Geometry Related Quotes

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

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.

Buckminster Fuller explained to me once that because our world is constructed from geometric relations like the Golden Ratio or the Fibonacci Series, by thinking about geometry all the time, you could organize and harmonize your life with the structure of the world.

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

result: every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for ‘distances’, the ‘distance’ being represented physically by means of the convention of two marks on a rigid body.

There is no reason, therefore, so far as I am able to perceive, to deny the ultimate and absolute philosophical validity of a theory of geometry which regards space as composed of points, and not as a mere assemblage of relations between non-spatial terms.

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

Fractals are a kind of geometry, associated with a man named Mandelbrot. Unlike ordinary Euclidean geometry that everybody learns in school—squares and cubes and spheres—fractal geometry appears to describe real objects in the natural world. Mountains and clouds are fractal shapes. So fractals are probably related to reality. Somehow. “Well, Mandelbrot found a remarkable thing with his geometric tools. He found that things looked almost identical at different scales.” “At different scales?” Grant said. “For example,” Malcolm said, “a big mountain, seen from far away, has a certain rugged mountain shape. If you get closer, and examine a small peak of the big mountain, it will have the same mountain shape. In fact, you can go all the way down the scale to a tiny speck of rock, seen under a microscope—it will have the same basic fractal shape as the big mountain.

We thus obtain the following result: every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for ‘distances’, the ‘distance’ being represented physically by means of the convention of two marks on a rigid body.

There is nothing in the world except empty curved space. Matter, charge, electromagnetism, and other fields are only manifestations of the bending of space. Physics is geometry.

The Golden Proportion, sometimes called the Divine Proportion, has come down to us from the beginning of creation. The harmony of this ancient proportion, built into the very structure of creation, can be unlocked with the 'key' ... 528, opening to us its marvelous beauty. Plato called it the most binding of all mathematical relations, and the key to the physics of the cosmos.

I am in doubt as to the propriety of making my first meditations in the place above mentioned matter of discourse; for these are so metaphysical, and so uncommon, as not, perhaps, to be acceptable to every one. And yet, that it may be determined whether the foundations that I have laid are sufficiently secure, I find myself in a measure constrained to advert to them. I had long before remarked that, in relation to practice, it is sometimes necessary to adopt, as if above doubt, opinions which we discern to be highly uncertain, as has been already said; but as I then desired to give my attention solely to the search after truth, I thought that a procedure exactly the opposite was called for, and that I ought to reject as absolutely false all opinions in regard to which I could suppose the least ground for doubt, in order to ascertain whether after that there remained aught in my belief that was wholly indubitable. Accordingly, seeing that our senses sometimes deceive us, I was willing to suppose that there existed nothing really such as they presented to us; and because some men err in reasoning, and fall into paralogisms, even on the simplest matters of geometry, I, convinced that I was as open to error as any other, rejected as false all the reasonings I had hitherto taken for demonstrations; and finally, when I considered that the very same thoughts (presentations) which we experience when awake may also be experienced when we are asleep, while there is at that time not one of them true, I supposed that all the objects (presentations) that had ever entered into my mind when awake, had in them no more truth than the illusions of my dreams. But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be somewhat; and as I observed that this truth, I think, therefore I am ["cogito ergo sum"], was so certain and of such evidence that no ground of doubt, however extravagant, could be alleged by the sceptics capable of shaking it, I concluded that I might, without scruple, accept it as the first principle of the philosophy of which I was in search

I forget if it was the Mathematician of Alexandria who said that geometry is beauty laid bare or the Father of Relativity who made the claim for physics,” Darger said. “She is, in either case, ravishing.

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

She was still getting organized, trying to get the books she'd taken out to fit into the shelf under the stroller. She would shove a book in, and then something, a juice cup, a Binky, or one disturbing Barbie-doll head, would fall out the other side. She would shove that back in, and then something else would leak out the other side. Her stroller was like a poorly designed clown car. I went over and helped. It was a good thing spatial relations were a strength of mine, because it required the geometry skills of Newton to get everything slotted into place.

Even if a particle could travel backward in time, information could not. Retrocausality will be replaced by something more sophisticated. There are no perfect symmetries, there is no pure randomness everything is an approximation of something else. Information may appear in a digital form but meaning never does. Spacetime is built up from approximations, not discrete ones and zeros, and the only constant may be ratios. Quantum entanglement and geometry; if we think of a particle as being at one pole of an expanding sphere that is not perfectly symmetrical, this surface would be "rippling" like the surface of the ocean (in the audio world this is called dithering), at the other pole is the entangled particle's pair and it is a property of the sphere that gives the illusion of connectivity. This is not a physical geometry, it is a computational geometry. Is spacetime a product of entanglement? Renate Loll believes that time is not perfectly symmetrical. Her computer models require causality. Possibly some form of quantum random walk in state space. If a photon is emitted by an electron inside of a clock on Earth and it travels to a clock four light years away, time stops for the clock on Earth and time jumps forward eight years for the distant clock also, the electron that will capture the photon becomes infinitely large relative to the photon but the electron that emitted it does not become infinitely small therefore, time is not perfectly symmetrical.

