Geometry In Nature 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.

Reading list (1972 edition)[edit] 1. Homer – Iliad, Odyssey 2. The Old Testament 3. Aeschylus – Tragedies 4. Sophocles – Tragedies 5. Herodotus – Histories 6. Euripides – Tragedies 7. Thucydides – History of the Peloponnesian War 8. Hippocrates – Medical Writings 9. Aristophanes – Comedies 10. Plato – Dialogues 11. Aristotle – Works 12. Epicurus – Letter to Herodotus; Letter to Menoecus 13. Euclid – Elements 14. Archimedes – Works 15. Apollonius of Perga – Conic Sections 16. Cicero – Works 17. Lucretius – On the Nature of Things 18. Virgil – Works 19. Horace – Works 20. Livy – History of Rome 21. Ovid – Works 22. Plutarch – Parallel Lives; Moralia 23. Tacitus – Histories; Annals; Agricola Germania 24. Nicomachus of Gerasa – Introduction to Arithmetic 25. Epictetus – Discourses; Encheiridion 26. Ptolemy – Almagest 27. Lucian – Works 28. Marcus Aurelius – Meditations 29. Galen – On the Natural Faculties 30. The New Testament 31. Plotinus – The Enneads 32. St. Augustine – On the Teacher; Confessions; City of God; On Christian Doctrine 33. The Song of Roland 34. The Nibelungenlied 35. The Saga of Burnt Njál 36. St. Thomas Aquinas – Summa Theologica 37. Dante Alighieri – The Divine Comedy;The New Life; On Monarchy 38. Geoffrey Chaucer – Troilus and Criseyde; The Canterbury Tales 39. Leonardo da Vinci – Notebooks 40. Niccolò Machiavelli – The Prince; Discourses on the First Ten Books of Livy 41. Desiderius Erasmus – The Praise of Folly 42. Nicolaus Copernicus – On the Revolutions of the Heavenly Spheres 43. Thomas More – Utopia 44. Martin Luther – Table Talk; Three Treatises 45. François Rabelais – Gargantua and Pantagruel 46. John Calvin – Institutes of the Christian Religion 47. Michel de Montaigne – Essays 48. William Gilbert – On the Loadstone and Magnetic Bodies 49. Miguel de Cervantes – Don Quixote 50. Edmund Spenser – Prothalamion; The Faerie Queene 51. Francis Bacon – Essays; Advancement of Learning; Novum Organum, New Atlantis 52. William Shakespeare – Poetry and Plays 53. Galileo Galilei – Starry Messenger; Dialogues Concerning Two New Sciences 54. Johannes Kepler – Epitome of Copernican Astronomy; Concerning the Harmonies of the World 55. William Harvey – On the Motion of the Heart and Blood in Animals; On the Circulation of the Blood; On the Generation of Animals 56. Thomas Hobbes – Leviathan 57. René Descartes – Rules for the Direction of the Mind; Discourse on the Method; Geometry; Meditations on First Philosophy 58. John Milton – Works 59. Molière – Comedies 60. Blaise Pascal – The Provincial Letters; Pensees; Scientific Treatises 61. Christiaan Huygens – Treatise on Light 62. Benedict de Spinoza – Ethics 63. John Locke – Letter Concerning Toleration; Of Civil Government; Essay Concerning Human Understanding;Thoughts Concerning Education 64. Jean Baptiste Racine – Tragedies 65. Isaac Newton – Mathematical Principles of Natural Philosophy; Optics 66. Gottfried Wilhelm Leibniz – Discourse on Metaphysics; New Essays Concerning Human Understanding;Monadology 67. Daniel Defoe – Robinson Crusoe 68. Jonathan Swift – A Tale of a Tub; Journal to Stella; Gulliver's Travels; A Modest Proposal 69. William Congreve – The Way of the World 70. George Berkeley – Principles of Human Knowledge 71. Alexander Pope – Essay on Criticism; Rape of the Lock; Essay on Man 72. Charles de Secondat, baron de Montesquieu – Persian Letters; Spirit of Laws 73. Voltaire – Letters on the English; Candide; Philosophical Dictionary 74. Henry Fielding – Joseph Andrews; Tom Jones 75. Samuel Johnson – The Vanity of Human Wishes; Dictionary; Rasselas; The Lives of the Poets

The harmony of the world is made manifest in Form and Number, and the heart and soul and all the poetry of Natural Philosophy are embodied in the concept of mathematical beauty.

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.

The Fibonacci Sequence turns out to be the key to understanding how nature designs... and is... a part of the same ubiquitous music of the spheres that builds harmony into atoms, molecules, crystals, shells, suns and galaxies and makes the Universe sing.

Of all the possible pathways of disorder, nature favors just a few.

Straight lines evidently belonged only to geometry, not to nature and life.

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.

The Golden Number is a mathematical definition of a proportional function which all of nature obeys, whether it be a mollusk shell, the leaves of plants, the proportions of the animal body, the human skeleton, or the ages of growth in man.

When the ancients discovered ‘Phi’, they were certain they had stumbled across God’s building block for the world.

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.

Simple shapes are inhuman. They fail to resonate with the way nature organizes itself or with the way human perception sees the world.

(Since algebra derives from the Arabic jabara = to bind together, fractal and algebra are etymological opposites!)

Well then – I see two ways of letting things take their course – Create one’s own sensations with the help of a flamboyant collision of rare words – not often, mind you – or else neatly draw the angles, the squares, the entire geometry of feelings – those of the moment, naturally.

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

And I cherish more than anything else the Analogies, my most trustworthy masters. They know all the secrets of Nature, and they ought to be least neglected in Geometry.

The water beneath the Temple was both actual and metaphorical, existing as springs and streams, as spiritual energy, and as a symbol of the receptive or lunar aspect of nature. The meaning of that principle is too wide and elusive for it to be given any one name, so in the terminology of ancient science it was given a number, 1,080. Its polar opposite, the positive, solar force in the universe, was also referred to as a number 666. These two numbers, which have an approximate golden-section relationship of 1:1.62, were at the root of the alchemical formula that expressed the supreme purpose of the Temple. Its polar opposite, the positive, solar force in the universe, was also referred to as a number 666. Not merely was it used to generate energy from fusion of atmospheric and terrestrial currents, but it also served to combine in harmony all the correspondences of those forces on every level of creation.

Rigid geometry forced in varied curves is "mother," / is "nature," is systematic violation." Muy Bueno. / This device is for you, the mutilated of no art.

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

With the sentiment of the stars and moon such nights I get all the free margins and indefiniteness of music or poetry, fused in geometry's utmost exactness.

