's limitless. The permanence that's change. The distance of the near at home. So--to roam, stay home. Then pursue to be caught, submit to conquer. Method--ancient--of Chinese. To pacify, inflame. Love, hate. Kiss, kill. In, out, up, down, start, stop. Ah . . . from pleasure, pain. Like circumcision of the heart. Judgement and mercy. Sin and grace. It little mattered; everything seemed to Furber to be magically right, and his heart grew fat with satisfaction. Therefore there is good in every evil; one must lower away to raise; seek what's found to mourn its loss; conceive in stone and execute in water; turn profound and obvious, miraculous and commonplace, around; sin to save; destroy in order to create; live in the sun, though underground. Yes. Doubt in order to believe--that was an old one--for this the square IS in the circle. O Phaedo, Phaedo. O endless ending. Soul is immortal after all--at last it's proved. Between dead and living there's no difference but the one has whiter bones. Furber rose, the mosquitoes swarming around him, and ran inside.

Fractal shapes were being expressed intuitively by artists long before they were recognized in science. Self-similar patterns appear in Celtic artefacts, like the spirals and circles within circles of the exquisitely crafted illuminated pages of the early 9th-century Book of Kells and the Densborough mirror made in the 1st century A.C. Mathematical awareness, particularly fractal awareness, reveals itself in the art of the Romans and the Egyptians, and in the work of the Aztec, Inca and Mayan civilizations of Central and South America. Shapes highly reminiscent of the Koch curve were used to depict waves by the Hellenic artist in a frieze in the ancient Greek town of Akrotiri.

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.

A mandala (Sanskrit for “circle”) is a circular pattern, considered a sacred geometry, that’s been used in spiritual practice since ancient times as an active visual meditation.

geometry is good for the mind; it trains you to think clearly and logically. It’s not the study of triangles, circles, and parallel lines per se that matters. What’s important is the axiomatic method, the process of building a rigorous argument, step by step, until a desired conclusion has been established.

When everything-EVERYTHING about life makes you want to grin, and it just gets sunnier and funnier until after a while you can only see the teeth in the smiles and then you feel... —well, not "on the edge" exactly, for the world has no edge; but as if you have always been over the edge, and the smiling and laughing is a sort of spastic reflex like crying or retching (really, it's all the same);— when you drink red wine in a cup and try to categorize the geometry of the gleam-patterns you see on the liquid's surface-and you may find, my friends, that you can almost do it: you agree with yourself upon the existence of a light-shape like the outline of a hemisphere drawn in concave at the equator; but another sip and it changes to a gleam-ring all around the rim of the wine circle; and another and it is reddish-black everywhere with the unsteady image of your face in it, your skin redder and your mouth blacker than the wine, and another and you see white specks swimming in the cup: they are not reflections at all, but bits of grease or rice or cereal, or maybe cheek-cells that got washed out of your mouth (the age-old question: is the imper-fection, the filth, in you or in the glass?); —but then your attention is diverted forever by the ugly purple stain around the edge of the cup where your lips have been; when everything is so confusing that you can never be sure whether or not your whore is a woman until she pulls her underpants down; when nothing is clear, and whore-chasing is a merry-go-round of death (if you don't catch a disease that will kill you, why, you will go around again, not because you want to die but because until you do everything remains unclear); when you get drunken crushes on women whose drunken mothers used to try to stab them; when the names of streets are like Nabokov's wearisome clever-ness; when only the pretty shapes of women have integrity and when you close your eyes still see them leaning and crossing their legs and milking their tits at you, THEN you may on occasion like Jimmy find yourself looking down a long black block, down the tunnels of infinity to a streedamp, a corner and a woman's waiting silhouette. —Or else like Jimmy you may have another drink

To be a castaway is to be a point perpetually at the centre of a circle. However much things may appear to change—the sea may shift from whisper to rage, the sky might go from fresh blue to blinding white to darkest black—the geometry never changes. Your gaze is always a radius. The circumference is ever great. In fact, the circles multiply. To be a castaway is to be caught in a harrowing ballet of circles. You are at the centre of one circle, while above you two opposing circles spin about. The sun distresses you like a crowd, a noisy, invasive crowd that makes you cup your ears, that makes you close your eyes, that makes you want to hide. The moon distresses you by silently reminding you of your solitude; you open your eyes wide to escape your loneliness. When you look up, you sometimes wonder if at the centre of a solar storm, if in the middle of the Sea of Tranquillity, there isn’t another one like you also looking up, also trapped by geometry, also struggling with fear, rage, madness, hopelessness, apathy.