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

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

One is ordinarily accustomed to study geometry divorced from any relation between its concepts and experience... This is satisfactory to the pure mathematician. He is satisfied if he can deduce his theorems from axioms correctly, that is, without errors of logic. The question as to whether Euclidean geometry is true or not does not concern him... The physicist is concerned with the question as to whether the theorems of geometry are true or not. That Euclidean geometry, from this point of view, affirms something more than the mere deductions derived logically from definitions may be seen from the following simple consideration.

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.

No one is alone in this world. No act is without consequences for others. It is a tenet of chaos theory that, in dynamical systems, the outcome of any process is sensitive to its starting point-or, in the famous cliche, the flap of a butterfly's wings in the Amazon can cause a tornado in Texas. I do not assert markets are chaotic, though my fractal geometry is one of the primary mathematical tools of "chaology." But clearly, the global economy is an unfathomably complicated machine. To all the complexity of the physical world of weather, crops, ores, and factories, you add the psychological complexity of men acting on their fleeting expectations of what may or may not happen-sheer phantasms. Companies and stock prices, trade flows and currency rates, crop yields and commodity futures-all are inter-related to one degree or another, in ways we have barely begun to understand. In such a world, it is common sense that events in the distant past continue to echo in the present.

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

Thus, the double unification given by the equivalence principle becomes a triple unification: All motions are equivalent once the effects of gravity are taken into account, gravity is indistinguishable from acceleration, and the gravitational field is unified with the geometry of space and time. When worked out in detail, this became Einstein's general theory of relativity, which he published in full form in 1915.

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

I will argue that five features are present in a wide range of genres, Western and non-Western, past and present, and that they jointly contribute to a sense of tonality: 1. Conjunct melodic motion. Melodies tend to move by short distances from note to note. 2. Acoustic consonance. Consonant harmonies are preferred to dissonant harmonies, and tend to be used at points of musical stability. 3. Harmonic consistency. The harmonies in a passage of music, whatever they may be, tend to be structurally similar to one another. 4. Limited macroharmony. I use the term “macroharmony” to refer to the total collection of notes heard over moderate spans of musical time. Tonal music tends to use relatively small

And barbarians were inventors not only of philosophy, but almost of every art. The Egyptians were the first to introduce astrology among men. Similarly also the Chaldeans. The Egyptians first showed how to burn lamps, and divided the year into twelve months, prohibited intercourse with women in the temples, and enacted that no one should enter the temples from a woman without bathing. Again, they were the inventors of geometry. There are some who say that the Carians invented prognostication by the stars. The Phrygians were the first who attended to the flight of birds. And the Tuscans, neighbours of Italy, were adepts at the art of the Haruspex. The Isaurians and the Arabians invented augury, as the Telmesians divination by dreams. The Etruscans invented the trumpet, and the Phrygians the flute. For Olympus and Marsyas were Phrygians. And Cadmus, the inventor of letters among the Greeks, as Euphorus says, was a Phoenician; whence also Herodotus writes that they were called Phoenician letters. And they say that the Phoenicians and the Syrians first invented letters; and that Apis, an aboriginal inhabitant of Egypt, invented the healing art before Io came into Egypt. But afterwards they say that Asclepius improved the art. Atlas the Libyan was the first who built a ship and navigated the sea. Kelmis and Damnaneus, Idaean Dactyli, first discovered iron in Cyprus. Another Idaean discovered the tempering of brass; according to Hesiod, a Scythian. The Thracians first invented what is called a scimitar (arph), -- it is a curved sword, -- and were the first to use shields on horseback. Similarly also the Illyrians invented the shield (pelth). Besides, they say that the Tuscans invented the art of moulding clay; and that Itanus (he was a Samnite) first fashioned the oblong shield (qureos). Cadmus the Phoenician invented stonecutting, and discovered the gold mines on the Pangaean mountain. Further, another nation, the Cappadocians, first invented the instrument called the nabla, and the Assyrians in the same way the dichord. The Carthaginians were the first that constructed a triterme; and it was built by Bosporus, an aboriginal. Medea, the daughter of Æetas, a Colchian, first invented the dyeing of hair. Besides, the Noropes (they are a Paeonian race, and are now called the Norici) worked copper, and were the first that purified iron. Amycus the king of the Bebryci was the first inventor of boxing-gloves. In music, Olympus the Mysian practised the Lydian harmony; and the people called Troglodytes invented the sambuca, a musical instrument. It is said that the crooked pipe was invented by Satyrus the Phrygian; likewise also diatonic harmony by Hyagnis, a Phrygian too; and notes by Olympus, a Phrygian; as also the Phrygian harmony, and the half-Phrygian and the half-Lydian, by Marsyas, who belonged to the same region as those mentioned above. And the Doric was invented by Thamyris the Thracian. We have heard that the Persians were the first who fashioned the chariot, and bed, and footstool; and the Sidonians the first to construct a trireme. The Sicilians, close to Italy, were the first inventors of the phorminx, which is not much inferior to the lyre. And they invented castanets. In the time of Semiramis queen of the Assyrians, they relate that linen garments were invented. And Hellanicus says that Atossa queen of the Persians was the first who composed a letter. These things are reported by Seame of Mitylene, Theophrastus of Ephesus, Cydippus of Mantinea also Antiphanes, Aristodemus, and Aristotle and besides these, Philostephanus, and also Strato the Peripatetic, in his books Concerning Inventions. I have added a few details from them, in order to confirm the inventive and practically useful genius of the barbarians, by whom the Greeks profited in their studies. And if any one objects to the barbarous language, Anacharsis says, "All the Greeks speak Scythian to me." [...]

