Geometry Universe 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.

You must follow me carefully. I shall have to controvert one or two ideas that are almost universally accepted. The geometry, for instance, they taught you at school is founded on a misconception.

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

The 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.

And so will I here state just plainly and briefly that I accept God. But I must point out one thing: if God does exist and really created the world, as we well know, he created it according to the principles of Euclidean geometry and made the human brain capable of grasping only three dimensions of space. Yet there have been and still are mathematicians and philosophers-among them some of the most outstanding-who doubt that the whole universe or, to put it more generally, all existence was created to fit Euclidean geometry; they even dare to conceive that two parallel lines that, according to Euclid, never do meet on earth do, in fact, meet somewhere in infinity. And so my dear boy, I’ve decided that I am incapable of understanding of even that much, I cannot possibly understand about God.

In reality the universe has no geometry.

Scientists say every action initiates an equal and opposite reaction. I say that's just the start. I say every action initiates a most unequal and upredictable chain reaction, that every filament of living becomes part of a larger weave, while remaining identifiable. That every line of latitude requires several stripes of longitude to obtain meaning. That every universe is part of a bigger heaven, a heaven of rhythm and geometry, where a heartbeat is the apex of a triangle.

If the human mind can understand the universe, it means the human mind is fundamentally of the same order as the divine mind. If the human mind is of the same order as the divine mind, then everything that appeared rational to God as he constructed the universe, it's “geometry,” can also be made to appear rational to the human understanding, and so if we search and think hard enough, we can find a rational explanation and underpinning for everything. This is the fundamental proposition of science.

The true horror of existence is not the certainty of death, nor the threat of hell, but the knowledge that we will likely go our entire lives as impossibly complex machines, walking about in an impossibly complex universe, and never truly discover what it was all for. I don’t know is brave. I’ll never know is heroic.

My task is to explain to you as quickly as possible my essence, that is, what sort of man I am, what I believe in, and what I hope for, is that right? And therefore I declare that I accept God pure and simple. But this, however, needs to be noted: if God exists and if he indeed created the earth, then, as we know perfectly well, he created it in accordance with Euclidean geometry, and he created human reason with a conception of only three dimensions of space. At the same time there were and are even now geometers and philosophers, even some of the most outstanding among them, who doubt that the whole universe, or, even more broadly, the whole of being, was created purely in accordance with Euclidean geometry; they even dare to dream that two parallel lines, which according to Euclid cannot possibly meet on earth, may perhaps meet somewhere in infinity. I, my dear, have come to the conclusion that if I cannot understand even that, then it is not for me to understand about God. I humbly confess that I do not have any ability to resolve such questions, I have a Euclidean mind, an earthly mind, and therefore it is not for us to resolve things that are not of this world. And I advise you never to think about it, Alyosha my friend, and most especially about whether God exists or not. All such questions are completely unsuitable to a mind created with a concept of only three dimensions. And so, I accept God, not only willingly, but moreover I also accept his wisdom and his purpose, which are completely unknown to us; I believe in order, in the meaning of life, I believe in eternal harmony, in which we are all supposed to merge, I believe in the Word for whom the universe is yearning, and who himself was 'with God,' who himself is God, and so on and so forth, to infinity. Many words have been invented on the subject. It seems I'm already on a good path, eh? And now imagine that in the final outcome I do not accept this world of God's, created by God, that I do not accept and cannot agree to accept. With one reservation: I have a childlike conviction that the sufferings will be healed and smoothed over, that the whole offensive comedy of human contradictions will disappear like a pitiful mirage, a vile concoction of man's Euclidean mind, feeble and puny as an atom, and that ultimately, at the world's finale, in the moment of eternal harmony, there will occur and be revealed something so precious that it will suffice for all hearts, to allay all indignation, to redeem all human villainy, all bloodshed; it will suffice not only to make forgiveness possible, but also to justify everything that has happened with men--let this, let all of this come true and be revealed, but I do not accept it and do not want to accept it! Let the parallel lines even meet before my own eyes: I shall look and say, yes, they meet, and still I will not accept it.