With quiet horror I knew then that the great transition in man’s history was not from illiteracy to literacy, nor from ground-bound to spacefaring, but that invisible, unremarked-of moment when the most dangerous threat to man’s existence ceased being nature, and became man himself.

From all this we concluded that the first two divisions of theoretical philosophy should rather be called guesswork than knowledge, theology because of its completely invisible and ungraspable nature, physics because of the unstable and unclear nature of the matter; hence there is no hope that philosophers will ever be agreed about them; and that only mathematics can provide sure and unshakable knowledge to its devotees, provided one approaches it rigorously. For its kind of proof proceeds by indisputable methods, namely arithmetic and geometry (tr. Toomer, p. 6).

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.

I am trying to explain as quickly as possible my essential nature, that is, what manner of man I am, what I believe in, and for what I hope, that's it, isn't it? And therefore I tell you that I accept God honestly and simply. But you must note this: If God exists and if He really did create the world, then, as we all know, He created it according to the geometry of only three dimensions in space. Yet there have been some very distinguished ones, who doubt whether the whole universe, or to speak more generally the whole of being, was only created in Euclid's geometry; they even dare to dream that two parallel lines, which according to Euclid can never meet on earth, may meet somewhere in infinity. I have come to the conclusion that, since I can't understand even that, I can't expect to understand about God. I acknowledge humbly that I have no faculty for settling such questions, I have a Euclidian earthly mind, and how could I solve problems that are not of this world? And I advise you never to think about it either, my dear Alyosha, especially about God, whether He exists or not. All such questions are utterly inappropriate for a mind created with a conception of only three dimensions. And so I accept God and am glad to, and what's more I accept His wisdom, His purpose - which are utterly beyond our ken; I believe in the underlying order and the meaning of life; I believe in the eternal harmony in which they say we shall one day be blended. I believe in the Word to Which the universe is striving, and Which Itself was "with God", and Which Itself is God and so on, and so on, to infinity.

Of the two alternatives - a curved manifold in a Euclidean space of ten dimensions or a manifold with non-Euclidean geometry and no extra dimensions - which is right? I would rather not attempt a direct answer, because I fear I should get lost in a fog of metaphysics. But I may say at once that I do not take the ten dimensions seriously; whereas I take the non-Euclidean geometry of the world very seriously, and I do not regard it as a thing which needs explaining away.

Our universe, extending immensely far beyond our present horizon, may itself be just one member of a possibly infinite ensemble. This ‘multiverse’ concept, though speculative, is a natural extension of current cosmological theories, which gain credence because they account for things that we do observe. The physical laws and geometry could be different in other universes, and this offers a new perspective on the seemingly special values that the six numbers take in ours.

The fractal structure nature has devised works so efficiently that, in most tissue, no cell is ever more than three or four cells away from a blood vessel. Yet the vessels and blood take up little space, no more than about five percent of the body.

The afternoon had passed to a ghostly gray. She was struck by the immensity of things, so much water and sky and forest, and after a time it occurred to her that she’d lived a life almost entirely indoors. Her memories were indoor memories, fixed by ceilings and plastered white walls. Her whole life had been locked to geometries: suburban rectangles, city squares. First the house she’d grown up in, then dorms and apartments. The open air had been nothing but a medium of transit, a place for rooms to exist.

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

Riemann concluded that electricity, magnetism, and gravity are caused by the crumpling of our three-dimensional universe in the unseen fourth dimension. Thus a "force" has no independent life of its own; it is only the apparent effect caused by the distortion of geometry. By introducing the fourth spatial dimension, Riemann accidentally stumbled on what would become one of the dominant themes in modern theoretical physics, that the laws of nature appear simple when expressed in higher-dimensional space. He then set about developing a mathematical language in which this idea could be expressed.

It was a hundred years later that Einstein gave a theory (general relativity) which said that the geometry of the universe is determined by its content of matter, so that no one geometry is intrinsic to space itself. Thus to the question, "Which geometry is true?" nature gives an ambiguous answer not only in mathematics, but also in physics

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.

Yarn is fundamental, knots are not. Similarly, Hilbert and Mike envisioned a natural order in which the geometry of fields is foremost and twists manifest themselves as particles.

With a geometry of sunbeams, the soul lays the foundations of nature.

You have to get old because of the geometry of spacetime.

Fractals, therefore, are also taken as a type of art, denoting the nature of the world around us, as well as the infinity and continuity that are reflected almost evidently in them.

With this, in a powerful sense, our Question has been answered. The world, insofar as we speak of the world of Chemistry, biology, astrophysics, engineering, and everyday life, does embody beautiful ideas. The Core, which governs those domains, is profoundly rooted in concepts of symmetry and geometry, as we have seen. And it works its will, in quantum theory, through music-like rules. Symmetry really does determine structure. A pure and perfect Music of the Spheres really does animate the soul of reality. Plato and Pythagoras: We salute you!

our civil rights have no dependance on our religious opinions, any more than our opinions in physics or geometry; that therefore the proscribing any citizen as unworthy the public confidence by laying upon him an incapacity of being called to offices of trust and emolument, unless he profess or renounce this or that religious opinion, is depriving him injuriously of those privileges and advantages to which, in common with his fellow citizens, he has a natural right; that it tends also to corrupt the principles of that very religion it is meant to encourage,

Mother Nature did not attend high school geometry courses or read the books of Euclid of Alexandria. Her geometry is jagged, but with a logic of its own and one that is easy to understand.

No very good sense can be given to the idea that the elements of Euclidean geometry may be found in nature because either everything is found in nature or nothing is. Euclidean geometry is a theory, and the elements of a theory may be interpreted only in terms demanded by the theory itself. Euclid’s axioms are satisfied in the Euclidean plane. Nature has nothing to do with it.

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.

Hatha yoga is a way of working with the body, a way of disciplining, purifying, and preparing it for higher levels of energy and for greater possibilities. Hatha yoga is not exercise. It is, instead, about understanding the mechanics of the body, creating a certain atmosphere, and then using physical postures to channel or drive your energy in specific directions. This is the aim of the various asanas, or postures. That kind of posture that allows you to access your higher nature is a yogasana. It is the science of aligning your inner geometry with the cosmic geometry.

Reverence for the natural environment, and experiencing the interconnectedness between all things has long guided me to create watercolor paintings of beauty and spirit. Life's continuing adventure has led me into an exciting exploration into the wisdom and symbolic imagery of Sacred Geometry. These paintings act as a bridge between this reality and a metaphorical world of healing, continuity, and transformation. I use multiple transparent watercolor glazes coupled with image overlapping techniques, and sacred geometry to produce visions of a multi-dimensional reality. It is my intention to create art that embodies the vibration of Universal Love and expresses the joy and gratitude I feel for the honor of being part of this earthwalk." ~Blessings, Francene~

The patterns of tiles created by the Moors are of secondary interest: it is the underlying group of symmetries which preserve aspects of the patterns that defines the geometry of the [Alhambra's] murals.