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

The Republic, Plato writes: The stars that decorate the sky, though we rightly regard them as the finest and most perfect of visible things, are far inferior, just because they are visible, to the true realities; that is, to the true relative velocities, in pure numbers and perfect figures, of the orbits and what they carry in them, which are perceptible to reason and thought but not visible to the eye … We shall therefore treat astronomy, like geometry, as setting us problems for solution, and ignore the visible heavens, if we want to make a genuine study of the subject …127 This separation of the absolute and eternal, which can be known by logos (reason), from the purely phenomenological, which is now seen as inferior, leaves an indelible stamp on the history of Western philosophy for the subsequent two thousand years.

The scientific principles that man employs to obtain the foreknowledge of an eclipse, or of anything else relating to the motion of the heavenly bodies, are contained chiefly in that part of science which is called trigonometry, or the properties of a triangle, which, when applied to the study of the heavenly bodies, is called astronomy; when applied to direct the course of a ship on the ocean, it is called navigation; when applied to the construction of figures drawn by rule and compass, it is called geometry; when applied to the construction of plans or edifices, it is called architecture; when applied to the measurement of any portion of the surface of the earth, it is called land surveying. In fine, it is the soul of science; it is an eternal truth; it contains the mathematical demonstration of which man speaks, and the extent of its uses is unknown.

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.

In the end, what is most satisfying about the picture of space given by loop quantum gravity is that it is completely relational. The spin networks do not live in space; their structure generates space. And they are nothing but a structure of relations, governed by how the edges are tied together at the nodes. Also coded in are rules about how the edges may knot and link with one another. It is also very satisfying that there is a complete correspondence between the classical and quantum pictures of geometry. In classical geometry the volumes of regions and the areas of the surfaces depend on the values of gravitational fields. They are coded in certain complicated collections of mathematical functions, known collectively as the metric tensor. On the other hand, in the quantum picture the geometry is coded in the choice of a spin network. These spin networks correspond to the classical description in that, given any classical geometry, one can find a spin network which describes, to some level of approximation, the same geometry (Figure 27).

For example, the central idea in Einstein's theory of general relativity is that gravity is not some mysterious, attractive force that acts across space but rather a manifestation of the geometry of the inextricably linked space and time. Let me explain, using a simple example, how a geometrical property of space could be perceived as an attractive force, such as gravity. Imagine two people who start to travel precisely northward from two different point on Earth's equator. This means that at their starting points, these people travel along parallel lines (two longitudes), which, according to the plane geometry we learn in school, should never meet. Clearly, however, these two people will meet at the North Pole. if these people did not know that they were really traveling on the curved surface of a sphere, they would conclude that they must have experienced some attractive force, since they arrived at the same point in spite of starting their motions along parallel lines. Therefore, the geometrical curvature of space can manifest itself as an attractive force.

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

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

The 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

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

If dimensions are virtual like the particles in quantum foam are virtual then, entanglement is information that is in more than one location (hologram). There are no particles, they may be wave packets but the idea of quantum is, a precise ratio of action in relationship to the environment. Feynman's path integral is not infinite, it is fractal. If you look at a star many light years away, the photon that hits your eye leaves the star precisely when the timing for the journey will end at your eye because the virtual dimension of the journey is zero distance or zero time. Wheeler said that if your eye is not there to receive the photon then it won't leave the star in the distant past. If the dimension in the direction of travel is zero, you have a different relationship then if it is zero time in terms of the property of the virtual dimensions. Is a particle really a wave packet? Could something like a "phase transition" involve dimensions that are more transitory then we imagined. Example; a photon as a two dimensional sheet is absorbed by an electron so that the photon becomes a part of the geometry of the electron in which the electrons dimensions change in some manner. Could "scale" have more variation and influence on space and time that our models currently predict? Could information, scale, and gravity be intimately related?

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.

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.

Hume begins by distinguishing seven kinds of philosophical relation: resemblance, identity, relations of time and place, proportion in quantity or number, degrees in any quality, contrariety, and causation. These, he says, may be divided into two kinds: those that depend only on the ideas, and those that can be changed without any change in the ideas. Of the first kind are resemblance, contrariety, degrees in quality, and proportions in quantity or number. But spatio-temporal and causal relations are of the second kind. Only relations of the first kind give certain knowledge; our knowledge concerning the others is only probable. Algebra and arithmetic are the only sciences in which we can carry on a long chain of reasoning without losing certainty. Geometry is not so certain as algebra and arithmetic, because we cannot be sure of the truth of its axioms. It is a mistake to suppose, as many philosophers do, that the ideas of mathematics 'must be comprehended by a pure and intellectual view, of which the superior faculties of the soul are alone capable'. The falsehood of this view is evident, says Hume, as soon as we remember that 'all our ideas are copied from our impressions'. The three relations that depend not only on ideas are identity, spatio-temporal relations, and causation. In the first two, the mind does not go beyond what is immediately present to the senses. (Spatio-temporal relations, Hume holds, can be perceived, and can form parts of impressions.) Causation alone enables us to infer some thing or occurrence from some other thing or occurrence: "'Tis only causation, which produces such a connexion, as to give us assurance from the existence or action of one object, that 'twas followed or preceded by any other existence or action.