Try to imagine this formless, liquid abyss of many waters and surfaces. With and within this awesome, abysmal substance, God Mother creates universes. We are made from this stuff, literally! And we literally live, move, and have our being in this fathomless, multi-dimensional matrix.

DOCTOR: Oh. My, God! Oh, it's bigger! RIVER: Well, yes. DOCTOR: On the inside, RIVER: We need to concentrate. DOCTOR: Than it is RIVER: I know where you're going with this, but I need you to calm down. DOCTOR: On the outside! RIVER: You've certainly grasped the essentials. DOCTOR: My entire understanding of physical space has been transformed! Three-dimensional Euclidean geometry has been torn up, thrown in the air and snogged to death! My grasp of the universal constants of physical reality has been changed forever.

sacred knowledge of the cosmos seems to be hidden within our souls and is shown within our artwork and creative expressions.

First I’ll tell you about the picture of the universe painted by modern physics: the geometry of the universe is not physical.” “Can you be a little less abstract?” “What if I put it this way: in the universe, apart from empty space, there is nothing.

Since time immemorial, the serpent has symbolized organic vitality. Serpents move in curves and so does energy. The serpent developed from the Light created on the first cosmic day. In the firmament it manifests as the life-breath that animates souls. In the physical universe it manifests electro-magnetism, the light in substance so to speak.

Don’t forget to be specific…Details. Put in all the details. The boys appreciate all that detailed daily life sh*t they don’t get anymore. If you’ve got a teacher you’re hot for, tell ‘em what her hair looks like, what her legs look like, what she eats for lunch. If she’s teaching you geometry, tell ‘em how she draws a bloody triangle on the blackboard. If you went down the shop for a bag of sweets yesterday, did you ride your pushee? Did you go by foot? Did you see a rainbow along the way? Did you buy gobstoppers or clinkers or caramels? If you had a good meat pie last week was it steak and peas or curry or mushroom and beef? You catchin’ my drift? Details.

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.

In Leipzig [in the 14th century], the university found it necessary to promulgate a rule against throwing stones at the professors. As late as 1495, a German statute explicitly forbade anyone associated with the university from drenching freshmen with urine.

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

Even as we sit, we dance – In this cosmic saga of numbers and geometric patterns, continually coming alive.

Three of the four forces (excluding gravity) are therefore united by quantum theory, giving us unification without geometry, which appears to contradict the theme of this book and everything we have considered so far.

The description of this proportion as Golden or Divine is fitting perhaps because it is seen by many to open the door to a deeper understanding of beauty and spirituality in life. That’s an incredible role for one number to play, but then again this one number has played an incredible role in human history and the universe at large.

This was the missing piece in the puzzle. The secret of wood that bound matter together was the Yang-Mills filed, not the geometry of Einstein. It appeared as though this, and not geometry, was the central lesson of physics.

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.

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 time will come, and come soon, in which we shall have a knowledge of God and mind that is not less certain than that of figures and numbers, and in which the invention of machines will be no more difficult than the construction of problems in geometry.

In an agential realist account, agency is cut loose from its traditional humanist orbit. Agency is not aligned with human intentionality or subjectivity. Nor does it merely entail resignification or other specific kinds of moves within a social geometry ofantihumanism. The space of agency is not only substantially larger than that allowed for in Butler's performative account, for example, but also, perhaps rather surprisingly, larger than what liberal humanism proposes. Significantly, matter is an agentive factor in its iterative materialization.

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

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.

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

Prototypes of inventions that use novel combinations of resonance, magnetism, states of matter, certain geometries or inward swirling motion to unlock the secrets of universal energy have already been built. They provide proof of new or rediscovered principles. In many variations of these inventions, a small input triggers a disproportionately large output of useable power." "These energy converters don't violate any laws of physics if they simply tap into a previously unrecognized source of power - background space. A flow of energy from that source can continue day and night, whether or not the sun shines or the wind blows.

