conversations.from: human
conversations.value: a description for a book titled The Big Questions about Professor, Questions, Urging with the genre: Politics & Social Sciences, Philosophy
conversations.from: assistant
conversations.value: Professor the author's first book drew on the wisdom of the great philosophers to solve our everyday problems, launching a movement that restored philosophy to what it once was: useful in all walks of life. Now, in The Big Questions, he takes the concept to the next level, applying centuries of philosophy and great literature to help answer central questions of modern existence. Urging us not to accept victimhood as the by-product of modern life, Professor the author uses specific case studies from his counselling practice to show how wisdom from the great thinkers can help us define our own philosophy, and thereby reclaim our sense of well-being. He asks questions that go to the heart of the human condition: How do we know what is right? What is love? How can we cope with change? Why can't we all get along? And most centrally, how can we use the centuries of wisdom that have come before us to help us answer these questions and feel at ease in the world?
conversations.from: human
conversations.value: the first five paragraphs for this book
conversations.from: assistant
conversations.value: The Big Questions
  The Egyptians based their cosmology on the human reproductive cycle. They believed that the sky goddess Nut gave birth to the Sun god Ra every year and that the changing altitude of the Sun with the seasons was its gestation in Nut's star-studded body. The Sun was said to be reborn every winter solstice, and returned inside Nut through her mouth at the spring equinox. In this way, Ra continually recreated himself, making the Universe an eternal, self-sustaining entity.
  Early civilizations told stories inspired by the patterns of stars in the night sky. They imagined the lines joining stars to form pictures of familiar or mythical characters. In Mesopotamia (modern-day Iraq) archaeologists have unearthed stone tablets and clay ledgers dating back to 1300BC, which detail many such "constellations" including the 12 signs of the zodiac. These zodiacal constellations were given special significance because they inhabited regions of the sky through which the Sun passed, and they were subsequently adopted by the Greeks—the Assyrian Hired Man and the Swallow became Aries and Pisces, for example, and the Goatfish and the Great Twins became Capricorn and Gemini. In Ancient Greece wandering minstrels would drift from village to village, recounting the star myths in exchange for food and lodgings. At the same time, philosophers would come up with their own fanciful tales to explain the nature of the Universe. One of the earliest was the philosopher Thales, in the sixth century BC. He put forward the idea that space was filled with water, in which the Earth floats, that earthquakes were caused by waves in this water, and the stars moved because they were caught in gentler currents.
  The Greek astronomer Claudius Ptolemy, who lived in the first century AD, compiled a list of 48 constellations, but since not all the sky could be seen from Greece, the regions around the South Pole remained uncharted until intrepid astronomers ventured far from Europe during the 16th and 17th centuries in order to chart the southern stars. Other new constellations were also proposed to fill gaps in Ptolemy's classical sky map. Inevitably this led to arguments as astronomers disagreed. In England, Edmond Halley proposed a constellation called Robur Carolinum (Charles' Oak), after the tree in which Charles II had hidden from the Roundheads following the battle of Worcester. While the King was delighted with the honor, some of Halley's fellow astronomers were not keen, and quietly discarded the constellation from their maps.
  Much later, in 1922, things were finally put on a firm footing when the International Astronomical Union ratified 88 constellations with defined boundaries, mostly based on the Greek model. These were not the only aspects of Greek astronomy to pass into modern usage. There was one Ancient Greek in particular who was not prepared to tell stories about the stars, or speculate about them; he realized that the first step on the way to true understanding was to measure them. That man was Hipparchus and he defined a system of classification for the stars still in use today.
  Even to a casual observer of the night sky, it is obvious that some stars are brighter than others. More than two millennia ago, Hipparchus meticulously compiled a catalog of 850 stars, recording each star's position and ranking its brightness. He had no equipment to measure the brightness; he simply made his estimates by eye. The brightest stars he called "first magnitude," the faintest he termed "sixth magnitude," and the rest he ranked in the categories between. Amazingly, astronomers still use this seemingly crude magnitude system today, although modern measuring devices have extended Hipparchus's original six classifications. At the top end of the scale, the very brightest stars are now given negative numbers; at the other end, the stars that can only be seen with the aid of a telescope are assigned magnitudes with numbers often much higher than six. From the surface of the Earth, the best telescopes can detect stars of between 24th and 27th magnitude, but in orbit, above the distorting effects of the Earth's atmosphere, the Hubble Space Telescope can detect 30th magnitude stars. Each magnitude category is about two and half times brighter than the previous one, so 30th magnitude is about 3.5 billion (3500 million) times fainter than the naked eye can see.
