Geometry Shapes Quotes

I did not know then that this is what life is - just when you master the geometry of one world, it slips away, and suddenly again, you're swarmed by strange shapes and impossible angles.

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

But then, Cap'n Crunch in a flake form would be suicidal madness; it would last about as long, when immersed in milk, as snowflakes sifting down into a deep fryer. No, the cereal engineers at General Mills had to find a shape that would minimize surface area, and, as some sort of compromise between the sphere that is dictated by Euclidean geometry and whatever sunken treasure related shapes that the cereal aestheticians were probably clamoring for, they came up with this hard -to-pin-down striated pillow formation.

Besides, don’t you know basic geometry, Strategist Sima?” I make a triangle with my fingers and look through it. “A triangle is the strongest shape.

He deals the cards to find the answer The sacred geometry of chance The hidden law of a probable outcome The numbers lead a dance I know that the spades Are the swords of a soldier I know that the clubs are weapons of war I know that diamonds Mean money for this art But that's not the shape of my heart

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.

The paradox of education: what we most admire we put in a box and make dull.

It is remarkable how much long-term advantage people like us have gotten by trying to be consistently not stupid, instead of trying to be very intelligent.” – Charlie Munger

Remember, successful investing is based more on minimizing regret than maximizing gains.

The flakes stuck in my eyelashes. They fell on my sleeves. Huge. Flowers and stars. They fell onto each other, held their shapes, became small piles of perfect asterisks and blooms tumbled together in their discrete geometries like children’s blocks.

Before Jeremiah knew God, God knew Jeremiah: “Before I shaped you in the womb, I knew all about you.” This turns everything we ever thought about God around. We think that God is an object about which we have questions. We are curious about God. We make inquiries about God. We read books about God. We get into late-night bull sessions about God. We drop into church from time to time to see what is going on with God. We indulge in an occasional sunset or symphony to cultivate a feeling of reverence for God. But that is not the reality of our lives with God. Long before we ever got around to asking questions about God, God had been questioning us. Long before we got interested in the subject of God, God subjected us to the most intensive and searching knowledge. Before it ever crossed our minds that God might be important, God singled us out as important. Before we were formed in the womb, God knew us. We are known before we know. This realization has a practical result: no longer do we run here and there, panicked and anxious, searching for the answers to life. Our lives are not puzzles to be figured out. Rather, we come to God, who knows us and reveals to us the truth of our lives. The fundamental mistake is to begin with ourselves and not God. God is the center from which all life develops. If we use our ego as the center from which to plot the geometry of our lives, we will live eccentrically.

If you think the value of a woman is only in the curve of her hips and the shape of her breasts, you do not understand how to read beneath her jagged lines, the sacred geometry that make up her glorious heart and her beautiful mind.

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]

When I touch her, my fingers don’t question what she is. My body knows who she is. The strange thing about strangers is that they are unknown and known. There is a pattern to her, a shape I understand, a private geometry that numbers mine. She is a maze where I got lost years ago, and now find the way out. She is the missing map. She is the place that I am. She is a stranger. She is the strange that I am beginning to love.

The Ark of the Covenant is a Golden Rectangle because its rectangular shape is in the proportions of the Golden Ratio.

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

In other words, if geometry was top-down mathematics, the method of indivisibles was bottom-up mathematics.

I did not know then that this is what life is—just when you master the geometry of one world, it slips away, and suddenly again, you’re swarmed by strange shapes and impossible angles.

The ultimate reason for teaching kids to write a proof is not that the world is full of proofs. It's that the world is full of non-proofs, and grown-ups need to know the difference. It's hard to settle for a non-proof once you've really familiarized yourself with the genuine article.

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.

the pattern appears so ethereally, that it is hard to remember that the shape is an attractor. It is not just any trajectory of a dynamical system. It is the trajectory toward which all other trajectories converge.

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.

We teach our children the mathematics of certainty—geometry and trigonometry—but not the mathematics of uncertainty, statistical thinking. And we teach our children biology but not the psychology that shapes their fears and desires. Even experts, shockingly, are not trained how to communicate risks to the public in an understandable way. And there can be positive interest in scaring people: to get an article on the front page, to persuade people to relinquish civil rights, or to sell a product.

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.