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.

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.

Foundational Principles: Nature is lazy, it likes to copy. Everything is information. Information is not stuff, it is relationships. All behaviors are constrained by relationships. All behaviors are emergent. Every engine takes advantage of a difference. Everything is an approximation of something else. Ratio may be the only thing that is discrete. The bending of spacetime is a variation of scale. Behavior is built from a quantum of action in a field. If dimensions are virtual in the same way that the dimensions of consciousness are virtual then the density of information in a field will affect the scale or bending of spacetime. Gravity and scale are related. Gravity and information are related. Information and scale are related. If it's relational, there's a geometry involved. Truth as a scale coordinate; truth lives in the macro world, the micro world is uncertain. Truth as a time coordinate; truth lives in the past, the future is uncertain. Information from the micro future is formed into a macro past. The process of formation involves entanglement. Coffee and cream; 1) separate, 2) complex, 3) homogeneous. Information appears to increase and then decrease.

It is impossible to consider the mechanism of our intellect and the progress of our science without arriving at the conclusion that between intellect and matter there is, in fact, symmetry, concord and agreement. On one hand, matter resolves itself more and more, in the eyes of the scholar, into mathematical relations, and on the other hand, the essential faculties of our intellect function with an absolute precision only when they are applied to geometry.

We will focus attention on binary relations as these are by far the most important. If R ⊆ S × S is a binary relation on S, it is common to use the notation aRb in place of (a, b) ∈ R.

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

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.

There is one are of work that should be mentioned here, referred to as 'automatic theorem proving'. One set of procedures that would come under this heading consists of fixing some formal system H, and trying to derive theorems within this system. We recall, from 2.9, that it would be an entirely computational matter to provide proofs of all the theorems of H one after the other. This kind of thing can be automated, but if done without further thought or insight, such an operation would be likely to be immensely inefficient. However, with the employment of such insight in the setting up of the computational procedures, some quite impressive results have been obtained. In one of these schemes (Chou 1988), the rules of Euclidean geometry have been translated into a very effective system for proving (and sometimes discovering) geometrical theorems. As an example of one of these, a geometrical proposition known as V. Thebault's conjecture, which had been proposed in 1938 (and only rather recently proved, by K.B. Taylor in 1983), was presented to the system and solved in 44 hours' computing time. More closely analogous to the procedures discussed in the previous sections are attempts by various people over the past 10 years or so to provide 'artificial intelligence' procedures for mathematical 'understanding'. I hope it is clear from the arguments that I have given, that whatever these systems do achieve, what they do not do is obtain any actual mathematical understanding! Somewhat related to this are attempts to find automatic theorem-generating systems, where the system is set up to find theorems that are regarded as 'interesting'-according to certain criteria that the computational system is provided with. I do think that it would be generally accepted that nothing of very great actual mathematical interest has yet come out of these attempts. Of course, it would be argued that these are early days yet, and perhaps one may expect something much more exciting to come out of them in the future. However, it should be clear to anyone who has read this far, that I myself regard the entire enterprise as unlikely to lead to much that is genuinely positive, except to emphasize what such systems do not achieve.

Adam: Adam was a young man whose anxiety turned into a monster. Where Shelly had a very mild case of social anxiety, Adam’s case could only be called severe. Over a period of several years, his underlying social fears developed into a full-blown school phobia. A quiet, unassuming person, Adam had never stood out in the classroom. Through elementary school and on into high school, he neither excelled nor failed his subjects. By no means a discipline problem, the “shy” Adam kept to himself and seldom talked in class, whether to answer a teacher’s question or chat with his buddies. In fact, he really had no friends, and the only peers he socialized with were his cousins, whom he saw at weekly family gatherings. Though he watched the other kids working together on projects or playing sports together, Adam never approached them to join in. Maybe they wouldn’t let him, he thought. Maybe he wasn’t good enough. Being rejected was not a chance he was willing to take. Adam never tried hard in school either. If he didn’t understand something, he kept quiet, fearful that raising his hand would bring ridicule. When he did poorly on an exam or paper, it only confirmed to him what he was sure was true: He didn’t measure up. He became so apprehensive about his tests that he began to feel physically ill at the thought of each approaching reminder of his inadequacy. Even though he had studied hard for a math test, for example, he could barely bring himself to get out of bed on the morning it was to take place. His parents, who thought of their child as a reserved but obedient boy who would eventually grow out of this awkward adolescent stage, did not pressure him. Adam was defensive and withdrawn, overwrought by the looming possibility that he would fail. For the two class periods preceding the math test, Adam’s mind was awash with geometry theorems, and his stomach churning. As waves of nausea washed over him, he began to salivate and swallowed hard. His eyes burned and he closed them, wishing he could block the test from his mind. When his head started to feel heavy and he became short of breath, he asked for a hall pass and headed for the bathroom. Alone, he let his anxiety overtake him as he stared into the mirror, letting the cool water flow from the faucet and onto his sweaty palms. He would feel better, he thought, if he could just throw up. But even when he forced his finger down his throat, there was no relief. His dry heaves made him feel even weaker. He slumped to the cold tile and began to cry. Adam never went back to math class that day; instead, he got a pass from the nurse and went straight home. Of course, the pressure Adam was feeling was not just related to the math test. The roots of his anxiety went much deeper. Still, the physical symptoms of anxiety became so debilitating that he eventually quit going to school altogether. Naturally, his parents were extremely concerned but also uncertain what to do. It took almost a year before Adam was sufficiently in control of his symptoms to return to school.