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

The basic consciousness structure is perfectly organized according to the principles of sacred geometry, even though the physical structure of the Universe looks to be very random.

The principal aim of the species will be to build a single great mirror which all will look into, and see only universe looking back.

The only things that can ever be universal, in a sense, are scaling things.

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

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.

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 secret of Giorgi’s universe was number, for it was built, so he believed, by its Architect as a perfectly proportioned Temple, in accordance with unalterable laws of 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 discovery of geometry had intoxicated them, and its a priori deductive method appeared capable of universal application. They would prove, for instance, that all reality is one, that there is no such thing as change, that the world of sense is a world of mere illusion; and the strangeness of their results gave them no qualms because they believed in the correctness of their reasoning.

Outside of the surrender of the incommunicable, the suspension amid our mute and unconsoled anxieties, life is merely a fracas on an unmapped terrain, and the universe a geometry stricken with epilepsy.

According to Thoth, because of the placement of the Great Pyramid on the Earth connecting into the Earth's huge geometrical field - specifically the octahedral field of the Earth, which is equivalent to our own fields - and because of the pyramid's mass and the geometries used in it, the white-light energy field spirals upward and becomes extremely strong, stretching all the way out to the center of the galaxy. The dark-light energy comes in from above, spirals through zero point and connects with the center of the Earth. In this way the Great Pyramid connects the center of the Earth to the center of our galaxy.

The mathematical order is beautiful precisely because it has no effect on the real world. Life isn't going to be easier, nor is anyone going to make a fortune, just because they know something about prime numbers. Of course, lots of mathematical discoveries have practical applications, no matter how esoteric they may seem. Research on ellipses made it possible to determine the orbits of the planets, and Einstein used non-Euclidean geometry to describe the form of the universe. Even prime numbers were used during the war to create codes—to cite a regrettable example. But those things aren't the goal of mathematics. The only goal is to discover the truth.

But who can quantify the algebra of space, or weigh those worlds that swim each in its place? Who can outdo the dark? And what computer knows how beauty comes to birth - shell star and rose? -Technicians by Jean Kenward

Galileo essentially started out from where Archimedes left off, proceeding in the same direction as defined by his Greek predecessor. This is true not only of Galileo but also of the other great figures of the so-called “scientific revolution,” such as Leibniz, Huygens, Fermat, Descartes, and Newton. All of them were Archimedes’ children. With Newton, the science of the scientific revolution reached its perfection in a perfectly Archimedean form. Based on pure, elegant first principles and applying pure geometry, Newton deduced the rules governing the universe. All of later science is a consequence of the desire to generalize Newtonian, that is, Archimedean methods.

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 geometry of a universe is very like the grammatical structure of a sentence. Just as a sentence has no structure and no existence apart from the relationships between the words, space has no existence apart from the relationships that hold between the things in the universe. If you change a sentence by taking some words out, or changing their order, its grammatical structure changes. Similarly, the geometry of space changes when the things in the universe change their relationships to one another.

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

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.

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.

Yet there have been and still are mathematicians and philosophers who doubt whether the whole universe, or to speak more widely, 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.

Everything is fields, and a particle is just a smaller version of a field. There is a harmonic relationship involved. Disturbing ideas like those of Einstein in 1905 and Feynman Pocono Conference in 1948. Here we go; 1) The universe is ringing like a bell. Neil Turok's Public Lecture: The Astonishing Simplicity of Everything. 2) The stuff of the universe is waves or fields. 3) Scale is relative, not fixed because all of these waves are ratios of one another. 4) The geometry is fractal. This could be physical or computational. 5) If the geometry is computational then, there is no point in speaking about the relationship of the pixels on the display.

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.