People who have more money are not necessarily happier—though some are. If anything, money alleviates sadness more than it inspires joy.

Taking more risk produces more return is inaccurate. Instead, taking more risk increases the range of potential outcomes

was reminded, again, of the shapes of women, the impossible geometry into which I was meant to fold myself.

In life and literature, the good guys fight for others. The bad guys fight for themselves. Heroes have context.

Microchip manufactures, similarly, cannot make their transistors too thin, or the performance of these devices will suffer from electon leakage due to tunneling effects.

We know that different regions in the brain process shape, colour, and motion. Benoit Mandelbrot has hypothesized that perhaps there is a specific circuit in the brain to deal with fractal complexity.

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.

On what may be the last page he wrote in his notebooks, Leonardo drew four right triangles with bases of differing lengths (fig. 143). Inside of each he fit a rectangle, and then he shaded the remaining areas of the triangle. In the center of the page he made a chart with boxes labeled with the letter of each rectangle, and below it he described what he was trying to accomplish. As he had done obsessively over the years, he was using the visualization of geometry to help him understand the transformation of shapes. Specifically, he was trying to understand the formula for keeping the area of a right triangle the same while varying the lengths of its two legs. He had fussed with this problem, explored by Euclid, repeatedly over the years. It was a puzzle that, by this point in his life, as he turned sixty-seven and his health faded, might seem unnecessary to solve. To anyone other than Leonardo, it may have been.

People often complain that no one likes facts and numbers and reason and science anymore, but as someone who talks about those things in public, I can tell you that's not true. People love numbers, and are impressed by them, sometimes more than they should be.

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.

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

Richer humans become ensnared in a “luxury trap,” in which formerly unimaginable inventions evolve from mind-blowing to luxuries to taken-for-granted to necessary. I can’t imagine living without a refrigerator or airplanes, but my great-grandparents certainly could. Over time, the amazing becomes normal.

Often people think of developments in computation as arising when we make our computers more blazingly fast, so they can compute more stuff, bigger data. It's actually just as important to prune away big parts of the data that aren't relevant to the problem at hand! The fastest computation is the one you don't do.

Priority one is risk management. It is to protect ourselves from the potential for loss, especially catastrophe. In the human mind, losses weigh much more heavily than gains, so elevating risk management is the right thing to do. This step is about building the proper mindset, one that values avoiding mistakes over demonstrating brilliance.

Simplification is the smart path toward effectively managing expectations. In general terms, met expectations lead to temporary happiness and unmet ones lead to temporary sadness. The human mind is wired to avoid losses more than it is to achieve gains, so minimizing regret is more important in this process than is maximizing future upside.

She had heard of the genetic code that could shape an eye or hand from passing proteins. Deoxyribonucleic acid. It contained the entire set of instructions for constructing a respiratory system and a digestive one, as well as the grip of an infant’s hand. Chess was like that. The geometry of a position could be read and reread and not exhausted of possibility. You saw deeply into this layer of it, but there was another layer beyond that, and another.

She had heard of the genetic code that could shape an eye or hand from passing proteins. Deoxyribonucleic acid. It contained the entire set of instructions for constructing a respiratory system and a digestive one, as well as the grip of an infant's hand. Chess was like that. The geometry of a position could be read and reread and not exhausted of possibility. You saw deeply into the layer of it, but there was another layer beyond that, and another, and another.

I’ve come to believe that there are four enduring sources of a joyful life. I call these Connection, Control, Competence, and Context. Connection is the need to belong. Control is the need to direct one’s own destiny. Competence is the need to be good at something worthwhile. Context is the need for a purpose outside of one’s self. At the nexus of money and meaning the “Four C’s” are what is being underwritten. They sit at the heart of funded contentment. Let’s look at each of these in turn.

Let us wonder, Carole, at the genius of hyperbolic geometry, where the sum of the angles adds up to tales than 180 degrees Let us wonder at how the ancient Egyptians worked out how to measure an irregularly-shaped field Let us wonder how X was just a rare letter until algebra came along and made it something special that can be unraveled to reveal its inner value You see, maths is a process of discovery, Carole, it is like the exploration of space, the planets were always there, it just took us a long time to find them

But it wouldn’t have half the power of a story in which Jamie and Claire truly conquer real evil and thus show what real love is. Real love has real costs—and they’re worth it. I’ve always said all my books have a shape, and Outlander’s internal geometry consists of three slightly overlapping triangles. The apex of each triangle is one of the three emotional climaxes of the book: 1) when Claire makes her wrenching choice at the stones and stays with Jamie, 2) when she saves Jamie from Wentworth, and 3) when she saves his soul at the abbey. It would still be a good story if I’d had only 1 and 2—but (see above), the Rule of Three. A story that goes one, two, three, has a lot more impact than just a one–two punch.