said that in non-Euclidean geometry space has a curvature, but this way of stating the matter is misleading, since it seems to imply a fourth dimension, which is not implied by these systems.

Albert Einstein was certainly the most important physicist of the twentieth century. Perhaps his greatest work was his discovery of general relativity, which is the best theory we have so far of space, time, motion, and gravitation. His profound insight was that gravity and motion are intimately related to each other and to the geometry of space and time. This idea broke with hundreds of years of thinking about the nature of space and time, which until then had been viewed as fixed and absolute. Being eternal and unchanging, they provided a background, which we used to define notions like position and energy.

Early education,” says President Thwing, “occupies itself with description (geometry, space, arithmetic, time, science, the world of nature). Later education with comparison and relations.

Early education,” says President Thwing, “occupies itself with description (geometry, space, arithmetic, time, science, the world of nature). Later education with comparison and relations.” If one asks, “Why not both together? Why learn facts at one time and their relations at another? Is it not the most vital possible way to learn facts to learn them in their relations?”—the answer that would be generally made reveals that most teachers are pessimists, that they have very small faith in what can be expected of the youngest pupils. The theory is that interpretative minds must not be expected of them. Some of us find it very hard to believe as little as this, in any child. Most children have such an incorrigible tendency for putting things together that they even put them together wrong rather than not put them together at all. Under existing educational conditions a child is more of a philosopher at six than he is at twenty-six. The third stage of education for which Dr. Thwing partitions off the human mind is the “stage in which a pupil becomes capable of original research, a discoverer of facts and relations” himself. In theory this means that when a man is thirty years old and all possible habits of originality have been trained out of him, he should be allowed to be original. In practice it means removing a man’s brain for thirty years and then telling him he can think.

Early education,” says President Thwing, “occupies itself with description (geometry, space, arithmetic, time, science, the world of nature). Later education with comparison and relations.” If one asks, “Why not both together? Why learn facts at one time and their relations at another? Is it not the most vital possible way to learn facts to learn them in their relations?”—the answer that would be generally made reveals that most teachers are pessimists, that they have very small faith in what can be expected of the youngest pupils. The theory is that interpretative minds must not be expected of them. Some of us find it very hard to believe as little as this, in any child. Most children have such an incorrigible tendency for putting things together that they even put them together wrong rather than not put them together at all. Under existing educational conditions a child is more of a philosopher at six than he is at twenty-six.

Descartes arrives at four precepts that “would prove perfectly sufficient for me, provided I took the firm and unwavering resolution never in a single instance to fail in observing them.” They amount to a kind of diagram for how to think. He writes: The first was never to accept anything for true which I did not clearly know to be such … to comprise nothing more in my judgment than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt. The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution. The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence. And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.

The process of objectifying the world through the primordial intuition of "repetition in time" and "following in time" gains in generality by the construction of mathematics from the same primordial intuition, without reference to direct applicability. In this way man has a ready-made supply of unreal causal sequences at his disposal, just waiting for an opportunity to be projected into reality. One should bear in mind that in mathematical systems with no time coordinate, all relations in practical applications clearly become causal relations in time; e.g. Euclidean geometry when applied to reality shows a causal connection between the results of different measurements made by means of the group of rigid bodies. Needless to say, in the application of a mathematical system, in general, only a fraction of the elements and substructures finds their correspondence in reality; the remainder plays the role of and unreal "physical hypothesis." Similarly, even with a limited development of method, the observed sequences no longer consist exclusively of phenomena evoked by man himself (acts without any direct instinctive aim, but carried out solely to complete the causal system into a more manageable one). The simplest example is the sound image (or written symbol) of number as a result of counting, or the sound image (or written symbol) of number as a result of measuring (this example shows how infinitely many causal sequences can be brought together under the viewpoint of one single law of causality on the basis of a mapping the numbers through mathematical induction.)

Various geometries are simply different ways of presenting relations among things; which we use depends on convenience.

According to quantum mechanics, at the Planck scale length, instead of a gradually undulating geometry, there should be wild fluctuations and loops and handles of spacetime branching off, the sort of topography that the futuristic Ike encountered. General relativity cannot be used in such untamed territory.