conversations.from: human
conversations.value: five more paragraphs
conversations.from: assistant
conversations.value: Yet MOND is not flawless; the hypothesis struggles to reproduce the motion of clusters of galaxies, requiring more matter than can be seen to make things work. So are we back to needing exotic dark matter? Possibly not, as astronomers think that there may be a substantial amount of normal matter hiding in galaxy clusters, in the form of warm gas. If they can detect this, using telescopes sensitive to the ultraviolet light it is believed to emit, it would solve MOND's difficulties with galaxy clusters.
  A particular battleground for the competing theories is the Bullet Cluster, where two galaxy clusters are colliding. Observations show that the gas between the galaxies has been stripped away, and yet weak lensing indicates that a strong curvature still persists around the galaxies. This can be interpreted as either the gigantic halos of dark matter needed to hold both clusters together, or as regions where MOND needs to be applied. Other galaxy cluster collisions show similar behavior, and likewise do not help astronomers decide between the dark matter and MOND theories. But very recently something has been found that may break the stalemate, an object that perplexes everyone: a most peculiar galaxy cluster collision known as Abell 520.
  In this particular cosmic smash-up, weak lensing results imply that somehow the dark matter has been separated from the galaxies. This is completely contrary to what dark matter theory says should happen, and the only way to explain the dark matter behaving like this is to propose that it is interacting with itself, producing a force that only dark matter feels. Such a hypothesis would solve a problem with the Bullet Cluster collision, as well: computer models cannot substantiate the speed at which the clusters are crashing together, but a previously unanticipated dark matter force could provide the extra kick needed.
  At present, the picture is more confusing than it has ever been. MOND does not work in all situations: it needs about twice the amount of matter that we see in the Universe; and dark matter needs a new, exclusive force to make its figures tally. So the bottom line is that no one knows what the dark matter is, or whether it even exists. It may be an illusion brought about by our misunderstanding of gravity. In the next few years, the welter of new experiments—the Large Hadron Collider, the dark matter detection devices, and the satellites poised to receive dark matter annihilation signals—promise new information that will surely reveal dark matter to us or disprove it once and for all.
  The discovery of the accelerating expansion not only solved the age crisis, by making the Universe appear younger than it actually is, it also gave cosmologists a way to "flatten" the Universe.
conversations.from: human
conversations.value: Delete the last sentence.
conversations.from: assistant
conversations.value: Yet MOND is not flawless; the hypothesis struggles to reproduce the motion of clusters of galaxies, requiring more matter than can be seen to make things work. So are we back to needing exotic dark matter? Possibly not, as astronomers think that there may be a substantial amount of normal matter hiding in galaxy clusters, in the form of warm gas. If they can detect this, using telescopes sensitive to the ultraviolet light it is believed to emit, it would solve MOND's difficulties with galaxy clusters.
  A particular battleground for the competing theories is the Bullet Cluster, where two galaxy clusters are colliding. Observations show that the gas between the galaxies has been stripped away, and yet weak lensing indicates that a strong curvature still persists around the galaxies. This can be interpreted as either the gigantic halos of dark matter needed to hold both clusters together, or as regions where MOND needs to be applied. Other galaxy cluster collisions show similar behavior, and likewise do not help astronomers decide between the dark matter and MOND theories. But very recently something has been found that may break the stalemate, an object that perplexes everyone: a most peculiar galaxy cluster collision known as Abell 520.
  In this particular cosmic smash-up, weak lensing results imply that somehow the dark matter has been separated from the galaxies. This is completely contrary to what dark matter theory says should happen, and the only way to explain the dark matter behaving like this is to propose that it is interacting with itself, producing a force that only dark matter feels. Such a hypothesis would solve a problem with the Bullet Cluster collision, as well: computer models cannot substantiate the speed at which the clusters are crashing together, but a previously unanticipated dark matter force could provide the extra kick needed.
  At present, the picture is more confusing than it has ever been. MOND does not work in all situations: it needs about twice the amount of matter that we see in the Universe; and dark matter needs a new, exclusive force to make its figures tally. So the bottom line is that no one knows what the dark matter is, or whether it even exists. It may be an illusion brought about by our misunderstanding of gravity. In the next few years, the welter of new experiments—the Large Hadron Collider, the dark matter detection devices, and the satellites poised to receive dark matter annihilation signals—promise new information that will surely reveal dark matter to us or disprove it once and for all.
conversations.from: human
conversations.value: the last five paragraphs
conversations.from: assistant
conversations.value: The hypothetical fabric of the Universe that, according to the General Theory of Relativity, is deformed by matter and produces gravity.