When I touch her, my fingers don’t question what she is. My body knows who she is. The strange thing about strangers is that they are unknown and known. There is a pattern to her, a shape I understand, a private geometry that numbers mine. She is a maze where I got lost years ago, and now find the way out. She is the missing map. She is the place that I am. She is a stranger. She is the strange that I am beginning to love. And you may say that only death has brought us to this. That one intensity must match another. That we have found each other because there is no one else, nothing else to find. It doesn’t matter – not the reasons for the death, nor the explanation of the love. It is happening, both together, and it is where we are, both together.

The nuggets themselves are pillow-shaped and vaguely striated to echo piratical treasure chests. Now, with a flake-type of cereal, Randy’s strategy would never work. But then, Cap’n Crunch in a flake form would be suicidal madness; it would last about as long, when immersed in milk, as snowflakes sifting down into a deep fryer. No, the cereal engineers at General Mills had to find a shape that would minimize surface area, and, as some sort of compromise between the sphere that is dictated by Euclidean geometry and whatever sunken-treasure-related shapes that the cereal-aestheticians were probably clamoring for, they came up with this hard-to-pin-down striated pillow formation. The important thing, for Randy’s purposes, is that the individual pieces of Cap’n Crunch are, to a very rough approximation, shaped

Music is carried by the vibrations of molecules of air, like waves upon an ocean. It perhaps uniquely captures and conveys the interior landscape of one human mind to another, holding our tears and sweat, pain and pleasure, packaged as paeans and preludes and etudes and nocturnes. It is the texturization of the deliquescence of time, the ebb and flow of mood and meaning. It ruminates, vacillates, contemplates, and stimulates. In music we organize and fantasize, arranging the elements of music-melody, rhythm, and harmony-into meaningful shapes and patterns. Its rhythms move our hands, feet and bodies to the pulses of the universe. Its harmonies breathe with the exploratory intricacies and curiosities of relationship and proportion, consonance, dissonance, assonance, and resonance. Its melodies flitter into flights of fancy, weaving woe and wonder.

If something is true and you try to disprove it, you will fail. We are trained to to think of failure as bad, but it's not all bad. You can learn from failure. You try to disprove the statement one way, and you hit a wall. You try another way, and you hit another wall. Each night you try, each night you fail, each night a new wall, and if you are lucky, those walls start to come together into a structure, and that structure is the structure of the proof of the theorem. For if you have really understood what's keeping you from disproving the theorem, you very likely understand, in a way inaccessible to you before, why the theorem is true. This is what happened to Bolyai, who bucked his father's well-meaning advice and tried, like so many before him, to prove that the parallel postulate followed from Euclid's other axioms. Like all the others, he failed. But unlike the others, he was able to understand the shape of his failure. What was blocking all his attempts to prove that there was no geometry without the parallel postulate was the existence of just such a geometry! And with each failed attempt he learned more about the features of the thing he didn't think existed, getting to know it more and more intimately, until the moment when he realized it was really there.

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

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.