Non-Euclidean geometry is a study of which the primary motive was logical and philosophical;

The prime advantage of defining the geometry of the pebble's surface in terms of general coordinates is that we can then proceed to define tensors that are true in any coordinate system, tensors that describe both the pebble's curvature (the Riemann curvature tensor) and physics (the field equations, for example).

When space-time is curved, even the straightest possible paths acquire bumps and wiggles, because they must adapt to changes in the local geometry. Putting these ideas together, we say that bodies respond to the metric field. These bumps and wiggles in a body's space-time trajectory-in more pedestrian language, changes in its direction and speed-provide, according to general relativity, an alternative and more accurate description of the effects formerly known as gravity.

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.

With a digital display having few pixels, symmetries are common but there is very little meaning because the image is very course grained. As we reiterate and begin breaking the symmetry of the individual pixels an image will begin to appear. Time is related to the process of reiteration and truth is related to the symmetry, with meaning being related to the image created. we do not know where the symmetry or the reiterations come from but the image is emergent. The idea of a quantum random walk in state space says that every complex event is statistically impossible and even though the probability space is very large, it is navigated and expressed, as I understand it, in a tree like structure or a fractal structure. This is a computational expression of the material world that looks very much like a display on a monitor. The decision engine is generating value. There is a bifurcation of the fitness into different dimensions and like the human brain which is said to have at least eleven dimensions, the dimensions are not constrained by a physical geometry, they are computational. Another way we can look at this would be to say that every behavior we can measure is constrained by a network of associations just like the nodes on the internet and the conservation laws become approximately true because of levels of description. All material expressions are constructed from a network of associations and are only reducible to some degree of resolution. If we are talking about information, then it is only reducible to some approximate explanation.

Insofar as the general theory of relativity follows Mach's ideas, it can be seen as expressing the distribution of masses in terms of properties of space itself. It all boils down to geometry-to the web of shortest distance between points in space-time. Here, all that exists can be expressed in terms of geometry. It is the motions occurring under the influence of all the masses of the universe that make up the elements of the geometrical web. The identity and the location of those masses can be read off the curvature of space-that is, off a geometrical quantity. Seen in this way, the general theory of relativity is a model for other, nongeometrical theories where the fields are added to space, just as we might add colors to a blank canvas; this is the way we treat electric or magnetic fields. But let us recall that those latter field theories were more realistic insofar as they quantize the field. It is only through the quantum effects that our hypothetical empty space becomes the physical vacuum.

Moreover, there is no known reason why the geometry of space and time should be described by the particular types of curved geometry defined by Riemann. There exist other more complicated varieties that could in principle have been employed by Nature. Only observation can at present tell us which mathematics is chosen by Nature for employment in particular situations. This may of course merely be a transient manifestation of our relative ignorance of the bigger picture in which everything that is not excluded is demanded.

Basically, we formulate geometry by inspecting the relations between displacements of material objects. Then we abstract to the realization that the goal of science “is not the things themselves . . . it is the relations between things; outside of these relations there is no reality knowable.

Moreover, as we saw in Problems with string theory,‡ while each string theory is internally consistent, Smolin presents a strong case against external consistency with evidence, concluding that “all the versions we can study in any detail disagree with observation”. He also maintains that it is externally inconsistent with the scientific tenet of relativity theory: “Einstein’s discovery that the geometry of space and time is dynamical has not been incorporated into string theory.” Without more positive results from these tests of reasonableness for a scientific (as distinct from a mathematical) conjecture, it is difficult to see how the speculated existence of other dimensions is any more tenable than the belief of many Buddhist schools that there are 31 distinct realms of existence. Furthermore, the so far untestable idea that the matter of the universe reduces not to fundamental particles but to strings of energy seems no more or less reasonable than the Upanishadic insight that prana (vital energy) is the essential substrate of all forms of energy and, in many interpretations, of all matter.

twisting doesn’t involve a trajectory between two points in space because ships don’t actually travel the intervening distance. Rather, it involves geometry. And there is an optimum twist geometry that relates any two points, that includes things like the gravitational effects of intervening masses such as stars, as well as the effects of time,

We tend to separate subjects or "areas of knowledge" in our heads, whereas in Universe everything is synergetically (holistically) related. In this case, we have separated geometry from evolution, if we are blocked, and that is why we cannot see a rather obvious answer.

use four-dimensional Riemannian geometry for his famed theory of relativity. Within seven decades, Theodr Kaluza at the University of Königsberg, Germany, would use five-dimensional Riemannian geometry to integrate both gravity and light. Light is now viewed as a vibration in the fifth dimension. Oskar Klein made several improvements, including the calculation of the size of the fifth dimension—the Planck length, which is 10-33 centimeters, much too small to detect experimentally. One hundred thirty years after Riemann’s famous lecture, physicists would extend the Kaluza-Klein constructs to develop ten-