By the sixth century B.C. Pythagoras and his students had embarked on the immense ordering task that attempted to find common numerical laws binding together astronomy, geometry, music, and arithmetic. Not surprisingly, their work was difficult to distinguish from religion, since it tried to accomplish similar goals: to find a way of expressing the structure of the universe. Two thousand years later, Kepler and then Newton were still on the same quest. Theoretical

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. 

This picture of matter curving space and curvaceous space dictating how matter and light will move has several striking features. It brings the non-Euclidean geometries that we talked about in the last chapter out from the library of pure mathematics into the arena of science. The vast collection of geometries describing spaces that are not simply the flat space of Euclid are the ones that Einstein used to capture the possible structures of space distorted by the presence of mass and energy.

The geometry of a universe is very like the grammatical structure of a sentence. Just as a sentence has no structure and no existence apart from the relationships between the words, space has no existence apart from the relationships that hold between the things in the universe. If you change a sentence by taking some words out, or changing their order, its grammatical structure changes. Similarly, the geometry of space changes when the things in the universe change their relationships to one another.

For what is it you and I are trying to do now? What I'm trying to do is to attempt to explain to you as quickly as possible the most important thing about me, that is to say, what sort of man I am, what I believe in what I hope for - that's it isn't it? And that's why I declare that I accept God plainly and simply. But there's this that has to be said: if God really exists and if he really has created the world, then, as we all know, he created it in accordance with the Euclidean geometry, and he created the human mind with the conception of only the three dimensions of space. And yet there have been and there still are mathematicians and philosophers, some of them indeed men of extraordinary genius, who doubt whether the whole universe, or, to put it more wildly, all existence was created only according to Euclidean geometry and they even dare to dream that two parallel lines which, according to Euclid can never meet on earth, may meet somewhere in infinity. I, my dear chap, have come to the conclusion that if I can't understand even that, then how can I be expected to understand about God? I humbly admit that I have no abilities for settling such questions. And I advise you too, Aloysha, my friend, never to think about it, and least of all about whether there is a God or not. All these problems which are entirely unsuitable to a mind created with the idea of only three dimensions. And so I accept God, and I accept him not only without reluctance, but what's more, I accept his divine wisdom and his purpose- which are completely beyond our comprehension.

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

Grossmann went home to think about the question. After consulting the literature, he came back to Einstein and recommended the non-Euclidean geometry that had been devised by Bernhard Riemann.11 Riemann (1826–1866) was a child prodigy who invented a perpetual calendar at age 14 as a gift for his parents and went on to study in the great math center of Göttingen, Germany, under Carl Friedrich Gauss, who had been pioneering the geometry of curved surfaces. This was the topic Gauss assigned to Riemann for a thesis, and the result would transform not only geometry but physics.

There have been and there still are mathematicians and philosophers, some of them indeed men of extraordinary genius, who doubt whether the whole universe, or, to put it more widely, all existence, was created only according to Euclidean geometry and they even dare to dream that wo parallel lines which, according to Euclid, can never meet on earth, may meet somewhere in infinity. I, my dear chap, have come to the conclusion that if I can’t understand even that, then how can I be expected to understand about God? I humbly admit that I have no abilities for settling such questions. I have a Euclidean, an earthly mind, and so how can I be expected to solve problems which are not of this world.

The Harmony of the World is the continuation of tye Cosmic Mystery, and the climax of his lifelong obsession. What Kepler attempted here is, simply, to bare the ultimate secret of the universe in an all-embracing synthesis of geometry, music, astrology, astronomy, and epistemology. It was the firat attempt of this kind since Plato, and it is the last to our day. After Kepler, fragmentation of experience sets in again, science is divorced from religion, religion from art, substance from form, matter from mind. The work is divided into five books. The first two deal with the concept of harmony in mathematics; the following three with the applications of this concept to music, astrology, and astronomy, in that order.

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.