A therapist who fears dependence will tell his patient, sometimes openly, that the urge to rely is pathologic. In doing so he denigrates a cardinal tool. A parent who rejects a child's desire to depend raises a fragile person. Those children, grown to adulthood, are frequently among those who come for help. Shall we tell them again that no one can find an art to lean on, that each alone must work to ease a private sorrow? Then we shall repeat and experiment already conducted; many know its result only too well. If patient and therapist are to proceed together down a curative path, they must allow limbic regulation and its companion moon, dependence, to make the revolutionary magic. Many therapists believe that reliance fosters a detrimental dependency. Instead, they say, patients should be directed to "do it for themselves" - as if they possess everything but the wit to throw that switch and get on with their lives. But people do not learn emotional modulation as they do geometry or the names of state capitals. They absorb the skill from living in the presence of an adept external modulator, and they learn it implicitly. Knowledge leaps the gap from one mind to the other, but the learner does not experience the transferred information as an explicit strategy. Instead, a spontaneous capacity germinates and becomes a natural part of the self, like knowing how to ride a bike or tie one's shoes. The effortful beginnings fade and disappear from memory. (171)

It is not very easy to see,” Mircea Eliade writes, “how the discovery that the primal laws of geometry were due to the empirical necessities of the irrigation of the Nile Delta can have any bearing on the validity or otherwise of those laws.” We can argue here in the same way. For it is really no easier to understand how the fact that the first emergence of the idea of God may possibly have been provoked by a particular spectacle, or have been linked to a particular experience of a sensible nature, could affect the validity of the idea itself. In each case the problem of its birth from experience and the problem of its essence or validity are distinct. The problems of surveying no more engendered geometry than the experience of storm and sky engendered the idea of God. He important thing is to consider the idea in itself; not the occasion of its birth, but its inner constitution. If the idea of God in the mind of man is real, then no fact accessible to history or psychology or sociology, or to any other scientific discipline, can really be its generating cause.

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

But surely beauty is no idea belonging to mensuration; nor has it anything to do with calculation and geometry. If it had, we might then point out some certain measures which we could demonstrate to be beautiful, either as simply considered, or as related to others; and we could call in those natural objects, for whose beauty we have no voucher but the sense, to this happy standard, and confirm the voice of our passions by the determination of our reason.

Geometry has two great treasures: one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel. —Johannes Kepler

Just solving certain theorems makes waves in the Platonic over-space. Pump lots of power through a grid tuned carefully in accordance with the right parameters—which fall naturally out of the geometry curve I mentioned, which in turn falls easily out of the Turing theorem—and you can actually amplify these waves, until they rip honking great holes in spacetime and let congruent segments of otherwise-separate universes merge. You really don’t want to be standing at ground zero when that happens.

The Philosophy of Tea is not mere aestheticism in the ordinary acceptance of the term, for it express cojointly with ethics and religion our whole point of view about man and nature. It is hygiene, for it enforces cleanliness; it is economics, for it shows comfort in simplicity rather than in the complex and costly; it is moral geometry, inasmuch as it defines our sense of proportion to the universe. It represents the true spirit of Eastern democracy by making all its votaries aristocrats in taste.

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''.

One day, soon after her disappearance, an attack of abominable nausea forced me to pull up on the ghost of an old mountain road that now accompanied, now traversed a brand new highway, with its population of asters bathing in the detached warmth of a pale-blue afternoon in late summer. After coughing myself inside out I rested a while on a boulder and then thinking the sweet air might do me good, walked a little way toward a low stone parapet on the precipice side of the highway. Small grasshoppers spurted out of the withered roadside weeds. A very light cloud was opening its arms and moving toward a slightly more substantial one belonging to another, more sluggish, heavenlogged system. As I approached the friendly abyss, I grew aware of a melodious unity of sounds rising like vapor from a small mining town that lay at my feet, in a fold of the valley. One could make out the geometry of the streets between blocks of red and gray roofs, and green puffs of trees, and a serpentine stream, and the rich, ore-like glitter of the city dump, and beyond the town, roads crisscrossing the crazy quilt of dark and pale fields, and behind it all, great timbered mountains. But even brighter than those quietly rejoicing colors - for there are colors and shades that seem to enjoy themselves in good company - both brighter and dreamier to the ear than they were to the eye, was that vapory vibration of accumulated sounds that never ceased for a moment, as it rose to the lip of granite where I stood wiping my foul mouth. And soon I realized that all these sounds were of one nature, that no other sounds but these came from the streets of the transparent town, with the women at home and the men away. Reader! What I heard was but the melody of children at play, nothing but that, and so limpid was the air that within this vapor of blended voices, majestic and minute, remote and magically near, frank and divinely enigmatic - one could hear now and then, as if released, an almost articulate spurt of vivid laughter, or the crack of a bat, or the clatter of a toy wagon, but it was all really too far for the eye to distinguish any movement in the lightly etched streets. I stood listening to that musical vibration from my lofty slope, to those flashes of separate cries with a kind of demure murmur for background, and then I knew that the hopelessly poignant thing was not Lolita's absence from my side, but the absence of her voice from that concord.

The science of Chaos teaches us that everything is interconnected, but the contemporary developments in neuroscience, getting started with the brain neurons and their multiple connections, reveal the topology of the brain, a miniature of the universal geometry of everything.

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.

Each week I plot your equations dot for dot, xs against ys in all manner of algebraical relation, and every week they draw themselves as commonplace geometry, as if the world of forms were nothing but arcs and angles. God’s truth, Septimus, if there is an equation for a curve like a bell, there must be an equation for one like a bluebell, and if a bluebell, why not a rose? Do we believe nature is written in numbers? Septimus We do. Thomasina Then why do your equations only describe the shapes of manufacture? Septimus I do not know. Thomasina Armed thus, God could only make a cabinet.

But by 1912, Einstein had come to appreciate that math could be a tool for discovering—and not merely describing—nature’s laws. Math was nature’s playbook. “The central idea of general relativity is that gravity arises from the curvature of spacetime,” says physicist James Hartle. “Gravity is geometry.

What he discovered was that these intervals were harmonious because the relationship between them was a precise and simple mathematical ratio. This series, which we now know as the harmonic series, confirmed for him that the elegance of the mathematics he had found in abstract geometry also existed in the natural world.