FDM (Fused Deposition Modeling) 3D printing is a type of additive manufacturing technology that works by extruding thermoplastic filament material layer by layer to build up a three-dimensional object. Here are some details defining FDM 3D printing: Process: FDM 3D printing involves melting a thermoplastic filament, usually ABS (Acrylonitrile Butadiene Styrene) or PLA (Polylactic Acid), and extruding it through a heated nozzle. The nozzle moves along a predetermined path, depositing the material layer by layer to create the desired object. Materials: FDM printers primarily use thermoplastic materials, which are available in various colors and types, each with its own properties such as strength, flexibility, and heat resistance. Common materials include ABS, PLA, PETG, TPU, and more. Layer Resolution: FDM printers have a layer resolution, which refers to the thickness of each layer of material deposited during printing. The layer resolution determines the level of detail and surface finish achievable in the printed object. Lower layer heights result in finer details but increase printing time. Build Volume: This refers to the maximum size of the object that can be printed in terms of length, width, and height. FDM printers come in various sizes, offering different build volumes to accommodate different project requirements. Support Structures: FDM printers often require support structures for overhanging or complex geometries. These supports are printed alongside the object and later removed manually or with tools after printing is complete. Heated Build Plate: Many FDM printers feature a heated build plate, which helps prevent warping and improves adhesion between the first layer of the print and the build surface. A heated build plate is particularly useful when printing materials like ABS. Dual Extrusion: Some FDM printers support dual extrusion, allowing for the simultaneous use of two different materials or colors during printing. This capability enables more complex prints with multiple colors or materials. Post-Processing: After printing, FDM-printed objects may require post-processing to improve surface finish or functionality. This can include sanding, painting, smoothing with acetone (for ABS), or other finishing techniques. FDM 3D printing is widely used in various industries, including prototyping, manufacturing, education, and hobbyist applications, due to its relatively low cost, ease of use, and versatility.

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.

The important point here is that music for the Greeks and the wider classical tradition was not so much understood as something performed, composed, practiced, or played; rather, music was interpreted as a mathematical discipline that sought to discover and formalize the symmetrical relations between sounds.6 It was an integral component to the mathematical disciplines that comprised the quadrivium: arithmetic, geometry, music, and astronomy. For the classical mind, arithmetic revealed “number in itself,” geometry revealed “number in space,” music revealed “number in time,” and astronomy revealed “number in space and time.” In this sense, music was an integral part of the Greek educational curriculum which functioned as a metaphor for this whole cosmic chain of interrelationships and harmonies. Indeed, Plato could say: “The whole choral art is also in our view the whole of education” (Laws, Bk II). The Greeks understood the nature of reality and its systems of relations in musical terms.

[Notice: Visual Enhancement in effect, co-relating known physical geometry, self-generated schematics, and acoustic co-locators and simulating them into Avatar visual cortex.] She translated that to: “Turning radar into vision while transmitting things directly into your brain without permission.

Analytic truths also take a blow from the fact that dictionaries need to be updated from time to time. The meaning and usage of words is not static and unchanging; there is nothing 'necessary' about it. Hence, trying to base analytic truths on definitions, which are supposedly immune to revision and absolutely certain, would seem to be risky business. This is true even of scientific terms. For example, at one point the statement 'Atoms are indivisible' would have been widely accepted as analytic, but not so today. At one point in history, Euclidean geometry and Newtonian physics appeared to provide analytic truths. However, the rise of Riemannian geometry and Einstein's relativity theory made analytic truths in those areas debatable.

Albert Einstein’s general theory of relativity explains the force of gravity as an illusion generated by the geometry of space-time. Because gravity is geometry, any story about the history of the universe is a story about its structure in disguise.

Even holding a position in the academic world is not a road to becoming more fulfilled or creative. In the absence of a strong women’s movement working in academia can be stifling, because you have to meet standards you do not have the power to determine and soon you begin to speak a language that is not your own. From this point of view it does not make any difference whether you teach Euclidean geometry or women’s history, though women’s studies still provide an enclave that, relatively speaking, allows us to be “more free.” But little islands are not enough. It is our relation to intellectual work and academic institutions that has to be changed. Women’s Studies are reserved to those who can pay or are willing to make a sacrifice, adding a school day to the workday in continuing education courses. But all women should have free access to school, for as long as studying is a commodity we have to pay for, or a step in the “job hunt,” our relation to intellectual work cannot be a liberating experience.

All greek civilization is a search for bridges to relate human misery and divine perfection. Their art, which is incomparable, their poetry, their philosophy, the sciences which they invented (geometry, astronomy, mechanics, physics, biology) are nothing but bridges.

Culture, which is not explicit knowledge, intellectual possession, but which is nonignorance, knowledge of ignorance, presence in me of the past as past, and which assures the communication between me and history, because the tradition is forgetful of origins, relation to an origin which is not possessed by the present, and which works in us and provokes geometry in advance, precisely because it is not possessed by thought. What I have in my presence in order to understand the past is a tradition, that is, a fullness made out of a certain emptiness (out of a certain 'forgetfulness'), a circumscribed negativity, which therefore makes a reference to the outside.

understanding of its relation to the occult, there are entire documentaries dedicated to this ‘sacred geometry’ and these ‘magical’ properties.