To do so he used something called a tensor. In Euclidean geometry, a vector is a quantity (such as of velocity or force) that has both a magnitude and a direction and thus needs more than a single simple number to describe it. In non-Euclidean geometry, where space is curved, we need something more generalized—sort of a vector on steroids—in order to incorporate, in a mathematically orderly way, more components. These are called tensors. A metric tensor is a mathematical tool that tells us how to calculate the distance between points in a given space. For two-dimensional maps, a metric tensor has three components. For three-dimensional space, it has six independent components. And once you get to that glorious four-dimensional entity known as spacetime, the metric tensor needs ten independent components.

There it was then. I would build a scientific religion, a religious science. I would reinvigorate man’s interest in the stars via spirituality. By chapel and telescope I would show reality in its nakedness, and convince the people of the Myriad of Man that we are but universe experiencing itself. At last we would look into the mirror of Being, and see only Universe looking back. “We would find wonder in the imponderable majesty of the little, in all of the cells and molecules and atoms and quarks inside one’s body, the beauty of it, the absurdity. That is where we would apprehend the transcendent. And in finding that transcendent, we would at last realise we are but matter, and finally seek to stop conquering more matter. The way out was not up, to some holy, blazing firmament, but down, into the skin, into the quiet absurdity of being.

Philosophy is one of those subjects, like astrophysics and neurosurgery, that are not for the fainthearted. To delve into the absolutes of the human experience, to seek to advance the progress of enlightenment first expounded by the likes of the revered Aristotle and Plato, to search for the answers to the profound questions of the universe, often at the risk of deadly reprisal from entrenched powers, requires not only brilliance and tenacity but a deep sense of purpose. But even among this select fraternity, [René] Descartes stands out. From him did we get practical discoveries like coordinates in geometry and the law of refraction of light. But what he really did was to shake loose the human mind from the shackles of centuries of stultifying religious orthodoxy by creating an entirely original approach to reasoning: the Cartesian method.

There have been and still are geometricians and philosophers, and even some of the most distinguished, who doubt whether the whole universe, or to speak more widely the whole of existence, 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 Euclidean 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 an idea of only three dimensions.

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.

Kids aren’t learning science right these days! The teachers teach it like it’s just supposed to be useful, like, here, learn this geometry so you can design a building, here, learn this chemistry so you can make a plastic bag. Of course kids don’t like it! No kid comes home from school and says, ‘I want to make plastic bags when I grow up!’ We already have plastic bags, and comfy chairs, and flying cars, we’ve had them for centuries, and they aren’t getting better because they work already so no one’s interested in replacing them, just making them cheaper, or with more games. That isn’t science! Science is figuring out where the universe is going! Science is noticing that the ants crawling up the picnic table like your sandwich better than your ba’sib’s and asking, ‘Why?’ Not ‘How is this useful?’ not ‘Can I make this into a plastic bag?’ but ‘Why?

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.

Kant distinguished between two types of truths: (1) analytic propositions, which derive from logic and “reason itself” rather than from observing the world; for example, all bachelors are unmarried, two plus two equals four, and the angles of a triangle always add up to 180 degrees; and (2) synthetic propositions, which are based on experience and observations; for example, Munich is bigger than Bern, all swans are white. Synthetic propositions could be revised by new empirical evidence, but not analytic ones. We may discover a black swan but not a married bachelor or (at least so Kant thought) a triangle with 181 degrees. As Einstein said of Kant’s first category of truths: “This is held to be the case, for example, in the propositions of geometry and in the principle of causality. These and certain other types of knowledge… do not previously have to be gained from sense data, in other words they are a priori knowledge.” Einstein

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.