  Albert Einstein's theory that allows observations from differently moving (but not accelerating) objects to be compared with one another.
  A massive ball of hydrogen and helium bound together by gravity and shining for most of its life with light generated by nuclear fusion.
  A possible way of unifying gravity with the other forces of nature, thus providing a single "theory of everything." If successful it will supply a quantum theory of gravity.
  Everything that exists around us in space. There are estimated to be around 500 billion galaxies in our own Universe and there may be other universes somehow separate from our own.
conversations.from: human
conversations.value: Swap the first and the last sentence.
conversations.from: assistant
conversations.value: There are estimated to be around 500 billion galaxies in our own Universe and there may be other universes somehow separate from our own.
  Albert Einstein's theory that allows observations from differently moving (but not accelerating) objects to be compared with one another.
  A massive ball of hydrogen and helium bound together by gravity and shining for most of its life with light generated by nuclear fusion.
  A possible way of unifying gravity with the other forces of nature, thus providing a single "theory of everything." If successful it will supply a quantum theory of gravity.
  Everything that exists around us in space.The hypothetical fabric of the Universe that, according to the General Theory of Relativity, is deformed by matter and produces gravity.
conversations.from: human
conversations.value: the first three paragraphs for this book
conversations.from: assistant
conversations.value: The Big Questions
  Ironically, the longest-term predictions come out OK: the science of climate prediction is not so sensitive to initial conditions as are short-term weather predictions. That's essentially because climate prediction deals more in generalities than specifics. The flap of a butterfly's wings might cause a storm in Texas, but another flap might calm a storm that was already blowing up. Over the 30-year average that constitutes a climate analysis, the number of storms evens out, and each butterfly becomes irrelevant.
  Lorenz used the equations of chaos theory to show this. When you look at a strange attractor, you see a particular shape. Applied to climate science, the shape you see indicates the future climate. The flowing line that moves around unpredictably, gradually creating the shape, is like noise on the signal; it is not the parameter of interest. That means running climate simulations, however chaotic their predictions might be over the short term, reveals a reliable broad-brush picture of what is coming. So, does chaos theory spell disaster? Quite the opposite: that flapping butterfly has been instrumental in warning us about the greatest threat faced by humanity: runaway climate change induced by human activity. Chaos is not always a problem.
  When Johnson wrote those words in 18th-century England, Isaac Newton's view of light as particles or "corpuscles" of energy reigned supreme. Within 20 years of Johnson's death, Thomas Young had "proved" light was a wave, not a particle. A century later, Albert Einstein showed light to be, once again, particles. Now we have to think of it as both—or neither. Light, the universal metaphor for understanding and revelation, is astonishingly opaque.
conversations.from: human
conversations.value: two more paragraphs. Start with the sentence 'It's not something to get depressed about—the world is as real as it has ever been'.
conversations.from: assistant
conversations.value: It's not something to get depressed about—the world is as real as it has ever been. What's more, unlike the ideas of universes run by supernatural gods, the simulation argument just might be open to testing. The first point to recognize is that it does indeed answer the question about fine-tuning. The simulation's creators must have a reason to create it. It seems sensible to suggest, therefore, that the overwhelming majority of simulations will have to work well enough to be interesting to their creators and users. Our experience with creating simulation environments suggests that this means populating them with beings that can enjoy their "existence," which, in turn, tends to involve an ability to interact with the simulated world and its inhabitants.
  A plausible simulation will therefore encourage the development of something we would regard as complex life. As we have seen with our look at the laws of nature, that gives a fairly narrow range of possibilities for the set-up. That at least provides a plausible explanation for the fine-tuning. Now we have to look for a scientific test for such an explanation. Again, this can be found within our own experience of creating simulations.
conversations.from: human
conversations.value: the last three paragraphs
conversations.from: assistant
conversations.value: A prediction of special relativity, borne out in experiments, that says the flow of time is altered by relative motion or by the presence of a gravitational field.
  The energy inherent in otherwise empty space–time. It is due to the Heisenberg uncertainty principle, which does not allow space–time to have zero energy for any finite period of time.