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.

We are Circles is an attempt to bring all of our choices into focus, discussion, analysis, and change. Into our center. Which is overwhelming! I know. So hopefully the simple metaphors and groupthink aspect of our evolving humanity circles up our collective choices into the here and now for love and life on Earth. We always start with a single choice.

Our choices are us. Envisioning choices as sea blue circles places a sticker across our days: every word, each coffee, every button, and choices in between. The blue stickers across our days are reflection points and moments of consideration. This is a choice is the breath before you look fully at loved ones or strangers and the breath before you change and reframe a pattern so as to not misplace a moment. The breath before you say I love you to you. Again and again. This is a choice.

I don’t get the point of this. Where in the real world will I have someone ask me to prove this is a circle? Look, it’s round. It’s a circle. Stop giving it an existential crisis, geometry!

Bella inspired this book through a series of questions and conversations that circled around how people live together. Growing up on Maui (and in California), she has observed the challenges of food production, housing, global economies, and environmental losses. She has watched people lose their options, and people fall through the cracks in systems. As a father, it is emotionally cutting to see pain and heartbreak in your daughter's eyes, to witness her start to realize social equations do not calculate toward equity. Her brilliance across these conversations has been the ever-present solutions, her ingenuity and moxie, and the art for creating visions for change. She inspires me to be more present with each choice that circles around her future.

I don’t know how to feel or think or love. I’m a character in a novel as yet unwritten, hovering in the air and undone before I’ve even existed, amongst the dreams of someone who never quite managed to breathe life into me. I’m always thinking, always feeling, but my thoughts lack all reason, my emotions all feeling. I’m falling through a trapdoor, through infinite space… in a directionless, empty fall. My soul is a black maelstrom, a great madness spinning about a vacuum, the swirling of a vast ocean around a hole in the void, and in the waters, more like whirlwinds than waters, float images of all I ever saw or heard in the world: houses, faces, books, boxes, snatches of music and fragments of voices, all caught up in a sinister, bottomless whirlpool. And I, I myself, am the centre that exists only because the geometry of the abyss demands it; I am the nothing around which all this spins, I exist so that it can spin, I am the centre that exists only because every circle has one.

Unit circle is one of the important math concepts that every student must learn and understand. There are numerous concepts related to Trigonometry and geometry that needs to understand basics before solving the problems. Unit circle is known as the foundation of projectile motion, sine, cosine, tangents, degrees and radians. If you are learning the concept of geometry and trigonometry then you must have a unit circle chart as reference sheet. Most of the school teachers us this sheet while teaching the concepts of applied mathematics. This basic circle will be helpful throughout your life. It is necessary to learn this Blank Unit Circle Printable by heart and to practice it regularly for a solid foundation.

...he will accompany us on a hike in the hills, leaping and whizzing back and forth, and coming when called as well as a dog. It is just that the organism, the whole pattern of nerve and muscle, is more complex and intelligent than logical systems of arithmetic, geometry and grammar - which are in fact nothing but inferior ritual. Life itself dances, for what else are trees, ferns, butterflies, and snakes but elaborate forms of dancing? Even wood and bones show, in their structure, the characteristic patterns of flowing water, which (as Lao-tzu pointed out in 400 B.C.) derives its incredible power by following gravity and seeking that "lowest level which all men abhor." When dance I do not think-count my steps, and some women say I have no sense of rhythm, but I have a daughter who (without ever having taken lessons in dancing) can follow me as if she were my shadow or I were hers. The whole secret of life and of creative energy consists in flowing with gravity. Even when he leaps and bounces our cat is going with it. This is the way the whole earth and everything in the universe beehives.* But man is making a mess of the earth be- *Harrumph! Excuse the pun, but it is important, because bees live in hexagonal as distinct from quadrilateral structures, and this is the natural way in which all things, such as bubbles and pebbles, congregate, nestling into each other by gravity. It will follow, because 2 x 6 is 12, that - as Buckminster Fuller has pointed out - as number-system to the base 12 (duodecimal) is closer to nature than one to the base of 10 (decimal). For 12 is divisible by both 2 and 3, whereas 10 is not. After all, we use the base 12 for measuring circles and spheres and time, and so can "think circles" around people who use only meters. The world is better duodecimal than decimated.