There is an art to the business of making sandwiches which it is given to few ever to find the time to explore in depth. It is a simple task, but the opportunities for satisfaction are many and profound: choosing the right bread for instance. The Sandwich Maker had spent many months in daily consultation and experiment with Grarp the baker and eventually they had between them created a loaf of exactly the consistency that was dense enough to slice thinly and neatly, while still being light, moist and having that fine nutty flavour which best enhanced the savour of roast Perfectly Normal Beast flesh. There was also the geometry of the slice to be refined: the precise relationships between the width and height of the slice and also its thickness which would give the proper sense of bulk and weight to the finished sandwich: here again, lightness was a virtue, but so too were firmness, generosity and that promise of succulence and savour that is the hallmark of a truly intense sandwich experience. The proper tools, of course, were crucial, and many were the days that the Sandwich Maker, when not engaged with the Baker at his oven, would spend with Strinder the Tool Maker, weighing and balancing knives, taking them to the forge and back again. Suppleness, strength, keenness of edge, length and balance were all enthusiastically debated, theories put forward, tested, refined, and many was the evening when the Sandwich Maker and the Tool Maker could be seen silhouetted against the light of the setting sun and the Tool Maker’s forge making slow sweeping movements through the air trying one knife after another, comparing the weight of this one with the balance of another, the suppleness of a third and the handle binding of a fourth. Three knives altogether were required. First there was the knife for the slicing of the bread: a firm, authoritative blade which imposed a clear and defining will on a loaf. Then there was the butter-spreading knife, which was a whippy little number but still with a firm backbone to it. Early versions had been a little too whippy, but now the combination of flexibility with a core of strength was exactly right to achieve the maximum smoothness and grace of spread. The chief amongst the knives, of course, was the carving knife. This was the knife that would not merely impose its will on the medium through which it moved, as did the bread knife; it must work with it, be guided by the grain of the meat, to achieve slices of the most exquisite consistency and translucency, that would slide away in filmy folds from the main hunk of meat. The Sandwich Maker would then flip each sheet with a smooth flick of the wrist on to the beautifully proportioned lower bread slice, trim it with four deft strokes and then at last perform the magic that the children of the village so longed to gather round and watch with rapt attention and wonder. With just four more dexterous flips of the knife he would assemble the trimmings into a perfectly fitting jigsaw of pieces on top of the primary slice. For every sandwich the size and shape of the trimmings were different, but the Sandwich Maker would always effortlessly and without hesitation assemble them into a pattern which fitted perfectly. A second layer of meat and a second layer of trimmings, and the main act of creation would be accomplished.

Arithmetic and number theory study patterns of number and counting. Geometry studies patterns of shape. Calculus allows us to handle patterns of motion. Logic studies patterns of reasoning. Probability theory deals with patterns of chance. Topology studies patterns of closeness ans position.

it is clear that in the study of beings this aim can be fulfilled by us perfectly only through successive examinations of them by one man after another,41 the later ones seeking the help of the earlier in that task, on the model of what has happened in the mathematical sciences. For if we suppose that the art of geometry did not exist in this age of ours, and likewise the art of astronomy, and a single person wanted to ascertain by himself the sizes of the 15 heavenly bodies, their shapes, and their distances from each other, that would not be possible for him—e.g. to know the proportion of the sun to the earth or other facts about the sizes of the stars—even though he were the most intelligent of men by nature, unless by a revelation or something resembling revelation.42 Indeed if he were told that the sun is about 150 or 160 times43 as great as the earth, he would think this statement madness on the part of the speaker, although this is a fact which has been demonstrated in 20 astronomy so surely that no one who has mastered that science doubts it.

Jumping to conclusions “is efficient if the conclusions are likely to be correct and the costs of an occasional mistake acceptable, and if the jump saves much time and effort.

All forces are multi-dimensional vibrations. The geometry and dynamics of dimensions will become important going forward. In a hologram, what is the location of the information? The holomovement is not static and the information is not perfectly distributed. Space and time are built up from entanglement. Consciousness is also entangled. The fractal nesting of event horizons For the photon, the dimensions involved are not the same as they are for the electron. The idea that all behaviors occupy the same spacetime is probably not correct. These forces and fields shape the spacetime they occupy in particular ways. In other words dimensions are dynamic and could also be virtual in some respect. There is a dynamic tension between space and time. Could the density of time create the pressure we call gravity? Does scale have a fractal quality? The fractal nesting of the relationship between space and time at different scales causes phase transitions as the influence of one force changes with respect to another. From this we get the astonishing variety of behaviors in the material world. I suspect that as the complexity increases, the dynamics of the dimensions is affected. The photon or electron is not a thing, it is a description of a relationship. The universe appears to be differentiating into a fractal computational geometry. Spacetime is fractal in the golden mean.

He spoke with determination about redress, but not revenge.

In the cycle of planning and adapting, demonstrating willpower (control before the fact) and resilience (control after the fact) is critical.

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.