I had better say something here about this question of age, since it is particularly important for mathematicians. No mathematician should ever allow himself to forget that mathematics, more than any other art or science, is a young man's game. To take a simple illustration at a comparatively humble level, the average age of election to the Royal Society is lowest in mathematics. We can naturally find much more striking illustrations. We may consider, for example, the career of a man who was certainly one of the world's three greatest mathematicians. Newton gave up mathematics at fifty, and had lost his enthusiasm long before; he had recognized no doubt by the time he was forty that his greatest creative days were over. His greatest idea of all, fluxions and the law of gravitation, came to him about 1666 , when he was twentyfour—'in those days I was in the prime of my age for invention, and minded mathematics and philosophy more than at any time since'. He made big discoveries until he was nearly forty (the 'elliptic orbit' at thirty-seven), but after that he did little but polish and perfect. Galois died at twenty-one, Abel at twenty-seven, Ramanujan at thirty-three, Riemann at forty. There have been men who have done great work a good deal later; Gauss's great memoir on differential geometry was published when he was fifty (though he had had the fundamental ideas ten years before). I do not know an instance of a major mathematical advance initiated by a man past fifty. If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself.

Long before being artists, we are artisans; and all fabrication, however rudimentary, lives on likeness and repetition, like the natural geometry which serves as its fulcrum. Fabrication works on models which it sets out to reproduce; and even when it invents, it proceeds, or imagines itself to proceed, by a new arrangement of elements already known. Its principle is that “we must have like to produce like.” In short, the strict application of the principle of finality, like that of the principle of mechanical causality, leads to the conclusion that “all is given.” Both principles say the same thing in their respective languages, because they respond to the same need.

In their later years, each (Einstein and Schrödinger) hoped to find a unified field theory that would fill in the gaps of quantum physics and unite the forces of nature. By extending general relativity to include all of the natural forces, such a theory would replace matter with pure geometry - fulfilling the dream of the Pythagoreans, who believed that "all is number".

What can you prove about space? How do you know where you are? Can space be curved? How many dimensions are there? How does geometry explain the natural order and unity of the cosmos? These are the questions behind the five geometric revolutions of world history. It started with a little scheme hatched by Pythagoras: to employ mathematics as the abstract system of rules that can model the physical universe.

Benoit Mandelbrot can be considered the Euclid of fractal geometry. He has collected the observations of mathematicians concerned with "monsters," or objects not definable by euclidean geometry. By combining the work of these mathematicians with his own insight, he has created a geometry of nature that thrives on asymmetry and roughness. Mandelbrot has said that "mountains are not cones, and clouds are not spheres.

Do you remember, Meir, that epigram quoted in the name of Rabbi Johanan ben Zaccai: 'There is no truth unless there be a faith on which it may rest'? Ironically enough the only sure principle I have achieved is this which I have known almost all my life. And it is so. For all truths rest ultimately on some act of faith, geometry on axioms, the sciences on the assumptions of the objective existence and orderliness of the world of nature. In every realm one must lay down postulates or he shall have nothing at all. So with morality and religion. Faith and reason are not antagonists. On the contrary, salvation is through the commingling of the two, the former to establish first premises, the latter to purify them of confusion and to draw the fullness of their implications. It is not certainty which one acquires so, only plausibility, but that is the best we can hope for.

The swirling lines of snow were composed of separate flakes, and each flake was a cluster of separate ice crystals--scientists had counted over a hundred of them in a single flake. Under the microscope each minuscule crystal, colorless and transparent, revealed a secret symmetry: six sides, the outward expression of an inward geometry of frozen molecules of water. But the real wonder was that no two crystals were precisely alike. In one of this father's camera magazines he had seen a stunning display of photomicrographs, and what was most amazing about the enlarged crystals was that each contained in its center a whole world of intricate six-sided designs, caused by microscopic air pockets. For no conceivable reason, Nature in a kind of exuberance created an inexhaustible outpouring of variations on a single form. A snowstorm was a fall of jewels, a delirium of hexagons--clearly the work of a master animator.

We can roll up two-dimensional graphene to make one-dimensional tubes, the so-called nanotubes. This can be done in many ways, giving nanotubes with different radii and pitches (see plate FF). Nanotubes that differ only slightly in geometry can have radically different physical properties. It is a triumph of quantum theory that these delicate properties can be predicted unambiguously, purely through calculation, and that they agree with experimental measurements.

With the invention of the fractals, what was once unrecognizable became recognized, and what was vague became distinguishable even in plain graphics.  One fact becomes clear: nature is full of beauty as the colours of infinity pictured in fractals.  They may appear vague in the initial view, but the beauty and precision become apparent the minute the colors appear, making us realize that with mystery comes the reality that the universe is not as alien as it may seem. 

What tends to be forgotten, amid all the cheerleading for today’s technology, is that people in ancient times might have lacked our current theoretical understanding of nature, but they were perfectly capable of noticing what worked and what didn’t, drawing rational conclusions on the basis of experience, and trying out new techniques to expand their ability to work with natural phenomena—even when their theories about the nature of those phenomena strike us as primitive or absurd.

On the SB5 Stanford-Binet intelligence test Isaiah’s reasoning scores were near genius levels. His abilities came naturally but were honed in his math classes. He was formally introduced to inductive reasoning in geometry, a tenth-grade subject he took in the eighth. His teacher, Mrs. Washington, was a severe woman who looked to be all gristle underneath her brightly colored pantsuits. Lavender, Kelly green, peach. She talked to the class like somebody had tricked her into it. “All

Before Einstein, geometry was thought to be part of the laws. Einstein revealed that the geometry of space is evolving in time, according to other, deeper laws. It is important to absorb this point completely. The geometry of space is not part of the laws of nature. There is therefore nothing in those laws that specifies what the geometry of space is. Thus, before solving the equations of Einstein's general theory of relativity, you don't have any idea what the geometry of space is. You find out only after you solve the equations.

The city: grime, glamor, geometries of glass, steel, and concrete. Intractable, it rises from nature, like proud Babel, only to lie arthwart our will, astride our being. Or so it often seems. Yet immanent in that gritty structure is another: invisible, imaginary, made of dreams and desire, agent of all our transformations. It is that other city I want here to invoke…Immaterial, that city in-formed history from the start, molding human space and time ever since time and space molded them selves to the wagging tongue. IHAB HASSAN, Cities of Mind, Urban Words

I really think I'd like to be a minister's wife when I grow up, Marilla. A minister mightn't mind my red hair because he wouldn't be thinking of such worldly things. But then of course one would have to be naturally good and I'll never be that, so I suppose there's no use in thinking about it. Some people are naturally good, you know, and others are not. I'm one of the others. Mrs. Lynde says I'm full of original sin. No matter how hard I try to be good I can never make such a success of it as those who are naturally good. It's a good deal like geometry, I expect. But don't you think the trying so hard ought to count for something?