Very often these luminous designs, rich in data, take the form of geometry. I speak from experience, having participated in more than seventy ayahuasca sessions since 2003, continuing to work with the brew for the valuable lessons it teaches me long after Supernatural was researched, written, and published. Here’s part of my account of the first time I drank ayahuasca in the Amazon: I raise the cup to my lips again. About two thirds of the measure that the shaman poured for me still remains, and now I drain it in one draught. The concentrated bittersweet foretaste, followed instantly by the aftertaste of rot and medicine, hits me like a punch in the stomach…. Feeling slightly apprehensive, I thank the shaman and wander back to my place on the floor…. Time passes but I don’t keep track of it. I’ve improvized a pillow from a rolled-up sleeping bag and I now find I’m swamped by a powerful feeling of weariness. My muscles involuntarily relax, I close my eyes, and without fanfare a parade of visions suddenly begins, visions that are at once geometrical and alive, visions of lights unlike any light I’ve ever seen—dark lights, a pulsing, swirling field of the deepest luminescent violets, of reds emerging out of night, of unearthly textures and colors, of solar systems revolving, of spiral galaxies on the move. Visions of nets and strange ladder-like structures. Visions in which I seem to see multiple square screens stacked side by side and on top of each other to form immense patterns of windows arranged in great banks. Though they manifest without sound in what seems to be a pristine and limitless vacuum, the images possess a most peculiar and particular quality. They feel like a drum-roll—as though their real function is to announce the arrival of something else.13 Other notes I made following my ayahuasca sessions in the Amazon refer to a “geometrical pulse,”14 to “a recurrence of the geometrical patterns,”15 to “a background of shifting geometrical patterns,”16 and to “complex interlaced patterns of geometry…. I zoom in for a closer view…. They’re rectangular, outlined in black, like windows. There’s a circle in the centre of each rectangle.”17

Rāgas are soliloquies and meditations, passionate melodies that draw circles and triangles in a mental space, a geometry of sounds that can turn a room into a fountain, a spring, a pool. What I learned from music—besides the pleasure of walking through those galleries of echoes and gardens of transparent trees, where sounds think and thoughts dance—was something that I also found in Indian poetry and thought: the tension between wholeness and emptiness, the continual coming and going between the two.

Discontinuity, bursts of noise, Cantor dusts—phenomena like these had no place in the geometries of the past two thousand years. The shapes of classical geometry are lines and planes, circles and spheres, triangles and cones. They represent a powerful abstraction of reality, and they inspired a powerful philosophy of Platonic harmony. Euclid made of them a geometry that lasted two millennia, the only geometry still that most people ever learn. Artists found an ideal beauty in them, Ptolemaic astronomers built a theory of the universe out of them. But for understanding complexity, they turn out to be the wrong kind of abstraction.

And amid all this confusion I, what’s truly I, am the centre that exists only in the geometry of the abyss: I’m the nothing around which everything spins, existing only so that it can spin, being a centre only because every circle has one. I, what’s truly I, am a well without walls but with the walls’ viscosity, the centre of everything with nothing around it. It’s not demons (who at least have a human face) but hell itself that seems to be laughing inside me, it’s the croaking madness of the dead universe, the spinning cadaver of physical space, the end of all worlds blowing blackly in the wind, formless and timeless, without a God who created it, without even its own self, impossibly whirling in the absolute darkness as the one and only reality, everything.

In geometry, whenever we had to find the area of a circle, pi * radius squared, I would get really hungry for pie. Square pie.

Geometry - a small circle of people put the whole world on the line.