is honest enough to admit that. I’m sorry if ignorance scares you. I’m sorry if you can’t stand the thought of dying before you’ve found the meta-yes. But that’s how this works, for now. And in the meantime, bookended by birth and death, riding about in bodies of carbon, loving, suffering, striving, for a short, short time, we get to be. If we’re forced to be in a mysterious universe, and that universe is mute on the subject of its own motivations for existing, I’ll still take living in honest ignorance over your metaphysical posturing any day. That’s real bravery. Not concocting blatantly contradictory stories to comfort oneself, not appealing to the vague transcendent every time you get your worldview in a twist. Ignorance. Brave, honest admission of one’s ignorance, and living with that ignorance in a kind, compassionate manner, treating each other well even if we know we eventually all go to dust and never happen again. We must try to be wise, to be good women, to be good men. The rite of adulthood is the admission

The cosmic sculptor had felt compelled to dot pupils onto the universe, yet had a tremendous terror of granting it sight. This balance of fear and desire resulted in the tininess of the stars against the hugeness of space, a declaration of caution above all. “See how the stars are points? The factors of chaos and randomness in the complex makeups of every civilized society in the universe get filtered out by the distance, so those civilizations can act as reference points that are relatively easy to manipulate mathematically.” “But there’s nothing concrete to study in your cosmic sociology, Dr. Ye. Surveys and experiments aren’t really possible.” “That means your ultimate result will be purely theoretical. Like Euclidean geometry, you’ll set up a few simple axioms at first, then derive an overall theoretic system using those axioms as a foundation.” “It’s all fascinating, but what would the axioms of cosmic sociology be?” “First: Survival is the primary need of civilization. Second: Civilization continuously grows and expands, but the total matter in the universe remains constant.” The

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.

Rμv– 1/2 gμv R = 8πTμv The left side of the equation starts with the term Rμv, which is the Ricci tensor he had embraced earlier. The term gμv is the all-important metric tensor, and the term R is the trace of the Ricci tensor called the Ricci scalar. Together, this left side of the equation—which is now known as the Einstein tensor and can be written simply as Gμv—compresses together all of the information about how the geometry of spacetime is warped and curved by objects. The right side describes the movement of matter in the gravitational field. The interplay between the two sides shows how objects curve spacetime and how, in turn, this curvature affects the motion of objects. As the physicist John Wheeler has put it, “Matter tells spacetime how to curve, and curved space tells matter how to move.”83 Thus is staged a cosmic tango, as captured by another physicist, Brian Greene: Space and time become players in the evolving cosmos. They come alive. Matter here causes space to warp there, which causes matter over here to move, which causes space way over there to warp even more, and so on. General relativity provides the choreography for an entwined cosmic dance of space, time, matter, and energy.84 At

I told him about how our second form teacher, Miss Crane, drew the tiniest chalk mark on the blackboard and explained that a point is “zero-dimensional,” meaning it doesn’t actually exist. But once you have two points—two nonexistent points—you can fill in the space between with lots and lots of points, and you get a line, which has length, so it’s now one dimension, which you could argue means it does now exist. Miss Crane dotted her chalk against the board, over and over, in a straight line, demonstrating how a series of nothings could become something. (Actually, you could also argue the line still doesn’t exist, it’s just a concept, but I’d learned by then not to add caveats to everything I said. This was, after all, a love letter.) I told Jack how I leaned forward that day in class as if I stood with my toes hanging over the very precipice of enlightenment. In my naivete, I believed Miss Crane was about to explain something that explained everything. Something I felt I almost already knew, but could not articulate; it was related to infinity and God, the ocean and space, the universe and my dad. Of course, I did not achieve enlightenment in my geometry lesson. Miss Crane put the chalk down and told us to take out our compasses and protractors. I told Jack that when I was with him, I felt like I was close to understanding what I had nearly understood that day.