Galois did not have a clear vision of the possible shapes lurking behind an equation, or of why the language he was developing would help reveal the symmetry of those shapes. Perhaps it was just as well, because the power of the language lay in its ability to create an abstraction – a mathematical description that was independent of any underlying geometry. What Galois could see was that every equation would have its own collection of permutations of the solutions which would preserve the laws relating these solutions, and that analysing the collection of permutations together revealed the secrets of each equation. He called this collection ‘the group’ of permutations associated with the equation. Galois discovered that it was the particular way in which these permutations interacted with each other that indicated whether an equation could be solved or not.

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

Nature deals in non-uniform shapes and rough edges. Take the human form. There is a certain symmetry about it, but it is, and has always been, indescribable in terms of Euclidean geometry.

Now: the way the inner dimensions are refracted has to do with the M-D geometries mentioned elsewhere in this volume. Plato was right in the sense that there are fundamental shapes, principles, or actions in the universe, and these are a direct reflection of the M-D geometries. In other words, a sphere is much more than it seems, and looks nothing like a sphere in the enfolded dimensions. Nor does it “act” like one. Particles and sub-fields are created by the same multi-dimensional interactions through “bending” the inter-dimensional angles primarily through gravity and consciousness.

As consciousness refracts the M-D shapes of the interactive geometries, resulting in our present 4-D interlink of manifest reality, so also comprehension, which is a function of consciousness, alters the M-D geometries.

What I like about stochastic gradient descent is how nuts it sounds. Imagine, for instance, that the president of the United States made decisions without any kind of global strategy; rather, the nations chief executive is surrounded by a crowd of shouting subordinates, each hollering for policy to be tweaked in a way that suits their own particular interest. And the president, every day, chooses one of those people at random to listen to, and changes course accordingly. That would be a ridiculous way for a person to run major world government, but it works pretty well in machine learning!

Along with numbers, shapes mattered too. In ancient Egypt, the measurement of lines and angles was of paramount importance. Each year surveyors had to redraw the boundaries of farmers' fields after the summer flooding of the Nile washed the borderlines away. That activity later gave its name to the study of shape in general: geometry, from the Greek ge, "earth," and metres, "measurer.

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.

Spiral shapes weave through everything. Magnetic fields are arranged in spirals. Mushroom spores propagate in spirals. And no matter what our race or religion, size or shape, we humans are made of the same all-pervading spiral geometry. This is very apparent in the swirling shape of heart muscles and skin pores.

Sadly, we know almost nothing about Euclid (c. 325-c. 265 BCE).32 We know even less about him than we do about Pythagoras, and what little we do know has been hotly contested by scholars. Euclid wrote at least ten books, only half of which have survived. A number of mutually consistent indications suggest that he lived after Aristotle and before Archimedes. He was one of the first mathematicians at the great library of Alexandria and there had gathered a group of talented mathematicians about him. Legends about him abound, many as (possibly apocryphal) insertions in other mathematicians' works. One tells that Ptolemy asked Euclid for a quick way to master geometry and received the reply, "There is no royal road to geometry." Another tells of a student who, after encountering the first proposition in the Elements, asked Euclid what practical use studying geometry could have. The mathematician allegedly turned to his slave and replied dismissively, "Slave, give this boy a threepence, since he must make gain of what he learns.

• Little fingers, little Moon (think of that tiny “fingernail clipping” shape). • Thumb joints, Full Moon—either almost, exactly, or just past. • Widdershins means right to left, moon-wise, opposite the Sun’s motion, so: • Right-hand backward-C-shape, the Moon is waxing, growing larger. • Left-hand C-shape, the Moon is waning, shrinking in size.

• Widdershins means right to left, moon-wise, opposite the Sun’s motion, so: • Right-hand backward-C-shape, the Moon is waxing, growing larger. • Left-hand C-shape, the Moon is waning, shrinking in size.

