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76d756b
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So, what is light? Is it a pure bombardment by particles (photons) or a pure wave? Really, it is neither. Light is a more complicated physical phenomenon than any single one of these concepts, which are based on classical physical models, can describe. To describe the propagation of light and to understand the phenomena like interference, we can and have to use the electromagnetic wave theory. When we want to discuss the interaction of ligh..
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Mario Livio |
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ba294d6
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While Euclid himself may not have been the greatest mathematician who ever lived, he was certainly the greatest teacher of mathematics.
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Mario Livio |
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57a9b45
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What can we conclude from all of these insights in terms of the role of symmetry in the cosmic tapestry? My humble personal summary is that we don't know yet whether symmetry will turn out to be the most fundamental concept in the workings of the universe. Some of the symmetries physicists have discovered or discussed over the years have later been recognized as being accidental or only approximate. Other symmetries, such as general covaria..
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Mario Livio |
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7696bb6
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Tegmark argues that "our universe is not just described by mathematics-it is mathematics" [emphasis added]. His argument starts with the rather uncontroversial assumption that an external physical reality exists that is independent of human beings. He then proceeds to examine what might be the nature of the ultimate theory of such a reality (what physicists refer to as the "theory of everything"). Since this physical world is entirely indep..
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Mario Livio |
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0ef7c28
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The result that Noether obtained was stunning. She showed that to every continuous symmetry of the laws of physics there corresponds a conservation law and vice versa.
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Mario Livio |
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deb8618
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Unfortunately, I do not find Tegmark's line of reasoning to be extremely compelling. The leap from the existence of an external reality (independent of humans) to the conclusion that, in Tegmark's words, "You must believe in what I call the mathematical universe hypothesis: that our physical reality is a mathematical structure," involves, in my opinion, a sleight of hand. When Tegmark attempts to characterize what mathematics really is, he ..
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Mario Livio |
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5b09f9f
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Noether's theorem fused together symmetries and conservation laws-these two giant pillars of physics are actually nothing but different facets of the same fundamental property.
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Mario Livio |
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199c9ef
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Based on these interviews, he compiled a list of ten dimensions of complexity-ten pairs of apparently antithetical characteristics that are often both present in the creative minds. The list includes: 1. Bursts of impulsiveness that punctuate periods of quiet and rest. 2. Being smart yet extremely naive. 3. Large amplitude swings between extreme responsibility and irresponsibility. 4. A rooted sense of reality together with a hefty dose of ..
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Mario Livio |
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bf8fe26
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This development had dramatic philosophical consequences. As in the case of the non-Euclidean geometries in the nineteenth century, there wasn't just one definitive set theory, but rather at least four! One could make different assumptions about infinite sets and end up with mutually exclusive set theories. For instance, once could assume that both the axiom of choice and the continuum hypothesis hold true and obtain one version, or that bo..
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Mario Livio |
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5b7fb25
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This was undoubtedly one of symmetry's greatest success stories. Glashow, Wienberg, and Salam managed to unmask the electromagnetic and weak forces by recognizing that underneath the differences in the strengths of these two forces (the electromagnetic force is about a hundred thousand times stronger within the nucleus) and the different masses of the messenger particles lay a remarkable symmetry. The forces of nature take the same form if ..
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Mario Livio |
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518ac3f
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The biggest stumbling block that has traditionally plagued all the unification endeavors has been the simple fact that on the face of it, general relativity and quantum mechanics really appear to be incomprehensible. Recall that the key concept of quantum theory is the uncertainty principle. When you try to probe positions with an ever-increasing magnification power, the momenta (or speeds) start oscillating violently. Below a certain minus..
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Mario Livio |
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818a6f1
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On the question of whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt-mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the feeble human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician-mathematics was God! The importance of the Pythagorean philosophy lies n..
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Mario Livio |
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4676420
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Quarks come in six "flavors" that were given the rather arbitrary names: up, down, strange, charm, top, and bottom. Protons, for instance, are made of two up quarks and one down quark, while neutrons consist of two down quarks and one up quark. Other than ordinary electric charge, quarks possess another type of charge, which has been fancifully called color, even though it has nothing to do with anything we can see. In the same way that the..
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Mario Livio |
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6b2ac87
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The forces of nature are color blind. Just as an infinite chessboard would look the same if we interchanged black and white, the force between a green quark and a red quark is the same as that between two blue quarks, or a blue quark and a green quark. Even if we were to use our quantum mechanical "palette" and replace each of the "pure" color states with a mixed-color state (e.g., "yellow" representing a mixture of red and green or "cyan" ..
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Mario Livio |
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ea020db
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The realization that symmetry is the key to the understanding of the properties of subatomic particles led to an inevitable question: Is there an efficient way to characterize all of these symmetries of the laws of nature? Or, more specifically, what is the basic theory of transformations that can continuously change one mixture of particles into another and produce the observed families? By now you have probably guessed the answer. The pro..
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Mario Livio |
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1ceecda
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In his book Opticks, Newton made it clear that he did not believe that the laws of nature by themselves were sufficient to explain the universe's existence-God was the creator and sustainer of all the atoms that make up the cosmic matter: "For it became him [God] who created them [the atoms] to set them in order. And if he did so, it's unphilosophical to seek for any other Origin of the World, or to pretend that it might arise out of a Chao..