The ramifications of Pythagoras' theorem have revolutionized twentieth century theoretical physics in many ways. For example, Minkowski discovered that Einstein's special theory of relativity could be represented by four-dimensional pseudo-Euclidean geometry where time is represented as the fourth dimension and a minus sign is introduced into Pythagoras' law. When gravitation is present, Einstein proposed that Minkowski's geometry must be "curved", the pseudo-Euclidean structure holding only locally at each point. A complex vector space having a natural generalization of the Pythagorean structure (defined over functions in an abstract space rather than geometrical points in the familiar Euclidean space) is known as Hilbert space and forms the basis of quantum mechanics. It is remarkable to think that the two pillars of twentieth century physics, relativity and quantum theory, both have their basis in mathematical structures based on a theorem formulated by an eccentric mathematician over two and a half thousand years ago.

We are trying to show not that mathematical thought rests upon the sensible but that it is creative...Non-Euclidean geometries contain Euclid's geometry as a particular case but not the inverse. What is essential to mathematical thought, therefore, lies in the moment where a structure is decentered, opens up to questioning, and reorganizes itself according to a new meaning which is nevertheless the meaning of this same structure. The truth of the result, its value independent of the content, consists in its not involving a change in which the initial relations dissolve, to be replaced by others in which they would be unrecognizable. Rather, the truth lies in a restructuring which, from one end to the other, is known to itself, is congruent with itself, a restructuring which was announced in the vectors of the initial structure by its style, so that each effective change is the fulfillment of an intention, and each anticipation receives from the structure the completion it needed.

The fact something is difficult does not mean to be confused.

The ordering of knowledge has changed with the centuries. All knowledge was once ordered in relation to the seven liberal arts— grammar, rhetoric, and logic, the trivium; arithmetic, geometry astronomy, and music, the quadrivium. Medieval encyclopedias reflected this arrangement. Since the universities were arranged according to the same system, and students studied according to it also, the arrangement was useful in education. [How to Read a Book (1972), P. 180]

Geometry is primarily a set of invariant rela- tionships between rays, points, and curvatures. It was Fludd’s conviction, just as it became for a thinker like René Thom, that this set of relation- ships, reified within the fundamental pneumatic realm, both proved and provided the rational undergirding of phenomena.

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

Michael Berry wrote a paper called Regular and Irregular Motion. He determined the predictability of the path of a struck billiards ball. The path after the first impact is easy to determine with basic geometry. The second path is a bit harder, but still relatively easy. By the 9th impact Berry determined that the gravitational pull of a person standing next to the table had enough of an influence on the ball path that it was necessary to be included in the calculations.

As you begin to befriend your inner silence, one of the first things you will notice is the superficial chatter on the surface level of your mind. Once you recognize this, the silence deepens. A distinction begins to emerge between the images that you have of yourself and your own deeper nature. Sometimes much of the conflict in our spirituality has nothing to do with our deeper nature but rather with the false surface constructs we build. We then get caught in working out a grammar and geometry of how these surface images and positions relate to each other; meanwhile our deeper nature remains unattended.

Dawkins’s memes are relatively trivial patterns with a lim- ited “life span”; an attractor, in its very definition, is possessed of a profound (deep) geometry that lies outside of temporal considerations.

The term “real” is a misnomer. You could say that the life forms that occupy earth are real. They appear as physical structures and they are based on mathematical formula and geometry. We are, in a sense, code. This code interacts with the physical senses, which decodes the formula and we are transformed to flesh and blood “life forms.” But at a quantum level we are, as I said, code. Underneath or beyond the quantum level—call it the pre-quantum level—we are infinite beings. We are Sovereign Integrals. What I am suggesting is that the term real, applies to the Sovereign Integral state of being and consciousness. The term is relative, however, because the Sovereign Integral consciousness is not considered real in the three-dimensional plane, and the human instrument is not real in the Domain of Unity. It is the reality of the dominant environment that determines the reality of the subjects and objects of that environment. The environment of three and four dimensional planes like earth, establishes a consensual, subjective reality—propagated by our unconscious and genetic mind. The environment of the infinite establishes a reality that has no mediation or signal transformation. It is elemental and core. There is no sensory “middleman” or propagation of perception. It is one and all in action. The individual is allowed to decide what is real and what is not. Illusion is a Russian doll; it has many layers, some of which transmit a sense of true awareness. I’m sure you’ve heard the phrase: Perception is reality. The subjective realm defines the seemingly objective. Illusion, the outermost doll, is the only one that is seen and therefore, known. However, if you open the outermost doll, you find another, albeit smaller one waits. This repeats seven times until you find the indivisible. We would call this core, innermost doll: the Sovereign Integral. This is where you want your imaginative faculty trained and honed in. This is where you want your life to progress, not satisfied that perception is reality, or that the outermost “doll” is real and worthy of your devotion. All of that said, it does not mean that the outer world—physical reality—is a waste of time or so mired in illusion that it isn’t worth developing. Quite the opposite. The outer world is your workshop, the place you can experiment, build things, create, try and fail, and so on. It is a place of severe challenges at time, but also of beauty and joy. The illusion isn’t in the physical things of this world. The illusion is in the self-perception of the individual life form.As long as the individual perceives themselves as a human instrument, maybe with a soul, maybe not, they will see all life as a place of separation and disunity. They will accept the illusion that life emits, which is one of separation. In this, they become lost, lost to themselves as infinite beings, and unable, as a result, to generate and sustain the vision into their true self.