What he found was the geometry of the universe. Looking at the bubbles made by the Wego’s propellers, he recalled his boarding school math teachers, who had taught him to measure a sphere’s volume in terms of pi. He also remembered that pi was an irrational number, a decimal that never ended. He asked himself how nature could ever make bubbles in such circumstances. Did nature approximate? The rules his teachers had taught him must be mistaken. Spheres ought to be understood in terms of the forces that made them. At the age of twenty-one, Bucky determined that the universe had no objects. Geometry described forces. It was an insight bound to shape Bucky’s entire worldview—informing every future invention—but

Historian Charles Mann has suggested that this mashed-up new biological phase we’ve induced be called the Homogenocene, as we have blended so many evolving populations, once geographically dispersed and isolated, into one homogenized genetic broth. In the 1970s Carl Sagan used to zip around Ithaca, New York, with a bumper sticker on his orange Porsche reading, “Reunite Gondwanaland!” referring to the time when all Earth’s continents were merged into one supercontinent. I guess, biologically, this is what we’ve now done. We’ve seriously rearranged the evolutionary geometry of the world—and

this way, we’ve changed the geometry of the planet. Before we came along, the world was discontinuous. Oceans, deserts, and mountain ranges formed impenetrable barriers, breaking Earth into separate regions where populations could evolve independently, and then be isolated or merged by continental drift and climate change. Now we’ve created pathways around all those borders, and to some degree the planet is one continuous habitat. Some

species. In this way, we’ve changed the geometry of the planet. Before we came along, the world was discontinuous. Oceans, deserts, and mountain ranges formed impenetrable barriers, breaking Earth into separate regions where populations could evolve independently, and then be isolated or merged by continental drift and climate change. Now we’ve created pathways around all those borders, and to some degree the planet is one continuous habitat. Some

Right now, looking out my kitchen window on a summer day on Capitol Hill, I see a complex, shifting scene composed of about 50 percent brick and 50 percent trees. It’s lovely, a riot of organic forms bouncing in the wind. The brick is festooned with lichen, ivy, and moss, its rigid geometry softened and blemished by hundreds of years of wind, rain, and life, and illuminated by splintered sunlight refracted through blowing branches and leaves. A squirrel skitters along a power line, balanced, at ease, “natural,” as if he’s been evolving to do this for a hundred thousand years. The trees are diverse, some deciduous and some evergreen. They look happy, at home, healthy, and strong. They are permanent residents, compared to any people. The birds and rodents that nest, chase, chatter, and squeal among them seem at home as well.

Dimensional Magic We are fractal. Our lungs, our circulatory system, our brains are like trees. They are fractal structures. Fractal geometry allows bounded curves of infinite length, and closed surfaces with an infinite area. It even allows curves with positive volume, and arbitrarily large groups of shapes with exactly the same boundary. This is exactly how our lungs manage to maximize their surface area.

The usefulness of mathematics allows us to build spaceships and investigate the geometry of our universe. Numbers may be our first means of communication with intelligent alien races. Some physicists have even speculated that an understanding of higher dimensions and of topology-the study of shapes and their interrelationships-may someday allow us to escape our universe, when it ends in either great heat or cold, and then we could call all of space-time our home.

We are fractal. Our lungs, our circulatory system, our brains are like trees. They are fractal structures. Fractal geometry allows bounded curves of infinite length, and closed surfaces with an infinite area. It even allows curves with positive volume, and arbitrarily large groups of shapes with exactly the same boundary. This is exactly how our lungs manage to maximize their surface area. Most natural objects-and that includes us human beings-are composed of many different types of fractals woven into each other, each with parts which have different fractal dimensions. For example, the bronchial tubes in the human lung have one fractal dimension for the first seven generations of branching, and a different fractal dimension from there on in.

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

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.