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Mario Livio |
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aeec569
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Moreover, Galileo argued that by pursuing science using the language of mechanical equilibrium and mathematics, humans could understand the divine mind.
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Mario Livio |
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7b77ed6
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Unlike classically spinning bodies, such as tops, however, where the spin rate can assume any value fast or slow, electrons always have only one fixed spin. In the units in which this spin is measured quantum mechanically (called Planck's constant) the electrons have half a unit, or they are "spin-1/2" particles. In fact, all the matter particles in the standard model-electrons, quarks, neutrinos, and two other types called muons and taus-a..
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Mario Livio |
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7ef51c6
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Galois's ideas, with all their brilliance, did not appear out of thin air. They addressed a problem whose roots could be traced all the way back to ancient Babylon. Still, the revolution that Galois had started grouped together entire domains that were previously unrelated. Much like the Cambrian explosion-that stunning burst of diversification in life forms on Earth-the abstraction of group theory opened windows into an infinity of truths...
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Mario Livio |
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a5b419c
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Indeed, the genius of Abel and Galois could be compared only to a supernova-an exploding star that for a short while outshines all the billions of stars in its host galaxy.
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Mario Livio |
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16a3d3e
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One of the string theory pioneers, the Italian physicist Daniele Amati, characterized it as "part of the 21st century that fell by chance into the 20th century." Indeed, there is something about the very nature of the theory at present that points to the fact that we are witnessing the theory's baby steps. Recall the lesson learned from all the great ideas since Einstein's relativity-put the symmetry first. Symmetry originates the forces. T..
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Mario Livio |
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ec80a30
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While composite faces tend, by construction, to also be more symmetric, Langlois found that even after the effects of symmetry have been controlled, averageness was still judged to be attractive. These findings argue for a certain level of prototyping in the mind, since averageness might well be coupled with a prototypical template.
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Mario Livio |
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c1672da
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Creators are hard-driving, focused, dominant, independent risk-takers.
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Mario Livio |
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c4570fd
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Tolerance of ambiguity is a necessary condition for creativity.
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Mario Livio |
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a5335b8
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The teleological argument, or argument from design, attempts to furnish evidence for God's existence from the apparent design of the world.
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Mario Livio |
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cea9b4d
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Accordingly, Pacioli's book also starts with a discussion of proportions in the human body, "since in the human body every sort of proportion and proportionality can be found, produced at the beck of the all-Highest through the inner mysteries of nature."
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Mario Livio |
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2ad6c20
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Supersymmetry is a subtle symmetry based on the quantum mechanical property spin.
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Mario Livio |
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9677a82
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The spirit of revolution and the power of free thought were Percy Shelley's biggest passions in life." One could use precisely the same words to describe Galois. On one of the pages that Galois had left on his desk before leaving for that fateful duel, we find a fascinating mixture of mathematical doodles, interwoven with revolutionary ideas. After two lines of functional analysis comes the word "indivisible," which appears to apply to the ..
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Mario Livio |
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510fd32
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Khong the nghi ngo gi rang dinh luat van vat hap dan cua Newton chinh la san pham cua mot thien tai. Song thien tai ay khong the hoat dong trong chan khong duoc. Mot so nen tang da duoc lat dat mot cach can than boi cac nha khoa hoc truoc do.
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thiên-tài
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Mario Livio |
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f7081d6
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The two solutions of the equation for the Golden Ratio are: x1 = (1+ Sqr5) / 2 x2 = (1 - Sqr5) / 2
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Mario Livio |
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f5070de
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Some of the greatest mathematical minds of all ages, from Pythagoras and Euclid in ancient Greece, through the medieval Italian mathematician Leonardo of Pisa and the Renaissance astronomer Johannes Kepler, to present-day scientific figures such as Oxford physicist Roger Penrose, have spent endless hours over this simple ratio and its properties. But the fascination with the Golden Ratio is not confined just to mathematicians. Biologists, a..
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golden-ratio
history-of-mathematics
history-of-science
sacred-geometry
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Mario Livio |
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34bdb27
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The strength of the familiar electromagnetic force between two electrons, for example, is expressed in physics in terms of a constant known as the fine structure constant. The value of this constant, almost exactly 1/137, has puzzled many generations of physicists. A joke made about the famous English physicist Paul Dirac (1902-1984), one of the founders of quantum mechanics, says that upon arrival to heaven he was allowed to ask God one qu..
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heaven
history-of-science
physics
quantum-mechanics
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Mario Livio |
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55a3fe1
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Cac cong cu toan hoc khong duoc lua chon mot cach tuy tien ma dung hon phai that chinh xac dua tren kha nang tien doan dung dan cua chung doi voi ket qua cua cac thi nghiem hoac quan sat co lien quan.
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hiệu-quả-đến-phi-lý
thí-nghiệm
tiên-đoán
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Mario Livio |
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3a6d149
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Van de la thong qua su ham hieu biet chay bong, su co chap buong binh, tri tuong tuong sang tao va mot quyet tam manh me, loai nguoi da tim ra nhung hinh thuc luan toan hoc thich hop cho viec lap mo hinh mot so luong lon cac hien tuong vat ly.
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hình-thức-luận
loài-người
sáng-tạo
tưởng-tượng
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Mario Livio |