An “infinite number”? For Leonardo, that was not just a figure of speech. When he spoke of the infinite variety in nature, and especially of phenomena such as flowing water, he was making a distinction based on his preference for analog over digital systems. In an analog system, there are infinite gradations. That applies to most of the things that fascinated him: sfumato shadows, colors, movement, waves, the passage of time, the flow of fluids. That is why he believed that geometry was better than arithmetic at describing nature, and even though calculus had not yet been invented, he seemed to sense the need for such a mathematics of continuous quantities.

Die Logik ist insofern die schwerste Wissenschaft, als sie es nicht mit Anschauungen, nicht einmal wie die Geometrie mit abstrakten sinnlichen Vorstellungen, sondern mit reinen Abstraktionen zu tun hat und eine Kraft und Geübtheit erfordert, sich in den reinen Gedanken zurückzuziehen, ihn festzuhalten und in solchem sich zu bewegen. Auf der andern Seite könnte sie als die leichteste angesehen werden, weil der Inhalt nichts als das eigene Denken und dessen geläufige Bestimmungen und diese zugleich die einfachsten und das Elementarische sind. Sie sind auch das Bekannteste, Sein, Nichts usf., Bestimmtheit, Größe usw., Ansichsein, Fürsichsein, Eines, Vieles usw.

An Act for establishing religious Freedom. Section 1 Whereas, Almighty God hath created the mind free; That all attempts to influence it by temporal punishments or burthens, or by civil incapacitations tend only to beget habits of hypocrisy and meanness, and therefore are a departure from the plan of the holy author of our religion, who being Lord, both of body and mind yet chose not to propagate it by coercions on either, as was in his Almighty power to do, That the impious presumption of legislators and rulers, civil as well as ecclesiastical, who, being themselves but fallible and uninspired men have assumed dominion over the faith of others, setting up their own opinions and modes of thinking as the only true and infallible, and as such endeavouring to impose them on others, hath established and maintained false religions over the greatest part of the world and through all time; That to compel a man to furnish contributions of money for the propagation of opinions, which he disbelieves is sinful and tyrannical; That even the forcing him to support this or that teacher of his own religious persuasion is depriving him of the comfortable liberty of giving his contributions to the particular pastor, whose morals he would make his pattern, and whose powers he feels most persuasive to righteousness, and is withdrawing from the Ministry those temporary rewards, which, proceeding from an approbation of their personal conduct are an additional incitement to earnest and unremitting labours for the instruction of mankind; That our civil rights have no dependence on our religious opinions any more than our opinions in physics or geometry, That therefore the proscribing any citizen as unworthy the public confidence, by laying upon him an incapacity of being called to offices of trust and emolument, unless he profess or renounce this or that religious opinion, is depriving him injuriously of those privileges and advantages, to which, in common with his fellow citizens, he has a natural right, That it tends only to corrupt the principles of that very Religion it is meant to encourage, by bribing with a monopoly of worldly honours and emoluments those who will externally profess and conform to it; That though indeed, these are criminal who do not withstand such temptation, yet neither are those innocent who lay the bait in their way; That to suffer the civil magistrate to intrude his powers into the field of opinion and to restrain the profession or propagation of principles on supposition of their ill tendency is a dangerous fallacy which at once destroys all religious liberty because he being of course judge of that tendency will make his opinions the rule of judgment and approve or condemn the sentiments of others only as they shall square with or differ from his own; That it is time enough for the rightful purposes of civil government, for its officers to interfere when principles break out into overt acts against peace and good order; And finally, that Truth is great, and will prevail if left to herself, that she is the proper and sufficient antagonist to error, and has nothing to fear from the conflict, unless by human interposition disarmed of her natural weapons free argument and debate, errors ceasing to be dangerous when it is permitted freely to contradict them.

The gnarled pine, I would have said, touch it. This is China. Horticulturalists around the world have come to study it. Yet no one has ever been able to explain why it grows like a corkscrew, just as no one can adequately explain China. But like that tree, there it is, old, resilient, and oddly magnificent. Within that tree are the elements in nature that have inspired Chinese artists for centuries: gesture over geometry, subtlety over symmetry, constant flow over static form. And the temples, walk and touch them. This is China. Don't merely stare at these murals and statues. Fly up to the crossbeams, get down on your hands and knees, and press your head to the floor tiles. Hide behind that pillar and come eye to eye with its flecks of paint. Imagine that you are the interior decorator who is a thousand years in age. Start with a bit of Tibetan Buddhism, plus a dash each of animism and Taoism. A hodgepodge, you say? No, what is in those temples is an amalgam that is pure Chinese, a lovely shabby elegance, a glorious new motley that makes China infinitely intriguing. Nothing is ever completely thrown away and replaced. If one period of influence falls out of favor, it is patched over. The old views still exist, one chipped layer beneath, ready to pop through with the slightest abrasion. That is the Chinese aesthetic and also its spirit. Those are the traces that have affected all who have traveled along China's roads.

Because a lake is conceived as having only two primary dimensions, you can't swim inside the lake, though that would seem to make geometric sense. Lederer asks why we say that something can be underwater or underground even though it's surrounded by, not beneath, the water or the ground. It's because water and ground are conceived as 2-D surfaces, not 3-D volumes, geologically improbable though that is. The dimensionality of an object is also the aspect of its geometry that modifiers "see" when they combine with it in a phrase. A big CD, for example, has to have an above-average diameter, not an above-standard thickness (that could only be a thick CD), and a big lake has to be one with an unusually large area, regardless of its depth; it can't be a few yards wide and a mile deep.

We are in the habit of visualizing .. the history of science as a steady, cumulative process,.. where each epoch adds some new item of knowledge to the legacy of the past, making the temple of science grow brick by brick to ever greater height.. In fact, .. the philosophy of nature evolved by occasional leaps and bounds alternating with delusional pursuits, culs-de-sac, regressions, periods of blindness and amnesia. The great discoveries .. were sometimes the unexpected by-products of a chase after quite different hares. At other times, the process of discovery consisted merely in the cleaning away of the rubbish that blocked the path, or in the rearranging of existing items of knowledge in a different pattern.. Europe knew less geometry in the fifteenth century than in Archimedes' time.