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

The result was not nearly as vivid to the layman as, say, E=mc2. Yet using the condensed notations of tensors, in which sprawling complexities can be compressed into little subscripts, the crux of the final Einstein field equations is compact enough to be emblazoned, as it indeed often has been, on T-shirts designed for proud physics students. In one of its many variations,82 it can be written as: Rμv– 1/2 gμv R = 8πTμv The left side of the equation starts with the term Rμv, which is the Ricci tensor he had embraced earlier. The term gμv is the all-important metric tensor, and the term R is the trace of the Ricci tensor called the Ricci scalar. Together, this left side of the equation—which is now known as the Einstein tensor and can be written simply as Gμv—compresses together all of the information about how the geometry of spacetime is warped and curved by objects. The right side describes the movement of matter in the gravitational field. The interplay between the two sides shows how objects curve spacetime and how, in turn, this curvature affects the motion of objects. As the physicist John Wheeler has put it, “Matter tells spacetime how to curve, and curved space tells matter how to move.”83 Thus is staged a cosmic tango, as captured by another physicist, Brian Greene: Space and time become players in the evolving cosmos. They come alive. Matter here causes space to warp there, which causes matter over here to move, which causes space way over there to warp even more, and so on. General relativity provides the choreography for an entwined cosmic dance of space, time, matter, and energy.

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!

Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different concepts. In some cases, the proofs generated the theorems-mathematicians examined what they could prove and from that they deduced the theorems. In others, as described by Archimedes in The Method, they first found the answer to a particular question they were interested in, and then they worked out the proof. Typically, the concepts were inventions. Prime numbers as a concept were an invention, but all the theorems about prime numbers were discoveries. The mathematicians of ancient Babylon, Egypt, and China never invented the concept of prime numbers, in spite of their advanced mathematics. Could we say instead that they just did not "discover" prime numbers? Not any more than we could say that the United Kingdom did not "discover" a single, codified, documentary constitution. Just as a country can survive without a constitution, elaborate mathematics could develop without the concept of prime numbers. And it did! Do we know why the Greeks invented such concepts as the axioms and prime numbers? We cannot be sure, but we could guess that this was part of their relentless efforts to investigate the most fundamental constituents of the universe. Prime numbers were the basic building blocks of matter. Similarly, the axioms were the fountain from which all geometrical truths were supposed to flow. The dodecahedron represented the entire cosmos and the golden ratio was the concept that brought that symbol into existence.

Here's a simplified version of what the Stanford group did. They started with a much-studied kind of string theory-a flat four-dimensional spacetime with a small six-dimensional geometry over each point. They chose the geometry of the six wrapped-up dimensions to be one of the Calabi-Yau spaces (see Chapter 8). As noted, there are at least a hundred thousand of these, and all you have to do is pick a typical one whose geometry depends on many constants. Then they wrapped large numbers of electric and magnetic fluxes around the six-dimensional spaces over each point. Because you can wrap only discrete units of flux, this tends to freeze out the instabilities. To further stabilize the geometry, you have to call on certain quantum effects not known to arise directly from string theory, but they are understood to some extent in supersymmetric gauge theories, so it is possible that they play a role here. Combining these quantum effects with the effects from the fluxes, you get a geometry in which all the moduli are stable. This can also be done so that there appears to be a negative cosmological constant in the four-dimensional spacetime. It turns out that the smaller we want the cosmological constant to be, the more fluxes we must wrap, so we wrap huge numbers of fluxes to get a cosmological constant that is tiny but still negative. (As noted, we don't know explicitly how to write the details of a string theory on such a background, but there's no reason to believe it doesn't exist.) But the point is to get a positive cosmological constant, to match the new observations of the universe's expansion rate. So the next step is to wrap other branes around the geometry, in a different way, which has the effect of raising the cosmological constant. Just as there are antiparticles, there are antibranes, and the Stanford group used them here. By wrapping antibranes, energy can be added so as to make the cosmological constant small and positive. At the same time, the tendency of string theories to flow into one another is suppressed, because any change requires a discrete step. Thus, two problems are solved at once: The instabilities are eliminated and the cosmological constant is small and positive.

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.

The discovery of the fractals had proven that in a world of chaos, there is a way to understand the natural world by using something as definite and precise as values and numbers.  Things and events that were beyond our understanding slowly emerge as distinctively clear as figures and images, leaving no room for mysteries and ambiguities, making the universe more understandable in the eyes of people.  With this, it presents a definite point of view in the way complex things are to be observed, which would represent irregularities and open up new paths for discovery. 