  The short-range force that affects all matter particles. It is responsible for, among other things, some forms of radiation.
conversations.from: human
conversations.value: Reverse the sentences.
conversations.from: assistant
conversations.value: It is responsible for, among other things, some forms of radiation..
  The short-range force that affects all matter particles. It is due to the Heisenberg uncertainty principle, which does not allow space–time to have zero energy for any finite period of time.
  The energy inherent in otherwise empty space–time.A prediction of special relativity, borne out in experiments, that says the flow of time is altered by relative motion or by the presence of a gravitational field
conversations.from: human
conversations.value: the first three paragraphs for this book. End with the sentence ' Barfield continues, describing how his new perception of the words in poems had an impact upon his perception of the world around him too:'.
conversations.from: assistant
conversations.value: The Big Questions
  This is because such weighty occurrences are felt more than they are thought; they are undergone rather than planned. They have the quality of ineffability, being beyond words. Again, this need not be put down to deliberate obfuscation and an indulgent fuzziness, for many common human experiences are difficult to describe, and are no less significant and real. "One must have musical ears to know the value of a symphony; one must have been in love one's self to understand a lover's state of mind," according to James. "Lacking the heart or ear, we cannot interpret the musician or the lover justly, and are even likely to consider him weak-minded or absurd."
  Another feature of mystical states is that they are limited in duration, though for the individual who takes them seriously they can have a lasting impact upon everyday life. "They modify the inner life of the subject between the times of their recurrence," James explains—a typical kind of modification being seeing the familiar and humdrum as weighty and significant. A phrase or place that was once thought unremarkable suddenly seizes the attention and claims center stage. It is understood.
  This experience then feeds back, so that the extraordinary enhances what had been ordinary before. Barfield continues, describing how his new perception of the words in poems had an impact upon his perception of the world around him too:
conversations.from: human
conversations.value: four more paragraphs
conversations.from: assistant
conversations.value: He could remain fully present to the turbulent cascade of events without being tossed around by the desires and fears it evoked within him. A still calm lay at the heart of his vision, a strange dropping away of familiar habits, the absence, at least momentarily, of anxiety and turmoil...
  Today, in the West, the story of the Buddha's discovery is relatively well known, though Buddhist teachers will never tire of pointing out that the middle way is a subtle way, with many potential wrong turnings and misunderstandings. If it were straightforward, a few hours under the bodhi tree would make enlightened beings of us all. But there is shocking aspect to this story.
  At first, the Buddha thought he should tell nobody about his enlightenment. He remained under the tree and decided to keep silent—Sakyamuni, or the Silent One from Sakka, as one of his names describes him. It is said that at the moment of his awakening, the whole cosmos sang of the new-found freedom. "The flowering trees bloomed; the fruit trees were weighed down by the burden of their fruit; the trunk lotuses bloomed on the trunks of trees. Suffering remained. The new Buddha knew that, in the same way that he had followed the path himself, that was the fate of every sentient creature. He could not do it for them. How could he even persuade them of the value of the path? As he had been in his youth, so most people were all their lives: they longed to be attached to the things that caused suffering. "If I taught the way, people would not understand it and that would be exhausting and disappointing for me..."
  So he decided to keep his secret to himself. That was the extent of his compassion, at least at first. The texts report that it was not until the Brahma appeared that the Buddha changed his mind. "The world will be lost, the world will not have a chance," the god of creation pleaded.
conversations.from: human
conversations.value: Add another sentence about singing, stopped.
conversations.from: assistant
conversations.value: He could remain fully present to the turbulent cascade of events without being tossed around by the desires and fears it evoked within him. A still calm lay at the heart of his vision, a strange dropping away of familiar habits, the absence, at least momentarily, of anxiety and turmoil... 
  Today, in the West, the story of the Buddha's discovery is relatively well known, though Buddhist teachers will never tire of pointing out that the middle way is a subtle way, with many potential wrong turnings and misunderstandings. If it were straightforward, a few hours under the bodhi tree would make enlightened beings of us all. But there is shocking aspect to this story. 