In this book, you will encounter various interesting geometries that have been thought to hold the keys to the universe. Galileo Galilei (1564-1642) suggested that "Nature's great book is written in mathematical symbols." Johannes Kepler (1571-1630) modeled the solar system with Platonic solids such as the dodecahedron. In the 1960s, physicist Eugene Wigner (1902-1995) was impressed with the "unreasonable effectiveness of mathematics in the natural sciences." Large Lie groups, like E8-which is discussed in the entry "The Quest for Lie Group E8 (2007)"- may someday help us create a unified theory of physics. in 2007, Swedish American cosmologist Max Tegmark published both scientific and popular articles on the mathematical universe hypothesis, which states that our physical reality is a mathematical structure-in other words, our universe in not just described by mathematics-it is mathematics.

The realization that symmetry is the key to the understanding of the properties of subatomic particles led to an inevitable question: Is there an efficient way to characterize all of these symmetries of the laws of nature? Or, more specifically, what is the basic theory of transformations that can continuously change one mixture of particles into another and produce the observed families? By now you have probably guessed the answer. The profound truth in the phrase I have cited earlier in this book revealed itself once again: "Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." The physicists of the 1960s were thrilled to discover that mathematicians had already paved the way. Just as fifty years earlier Einstein learned about the geometry tool-kit prepared by Riemann, Gell-Mann and Ne'eman stumbled upon the impressive group-theoretical work of Sophus Lie.

You too can make the golden cut, relating the two poles of your being in perfect golden proportion, thus enabling the lower to resonate in tune with the higher, and the inner with the outer. In doing so, you will bring yourself to a point of total integration of all the separate parts of your being, and at the same time, you will bring yourself into resonance with the entire universe.... Nonetheless the universe is divided on exactly these principles as proven by literally thousands of points of circumstantial evidence, including the size, orbital distances, orbital frequencies and other characteristics of planets in our solar system, many characteristics of the sub-atomic dimension such as the fine structure constant, the forms of many plants and the golden mean proportions of the human body, to mention just a few well known examples. However the circumstantial evidence is not that on which we rely, for we have the proof in front of us in the pure mathematical principles of the golden mean.

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.

Now, if on the one hand it is very satisfactory to be able to give a common ground in the theory of knowledge for the many varieties of statements concerning space, spatial configurations, and spatial relations which, taken together, constitute geometry, it must on the other hand be emphasised that this demonstrates very clearly with what little right mathematics may claim to expose the intuitional nature of space. Geometry contains no trace of that which makes the space of intuition what it is in virtue of its own entirely distinctive qualities which are not shared by “states of addition-machines” and “gas-mixtures” and “systems of solutions of linear equations”. It is left to metaphysics to make this “comprehensible” or indeed to show why and in what sense it is incomprehensible. We as mathematicians have reason to be proud of the wonderful insight into the knowledge of space which we gain, but, at the same time, we must recognise with humility that our conceptual theories enable us to grasp only one aspect of the nature of space, that which, moreover, is most formal and superficial.

The influence of geometry upon philosophy and scientific method has been profound. Geometry, as established by the Greeks, starts with axioms which are (or are deemed to be) self-evident, and proceeds, by deductive reasoning, to arrive at theorems that are very far from self-evident. The axioms and theorems are held to be true of actual space, which is something given in experience. It thus appeared to be possible to discover things about the actual world by first noticing what is self-evident and then using deduction. This view influenced Plato and Kant, and most of the intermediate philosophers. When the Declaration of Independence says 'we hold these truths to be self-evident', it is modelling itself on Euclid. The eighteenth-century doctrine of natural rights is a search for Euclidean axioms in politics.8 The form of Newton's Principia, in spite of its admittedly empirical material, is entirely dominated by Euclid. Theology, in its exact scholastic forms, takes its style from the same source. Personal religion is derived from ecstasy, theology from mathematics; and both are to be found in Pythagoras.

One method that Einstein employed to help people visualize this notion was to begin by imagining two-dimensional explorers on a two-dimensional universe, like a flat surface. These “flatlanders” can wander in any direction on this flat surface, but the concept of going up or down has no meaning to them. Now, imagine this variation: What if these flatlanders’ two dimensions were still on a surface, but this surface was (in a way very subtle to them) gently curved? What if they and their world were still confined to two dimensions, but their flat surface was like the surface of a globe? As Einstein put it, “Let us consider now a two-dimensional existence, but this time on a spherical surface instead of on a plane.” An arrow shot by these flatlanders would still seem to travel in a straight line, but eventually it would curve around and come back—just as a sailor on the surface of our planet heading straight off over the seas would eventually return from the other horizon. The curvature of the flatlanders’ two-dimensional space makes their surface finite, and yet they can find no boundaries. No matter what direction they travel, they reach no end or edge of their universe, but they eventually get back to the same place. As Einstein put it, “The great charm resulting from this consideration lies in the recognition that the universe of these beings is finite and yet has no limits.” And if the flatlanders’ surface was like that of an inflating balloon, their whole universe could be expanding, yet there would still be no boundaries to it.10 By extension, we can try to imagine, as Einstein has us do, how three-dimensional space can be similarly curved to create a closed and finite system that has no edge. It’s not easy for us three-dimensional creatures to visualize, but it is easily described mathematically by the non-Euclidean geometries pioneered by Gauss and Riemann. It can work for four dimensions of spacetime as well. In such a curved universe, a beam of light starting out in any direction could travel what seems to be a straight line and yet still curve back on itself. “This suggestion of a finite but unbounded space is one of the greatest ideas about the nature of the world which has ever been conceived,” the physicist Max Born has declared.

It was in postwar Paris that Mandelbrot began this quest in earnest. Uncle Szolem urged him to attend the École Normale Supérieure, France’s most rarefied institution of higher learning, where Mandelbrot had earned entry at the age of twenty (one of only twenty Frenchmen to do so). But the aridly abstract style of mathematics practiced there was uncongenial to him. At the time, the École Normale—dite normale, prétendue supérieure, says the wag—was dominated in mathematics by a semisecret cabal called Bourbaki. (The name Bourbaki was jocularly taken from a hapless nineteenth-century French general who once tried to shoot himself in the head but missed.) Its leader was André Weil, one of the supreme mathematicians of the twentieth century (and the brother of Simone Weil). The aim of Bourbaki was to purify mathematics, to rebuild it on perfectly logical foundations untainted by physical or geometric intuition. Mandelbrot found the Bourbaki cult, and Weil in particular, “positively repellent.” The Bourbakistes seemed to cut off mathematics from natural science, to make it into a sort of logical theology. They regarded geometry, so integral to Mandelbrot’s Keplerian dream, as a dead branch of mathematics, fit for children at best.