In the age of computer simulation, when flows in everything from jet turbines to heart valves are modeled on supercomputers, it is hard to remember how easily nature can confound an experimenter. In fact, no computer today can completely simulate even so simple a system as Libchaber's liquid helium cell. Whenever a good physicist examines a simulation, he must wonder what bit of reality was left out, what potential surprise was sidestepped. Libchaber liked to say that he would not want to fly in a simulated airplane-he would wonder what had been missed. Furthermore, he would say that computer simulations help to build intuition or to refine calculations, but they do not give birth to genuine discovery. This, at any rate, is the experimenter's creed. His experiment was so immaculate, his scientific goals so abstract, that there were still physicists who considered Libchaber's work more philosophy or mathematics than physics. He believed, in turn, that the ruling standards of his field were reductionist, giving primacy to the properties of atoms. "A physicist would ask me, How does this atom come here and stick there? And what is the sensitivity to the surface? And can you write the Hamiltonian of the system? "And if I tell him, I don't care, what interests me is this shape, the mathematics of the shape and the evolution, the bifurcation from this shape to that shape to this shape, he will tell me, that's not physics, you are doing mathematics. Even today he will tell me that. Then what can I say? Yes, of course, I am doing mathematics. But it is relevant to what is around us. That is nature, too." The patterns he found were indeed abstract. They were mathematical. They said nothing about the properties of liquid helium or copper or about the behavior of atoms near absolute zero. But they were the patterns that Libchaber's mystical forbears had dreamed of. They made legitimate a realm of experimentation in which many scientists, from chemists to electrical engineers, soon became explorers, seeking out the new elements of motion. The patterns were there to see the first time eh succeeded in raising the temperature enough to isolate the first period-doubling, and the next, and the next. According to the new theory, the bifurcations should have produced a geometry with precise scaling, and that was just what Libchaber saw, the universal Feigenbaum constants turning in that instant from a mathematical ideal to a physical reality, measurable and reproducible. He remembered the feeling long afterward, the eerie witnessing of one bifurcation after another and then the realization that he was seeing an infinite cascade, rich with structure. It was, as he said, amusing.

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

The pits and tangles are more than blemishes distorting the classic shapes of Euclidian geometry. They are often the keys to the essence of a thing

Returning to bed, Rachel strokes Zachariah's black curls as he drifts into sleep and appreciates the shape and fractal geometry there, the self-similarity and infinity of scale. She breathes in at his scalp, then presses her ear to his, listening for the clamour of voices within, to the long line of fighting men who made him, his head a seashell. There is a template for the fighting man. Rachel listens across three times nine countries, as the fairy-tale saying goes, across three times nine countries in the thirtieth tsardom . . .

Eventually, four billion years later, again came something completely new: novel, stark, inorganic geometries began quickly remaking the surface. Suddenly, the nightside lit up in bright, spreading webs. Then there was a curious anti-accretion: some pieces of Earth, in seeming defiance of the laws of gravity, started launching themselves back out into the surrounding space from whence everything had once quickly fallen. To Distant Climes I don’t know if space junkies are born or made, but my timing was good.

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

While we all share similar concerns and pose similar questions, we don’t figure it out collectively. Money is an isolated—and isolating—affair. We figure it out on our own. What are we all up against? I see three broad challenges currently: We have more control over our own finances. We are wired to make bad money decisions. We have less room for error.

One of my investment heroes, Charlie Munger, once remarked: “Invert. Always invert.” By this he meant take the time to think differently about common problems. Conventional thinking leads to conventional outcomes.

the shape of her eyes and the lines of mascara around it were perfect cosmic geometry.

Between Jeans and Pearson comes a notice from one John Butler Burke, who believed he had observed spontaneous generation of microscopic life in a vat of beef bouillon by exposure to the recently discovered element radium.

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.

I was reminded, again, of the shapes of women, the impossible geometry into which I was meant to fold myself.

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

The scientist is another who prepares, who makes ready, working day and night, sleeping and awake, for inspiration. As Pythagoras knew, the god may speak in the forms of geometry as well as in the shapes of dreams; in the harmony of pure thought as well as in the harmony of sounds; in numbers as well as in words.

Each year surveyors had to redraw the boundaries of farmers’ fields after the summer flooding of the Nile washed the borderlines away. That activity later gave its name to the study of shape in general: geometry, from the Greek gē, “earth,” and metrēs, “measurer.

Virahanka sequence,

I did not know then that this is what life is- just when you master the geometry of one world, it slips away, and suddenly again, you're swarmed by strange shapes and impossible angles.

Geometry is the cilantro of math.

The neighborhood was deserted, and for good reason. Half of the buildings had been torn down; what formerly must have been a row of dwellings was now a row of holes, homes displaying their innards—beams, rubble, clods of exposed wire—empty spaces held between torn walls and floors, their jagged edges like claws, ripped while clinging to remain whole. In this disorder I could not help but see my mother’s hand. It was as if her death had reached over the ocean, anticipating me, contriving to remove her traces before I arrived. I saw in the pulverized dust an analogue of what my mother had become, something that could be scooped into a plastic bag and carried on my back. I saw in the slabs of broken wood a suggestion of geometry—lines, rectangles, regular shapes, broken and reverted to their original material form.