  At first, the Buddha thought he should tell nobody about his enlightenment. He remained under the tree and decided to keep silent—Sakyamuni, or the Silent One from Sakka, as one of his names describes him. It is said that at the moment of his awakening, the whole cosmos sang of the new-found freedom. "The flowering trees bloomed; the fruit trees were weighed down by the burden of their fruit; the trunk lotuses bloomed on the trunks of trees. " But then the singing stopped. Suffering remained. The new Buddha knew that, in the same way that he had followed the path himself, that was the fate of every sentient creature. He could not do it for them. How could he even persuade them of the value of the path? As he had been in his youth, so most people were all their lives: they longed to be attached to the things that caused suffering. "If I taught the way, people would not understand it and that would be exhausting and disappointing for me... "
  So he decided to keep his secret to himself. That was the extent of his compassion, at least at first. The texts report that it was not until the Brahma appeared that the Buddha changed his mind. "The world will be lost, the world will not have a chance," the god of creation pleaded.
conversations.from: human
conversations.value: the last three paragraphs
conversations.from: assistant
conversations.value: But that does not seem right. If it were so, then the electronics inside your computer would mean that the computer was experiencing itself, much as we experience ourselves, if to a lesser degree. If you really believed that to be so, you might think twice before shutting your computer down or commanding it to open a new application. Maybe it doesn't want to be turned off or spend hours poring over yet another spreadsheet?
  That the thought is ridiculous highlights the fact that existing is different from living. It's Aristotle's main point again. A computer has a body. It does not have a soul. Therein lies every difference.
  Aristotle's ideas are helpful and, to me, persuasive. "We must not heed those who advise us to think as human beings since we are human and to think mortal things since we are mortal," he urges his followers, "but we must be like immortals in so far as possible and do everything toward living in accordance with the best thing in us." This side of the grave, we are definitely mortals, he argues, and so advises a certain humility, to be like immortals in so far as is possible. But we are also ensouled bodies. Even if an afterlife is hard to accept, with what we know about human biology, that seems to be no reason to give up on "the best thing in us." And that best thing is well captured by the word "soul."
conversations.from: human
conversations.value: Delete the last sentence.
conversations.from: assistant
conversations.value: But that does not seem right. If it were so, then the electronics inside your computer would mean that the computer was experiencing itself, much as we experience ourselves, if to a lesser degree. If you really believed that to be so, you might think twice before shutting your computer down or commanding it to open a new application. Maybe it doesn't want to be turned off or spend hours poring over yet another spreadsheet?
  That the thought is ridiculous highlights the fact that existing is different from living. It's Aristotle's main point again. A computer has a body. It does not have a soul. Therein lies every difference.
  Aristotle's ideas are helpful and, to me, persuasive. "We must not heed those who advise us to think as human beings since we are human and to think mortal things since we are mortal," he urges his followers, "but we must be like immortals in so far as possible and do everything toward living in accordance with the best thing in us." This side of the grave, we are definitely mortals, he argues, and so advises a certain humility, to be like immortals in so far as is possible. But we are also ensouled bodies. Even if an afterlife is hard to accept, with what we know about human biology, that seems to be no reason to give up on "the best thing in us." And that best thing is well captured by the word "soul.
conversations.from: human
conversations.value: the first five paragraphs for this book
conversations.from: assistant
conversations.value: The Big Questions
  The quest for a proof of the parallel postulate tripped up many of the greatest mathematicians over the years. One of the potential pitfalls was falling prey to a kind of circular thinking, attempting to prove the postulate by ultimately appealing to the postulate itself; that is, assuming what had to be proved.
  In the effort to solve the question, the efforts of the 18th-century mathematician Giovanni Saccheri are significant. He argued that in thinking of lines passing through a given point in relation to a given line there are logically just three cases to consider:
  The key to this thinking lay in looking beyond a flat plane to a curved surface. It was not a sphere which had a constant positive roundness, or curvature, but a surface with a constant negative curvature. Lobachevsky and Bolyai had, in effect, created the geometry of the "pseudosphere," or, to give it its more accepted name, hyperbolic geometry. The lines on a pseudosphere are "geodesics," that is, the curved lines of shortest length that join two points. This shape, due to Eugenio Beltrami, provided a real model for a geometry that was not Euclid's.
  The theorems of the new geometry did not contradict all of Euclid's theorems, for Lobachevsky and Bolyai kept Euclid's other four postulates. But one matter that was affected was the sum of the angles of a triangle. In Euclid's geometry, as we have seen, the fact that the angle sum in a triangle is 180° is an equivalent of his parallel postulate. When the new postulate of Lobachevsky and Bolyai was substituted, the angle sum of triangle could not equal 180°.