But if some mind very different from ours were to look upon some property of some curved line as we do on the evenness of a straight line, he would not recognize as such the evenness of a straight line; nor would he arrange the elements of his geometry according to that very different system, and would investigate quite other relationships as I have suggested in my notes. We fashion our geometry on the properties of a straight line because that seems to us to be the simplest of all. But really all lines that are continuous and of a uniform nature are just as simple as one another. Another kind of mind which might form an equally clear mental perception of some property of any one of these curves, as we do of the congruence of a straight line, might believe these curves to be the simplest of all, and from that property of these curves build up the elements of a very different geometry, referring all other curves to that one, just as we compare them to a straight line. Indeed, these minds, if they noticed and formed an extremely clear perception of some property of, say, the parabola, would not seek, as our geometers do, to rectify the parabola, they would endeavor, if one may coin the expression, to parabolify the straight line.

To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The definition of the Golden Ratio emerged originally from the axioms of Euclidean geometry; the definition of the Fibonacci sequence from the axioms of the theory of numbers. Yet the fact that the ratio of successive Fibonacci numbers converges to the Golden Ratio was imposed on us-humans had not choice in the matter. Therefore, mathematical objects, albeit imaginary, do have real properties. Second, the explanation of the unreasonable power of mathematics cannot be based entirely on evolution in the restricted sense. For example, when Newton proposed his theory of gravitation, the data that he was trying to explain were at best accurate to three significant figures. Yet his mathematical model for the force between any two masses in the universe achieved the incredible precision of better than one part in a million. Hence, that particular model was not forced on Newton by existing measurements of the motions of planets, nor did Newton force a natural phenomenon into a preexisting mathematical pattern. Furthermore, natural selection in the common interpretation of that concept does not quite apply either, because it was not the case that five competing theories were proposed, of which one eventually won. Rather, Newton's was the only game in town!

As Wheeler explained at the time, “Nature at the quantum level is not a machine that goes its inexorable way. Instead what answer we get depends on the question we put, the experiment we arrange, the registering device we choose. We are inescapably involved in bringing about that which appears to be happening.” Posing an example, he set up a fascinating mind-experiment. Utilizing the fact that a strong mass or gravity warps space-time, he imagined a small, distant light source like a quasar, whose bits of light must traverse the vicinity of a foreground massive galaxy en route to our eyes. If the geometry is correct—if the distant quasar, the intermediate massive galaxy, and our Earth are all on a perfectly straight line—each photon’s path will be warped to pass either above or below that galaxy. (The photon cannot go straight through the foreground galaxy because the galaxy’s mass has altered the actual geometry of space-time so that the shortest “highway” from the quasar to Earth is no longer a seemingly straight line. In any case, the material in the foreground galaxy would block the quasar’s light from penetrating it, even if it tried to travel that way.) Then it will continue for billions of more years before reaching our telescopes here on Earth (see Figure 8-4). If they really had a 50/50 chance of taking either route, which path did each photon traverse? Wheeler’s conclusion: The event, billions of years ago, didn’t really happen until we observe it today. Only now will a particular photon pass above or below the foreground galaxy billions of years ago. In other words, the past isn’t something that has already irrevocably occurred. Rather, long-ago events depend on the present observer. Until they’re observed at this moment, the events didn’t really unfold, but lurked in a blurry probabilistic state, all ready to become an actual “past” occurrence only upon our current observation. This astonishing possibility is called retrocausality.

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

My bedroom is separated from the main body of my house so that I have to go outside and cross some pseudo-Japanese stepping stones in order to go to sleep at night. Often I get rained on a little bit on my way to bed. It’s a benediction. A good night kiss. Romantic? Absolutely. And nothing to be ashamed of. If reality is a matter of perspective, then the romantic view of the world is as valid as any other - and a great deal more rewarding. It makes of life and unpredictable adventure rather that a problematic equation. Rain is the natural element for romanticism. A dripping fir is a hundred times more sexy than a sunburnt palm tree, and more primal and contemplative, too. A steady, wind-driven rain composed music for the psyche. It not only nurtures and renews, it consecrates and sanctifies. It whispers in secret languages about the primordial essence of things. Obviously, then, the Pacific Northwest's customary climate is perfect for a writer. It's cozy and intimate. Reducing temptation (how can you possibly play on the beach or work in the yard?), it turns a person inward, connecting them with what Jung called "the bottom below the bottom," those areas of the deep unconscious into which every serious writer must spelunk. Directly above my writing desk there is a skylight. This is the window, rain-drummed and bough-brushed, through which my Muse arrives, bringing with her the rhythms and cadences of cloud and water, not to mention the latest catalog from Victoria's Secret and the twenty-three auxiliary verbs. Oddly enough, not every local author shares my proclivity for precipitation. Unaware of the poetry they're missing, many malign the mist as malevolently as they non-literary heliotropes do. They wring their damp mitts and fret about rot, cursing the prolonged spillage, claiming they're too dejected to write, that their feet itch (athlete's foot), the roof leaks, they can't stop coughing, and they feel as if they're slowly being digested by an oyster. Yet the next sunny day, though it may be weeks away, will trot out such a mountainous array of pagodas, vanilla sundaes, hero chins and god fingers; such a sunset palette of Jell-O, carrot oil, Vegas strip, and Kool-Aid; such a sea-vista display of broad waters, firred islands, whale spouts, and boat sails thicker than triangles in a geometry book, that any and all memories of dankness will fizz and implode in a blaze of bedazzled amnesia. "Paradise!" you'll hear them proclaim as they call United Van Lines to cancel their move to Arizona.