By contrast, the three other features—conjunct melodic motion, limited macroharmony, and centricity—are common to virtually all human music. This near universality may be attributable, at least in part, to features of our biological inheritance.10

Quantum uncertainty across the horizon of the black hole generates fluctuations of the horizon geometry. But "fluctuations" imply "probability," and probability implies thermodynamics, and therefore temperature. Concealing from us a part of the universe, a black hole makes its quantum fluctuations detectable in the form of heat.

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.

The sphere is a symbol of the cosmos and the totality of manifest creation. Very large and very small things in nature tend to be spherical. Einstein discovered that a point in four dimensions (i.e. you here and now) is a sphere expanding at the speed of light, and all we can see of the entire universe is inside an event-horizon sphere. The cube represents the earth.

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

In Euclid's Elements we meet the concept which later plays a significant role in the development of science. The concept is called the "division of a line in extreme and mean ratio" (DEMR). ...the concept occurs in two forms. The first is formulated in Proposition 11 of Book II. ...why did Euclid introduce different forms... which we can find in Books II, VI and XIII? ...Only three types of regular polygons can be faces of the Platonic solids: the equilateral triangle... the square... and the regular pentagon. In order to construct the Platonic solids... we must build the two-dimensional faces... It is for this purpose that Euclid introduced the golden ratio... (Proposition II.11)... By using the "golden" isosceles triangle...we can construct the regular pentagon... Then only one step remains to construct the dodecahedron... which for Plato is one of the most important regular polyhedra symbolizing the universal harmony in his cosmology.

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

Merchant Bashir got up and plodded to a pile of rugs. He grabbed a kilim and unrolled it across the floor. A mosaic of black, yellow, and maroon geometries glimmered. “He taught me rug weaving. It’s a nomadic art, he said. Pattern making carries the past into the future.” Bashir pointed to a recurrent cross motif that ran down the kilim’s center. “The four corners of the cross are the four corners of the universe. The scorpion here”—he toed a many-legged symmetric creature woven in yellow—“represents freedom. Sharif taught me this and more. He was a natural at symbols. I asked him why he went to Turkey. He looked at me and said, ‘To learn to weave the best kilim in the world.

It’s been sitting in their collections, while those Sufis spend all their time searching for radical numerology in sacred geometry. You let some people read Marx…

In Ernie’s book I Don’t Want to Live on the Moon, the cover shows stars shining right through a crescent moon. And in a “C in space” segment, the moon looks surprisingly happy, despite the fact that these stars are shining through it. OK, yes, the moon having a face and emotions is not astronomically accurate either, but that is still no excuse for teaching children inaccurate geometry. I expect more from a supposedly “educational” program. The only explanation I can think of is that, in the extended Sesame Street universe, there are Muppet bases on the moon, and those are the dots of light we are seeing.

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.

Barabási and his team pointed out that scale-free networks also embody a compromise bearing the stamp of natural selection: They are inherently resistant to random failures, yet vulnerable to deliberate attack against their hubs. Given that mutations occur at random, natural selection favors designs that can tolerate haphazard insults. By their very geometry, scale-free networks are robust with respect to random failures, because the vast majority of nodes have few links and are therefore expendable. Unfortunately, this evolutionary design has a downside. When hubs are selectively targeted (something that random mutation could never do), the integrity of the network degrades rapidly—the size of the giant component collapses and the average path length swells, as nodes become isolated, cast adrift on their own little islands.

The geometry of a universe is very like the grammatical structure of a sentence. Just as a sentence has no structure and no existence apart from the relationships between the words, space has no existence apart from the relationships that hold between the things in the universe. If you change a sentence by taking some words out, or changing their order, its grammatical structure changes. Similarly, the geometry of space changes when the things in the universe change their relationships to one another.