  Lobachevsky, a professor at the University of Kazan, and Bolyai, an army officer, published their results in the early 1830s. Instead of recognition for the magnitude of their accomplishment, their works were ignored. Here, the great Carl Friedrich Gauss came to the rescue—albeit posthumously. Unknown for his work in this field, Gauss had come to similar conclusions but feared the criticism of those who revered Euclid. So he did not publish his results. But when he died in 1855 and his correspondence on non-Euclidean geometry was published, his prestige ensured that mathematicians would also read the connected papers of Lobachevsky and Bolyai.
conversations.from: human
conversations.value: Add another sentence about Theorems, Proposition, proved, maintained.
conversations.from: assistant
conversations.value: The Big Questions
  The quest for a proof of the parallel postulate tripped up many of the greatest mathematicians over the years. One of the potential pitfalls was falling prey to a kind of circular thinking, attempting to prove the postulate by ultimately appealing to the postulate itself; that is, assuming what had to be proved. 
  In the effort to solve the question, the efforts of the 18th-century mathematician Giovanni Saccheri are significant. He argued that in thinking of lines passing through a given point in relation to a given line there are logically just three cases to consider:
  The key to this thinking lay in looking beyond a flat plane to a curved surface. It was not a sphere which had a constant positive roundness, or curvature, but a surface with a constant negative curvature. Lobachevsky and Bolyai had, in effect, created the geometry of the "pseudosphere," or, to give it its more accepted name, hyperbolic geometry. The lines on a pseudosphere are "geodesics," that is, the curved lines of shortest length that join two points. This shape, due to Eugenio Beltrami, provided a real model for a geometry that was not Euclid's. 
  The theorems of the new geometry did not contradict all of Euclid's theorems, for Lobachevsky and Bolyai kept Euclid's other four postulates. Theorems proved without recourse to the parallel postulate were still maintained (those up to Proposition 29). But one matter that was affected was the sum of the angles of a triangle. In Euclid's geometry, as we have seen, the fact that the angle sum in a triangle is 180° is an equivalent of his parallel postulate. When the new postulate of Lobachevsky and Bolyai was substituted, the angle sum of triangle could not equal 180°. 
  Lobachevsky, a professor at the University of Kazan, and Bolyai, an army officer, published their results in the early 1830s. Instead of recognition for the magnitude of their accomplishment, their works were ignored. Here, the great Carl Friedrich Gauss came to the rescue—albeit posthumously. Unknown for his work in this field, Gauss had come to similar conclusions but feared the criticism of those who revered Euclid. So he did not publish his results. But when he died in 1855 and his correspondence on non-Euclidean geometry was published, his prestige ensured that mathematicians would also read the connected papers of Lobachevsky and Bolyai.
conversations.from: human
conversations.value: five more paragraphs
conversations.from: assistant
conversations.value: By adjusting ratios, music can generate harmony or discord, with the former a particularly central aspect of the pleasure we derive from it. Scholars of music have identified other mathematical principles in the way music is structured which govern its reception, and symmetry—a mathematical principle, if ever there was one—figures strongly. In music, strength and appeal often lies in the repeated pattern; a singular occurrence would sound impoverished by comparison.
  In one curious case, mathematics and musical composition are clearly intertwined: in campanology, otherwise known as bell-ringing. Here, the practicalities of ringing bells in a certain order determine the nature of compositions. The mathematical patterns of the "changes" have much in common with permutation group theory in modern algebra!
  In discussing music, we have considered proportion and symmetry. Those factors are vital ingredients in the mathematics of shape and structure, too. Often they are most appealing at their simplest, as in an example of a humble rectangle and its properties.
  But there are other ways of analyzing a rectangle, too, in which we appreciate its proportions. If we fold a rectangular piece of paper in half and cut along the fold we are left with two rectangular pieces.
  The root 2 rectangle can be analyzed further by "taking out squares." If we take a square from the root 2 rectangle with actual sides 1 and... we are left with a rectangle with short side 0.4142 and long side equal to 1.
conversations.from: human
conversations.value: the last four paragraphs
conversations.from: assistant
conversations.value: The attempt to analyze the motion of three bodies (such as the Moon, Earth, Sun) which are mutually attracted to each other.
  Also known as "rubber sheet" geometry, which concentrates on such properties of surfaces as those which are preserved when they are transformed by stretching and contracting, but not cutting. The measurement of lengths and angles has no part to play in topology as it does in ordinary geometry.
  Two prime numbers separated by at most one number: for example, the twins 11 and 13. It is not known whether there is an infinity of these twins.
  Applies to game theory. What one player wins (+w) the other loses (–w) so the net sum (+w)+(–w) is zero.