That the line does not consist of points, nor the plane of lines, follows from their concepts, for the line is the point existing outside of itself relating itself to space, and suspending itself and the plane is just as much the suspended line existing outside of itself.-Here the point is represented as the first and positive entity, and taken as the starting point. The converse, though, is also true: in as far as space is positive, the plane is the first negation and the line is the second, which, however, is in its truth the negation relating self to self, the point. The necessity of the transition is the same.- The other configurations of space considered by geometry are further qualitative limitations of a spatial abstraction, of the plane, or of a limited spatial whole. Here there occur a few necessary moments, for example, that the triangle is the first rectilinear figure, that all other figures must, to be determined, be reduced to it or to the square, and so on.-The principle of these figures is the identity of the understanding, which determines the figurations as regular, and in this way grounds the relationships and sets them in place, which it now becomes the purpose of science to know. Negativity, which as point relates itself to space and in space develops its determinations as line and plane, is, however, in the sphere of self-externality equally for itself and appearing indifferent to the motionless coexistence of space. Negativity, thus posited for itself is time. Time, as the negative unity of being outside of itself, is just as thoroughly abstract, ideal being: being which, since it is, is not, and since it is not, is If these determinations (of Kant, the forms of intuition or sensation) are applied to space and time, then space is abstract objectivity, whereas time is abstract subjectivity (“the pure I=I of self-consciousness” but still the concept is in its pure externality). Time is just as continuous as space, for it is abstract negativity relating itself to itself and in this abstraction there is as yet no real difference. In time, it is said, everything arises and passes away, or rather, there appears precisely the abstraction of arising and falling away. If abstractions are made from everything, namely, from the fullness of time just as much as from the fullness of space, then there remains both empty time and empty space left over; that is, there are then posited these abstractions of exteriority.-But time itself is this becoming, this existing abstraction, the Chronos who gives birth to everything and destroys his offspring.-That which is real, however, is just as identical to as distinct from time. Everything is transitory that is temporal, that is, exists only in time or, like the concept, is not in itself pure negativity. To be sure, this negativity is in everything as its immanent, universal essence, but the temporal is not adequate to this essence, and therefore relates to this negativity in terms of its power. Time itself is eternal, for it is neither just any time, nor the moment now, but time as time is its concept. The concept, however, in its identity with itself I= I, is in and for itself absolute negativity and freedom. Time, is not, therefore, the power of the concept, nor is the concept in time and temporal; on the contrary, the concept is the power of time, which is only this negativity as externality.-The natural is therefore subordinate to time, insofar as it is finite; that which is true, by contrast, the idea, the spirit, is eternal. Thus the concept of eternity must not be grasped as if it were suspended time, or in any case not in the sense that eternity would come after time, for this would turn eternity into the future, in other words into a moment of time.

As a boy, I had the privilege of realizing that nature only moves and grows in precise, turbulent, spiraling flows. As an adult, I learned that human technology, in the main, tries to suppress turbulence. Nature doesn't waste the opportunity. It exploits the energy that is rolled up in turbulence. Birds, insects, fish, and the human heart clearly demonstrate the advantage of this strategy. Humans insist on traveling in straight lines and guzzle energy. Nature travels in spirals and sips energy. Truly grasping the significance of this simple fact throws open the door to reinventing the industrial world and gives us the tools to rescue our ailing planet, populations, and economy. By adapting and applying nature's spiraling geometries, I am confident that we can halve the world's energy consumption-without sacrifice.

Mantra of Geometry In Wheeler's poem, "matter" is a little too poetic. Matter can have several properties (for example, electric charge), but the curvature of space-time responds only to the total density of energy and momentum. So we should say instead: Energy-momentum tells space-time how to curve. Also, forces other than gravity influence how matter moves. Those forces will lead to deviations from the straightest possible (geodesic) paths. What we should say is therefore: Space-time tells energy-momentum what straight is (in space-time). And so, putting it all together: Energy-momentum tells space-time how to curve. Space-time tells energy-momentum what straight is (in space-time). And now comes the Core Theory of electromagnetism: Electric charge tells electromagnetic property space how to curve. Electromagnetic property space tells electric charge what straight is (in electromagnetic property space). And of the weak force: Weak charge tells weak property space how to curve. Weak property space tells weak charge what straight is (in weak property space). And of the strong force: Strong charge tells strong property space how to curve. Strong property space tells strong charge what straight is (in strong property space). In the full Core Theory, including all four forces, matter has four kinds of properties: energy momentum, electric charge, weak charge, and strong charge. Particles of matter propagate through a more complex space than Wheeler allowed for, which includes electromagnetic, weak, and strong property spaces atop ordinary space-time. But matter follows, according to the Core, the same yin principle, adapted to this more complex environment: Keep going as straight as you can!

AN ANSWER TO OUR QUESTION Places of worship embody the aspirations of their architects, and the communities they represent, to ideal beauty. Their chosen means of expression feature color, geometry, and symmetry. Consider, in particular. the magnificent plate HH. Here the local geometry of the ambient surfaces and the local patterns of their color change as our gaze surveys them. It is a vibrant embodiment of anamorphy and anachromy-the very themes that our unveiling of Nature's deep design finds embodied at Nature's core. Does the world embody beautiful ideas? There is our answer, before our eyes: Yes. Color and geometry, symmetry, anachromy, and anamorphy, as ends in themselves, are only one branch of artistic beauty. Islam's injunction against representational art played an important part in bringing these forms of beauty to the fore, as did the physical constraint of structural stability (we need columns to support the weight of ceilings, and the arches and domes to distribute tension). Depictions of human faces, bodies, emotions, landscapes, historic scenes, and the like, when they are allowed, are far more common subjects for art than those austere beauties. The world does not, in its deep design, embody all forms of beauty, nor the ones that people without special study, or very unusual taste, find most appealing. But the world does, in its deep design, embody some forms of beauty that have been highly prized for their own sake, and have been intuitively associated with the divine.

Heraclitus theorized, by observing the natural world, that everything in existence was created from flow in nature-which physicists now agree to be true. Meanwhile, Plato saw particular angles and proportions everywhere and developed the science of geometry. More than two thousand years later, Einstein echoed Plato's understanding that "God ever geometrizes.

Leonardo da Vinci was perhaps the greatest biomimic of all time. Not only did he precisely adhere to nature's proportions in his art but also spent the last ten years of his life studying-even obsessing over-the geometry and motions of natural flow. Based on years of observing birds in flight, Leonardo, the world's first fluid dynamicist, designed flying wings, a helicopter, and numerous other machines. His understanding of how the human heart actually operates-through manipulation of whirlpools-has only been rediscovered in the past decade. Even five hundred years later, the depth of Leonardo's insights into the secrets of nature's form and function cause scientists to marvel.

The whole known universe is made of and according to nature's spiraling geometries-and nature uses them exclusively to move energy.

The few inches of air shrank. Matthew did not know who had first leaned toward the other, but did it really matter? One leaned and one met, and that was both the geometry and poetry of their kiss. Though Matthew had never before done this, it seemed a natural act. What was most alarming was the speed of his heart, which if it had been a horse might have reached Boston by first star. Something inside him seemed molten, like blue-flamed glass being changed and reshaped by the power of a breath. It was both strengthening and weakening, thrilling and frightening—again that conjunction of God and Devil that seemed to be at the essence of all things. It was a moment he would remember the rest of his life. Their lips remained sealed together, melded by bloodheat and heartbeat. Who drew away first was also unknown to Matthew, as time had slipped its boundaries like rain and river.