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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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344e400
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Jacques was so impressed with the beauty of the curve known as a logarithmic spiral (Figure 37; the name was derived from the way in which the radius grows as we move around the curve clockwise) that he asked that this shape, and the motto he assigned to it: "Eadem mutato resurgo" (although changed, I rise again the same), be engraved on his tombstone. The motto describes a fundamental property unique to the logarithmic spiral-it does not a..
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Mario Livio |
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ff5b846
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Supporters of the "modified Platonic view" of mathematics like to point out that, over the centuries, mathematicians have produced (or "discovered") numerous objects of pure mathematics with absolutely no application in mind. Decades later, these mathematical constructs and models were found to provide solutions to problems in physics. Penrose tilings and non-Euclidean geometries are beautiful testimonies to this process of mathematics unex..
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Mario Livio |
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7e20552
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The curriculum for the education of statesmen at the time of Plato included arithmetic, geometry, solid geometry, astronomy, and music-all of which, the Pythagorean Archytas tells us, fell under the general definition of "mathematics." According to legend, when Alexander the Great asked his teacher Menaechmus (who is reputed to have discovered the curves of the ellipse, the parabola, and the hyperbola) for a shortcut to geometry, he got the..
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Mario Livio |
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1d21842
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Nature loves logarithmic spirals. From sunflowers, seashells, and whirlpools, to hurricanes and giant spiral galaxies, it seems that nature chose this marvelous shape as its favorite "ornament." The constant shape of the logarithmic spiral on all size scales reveals itself beautifully in nature in the shapes of minuscule fossils or unicellular organisms known as foraminifera. Although the spiral shells in this care are composite structures ..
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Mario Livio |
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947cb28
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Gell-Mann and Ne'eman discovered that one such simple Lie group, called "special unitary group of degree 3," or SU(3), was particularly well suited for the "eightfold way"-the family structure the particles were found to obey. The beaty of the SU(3) symmetry was revealed in full glory via its predictive power. Gell-Mann and Ne'eman showed that if the theory were to hold true, a previously unknown tenth member of a particular family of nine ..
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Mario Livio |
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ec8f06f
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As we shall see throughout this book, the unifying powers of group theory are so colossal that historian of mathematics Eric Temple Bell (1883-1960) once commented, "When ever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos."
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Mario Livio |
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15a240f
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In the late 1960's, physicists Steven Weinberg, Abdus Salam, and Sheldon Glashow conquered the next unification frontier. In a phenomenal piece of scientific work they showed that the electromagnetic and weak nuclear forces are nothing but different aspects of the same force, subsequently dubbed the electroweak force. The predictions of the new theory were dramatic. The electromagnetic force is produced when electrically charged particles e..
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Mario Livio |
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026d1a7
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In an old joke, a physicist and a mathematician are asked what they would do if they needed to iron their pants, but although they are in possession of an iron, the electric outlet is in the adjacent room. Both answer that they would take the iron to the second room and plug it in there. Now they are asked what they would do if they were already in the room in which the outlet is located. They physicist answers that he would plug the iron i..
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Mario Livio |
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41080c2
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There were also many cases of feedback between physics and mathematics, where a physical phenomenon inspired a mathematical model that later proved to be the explanation of an entirely different physical phenomenon. An excellent example is provided by the phenomenon known as Brownian motion. In 1827, British botanist Robert Brown (1773-1858) observed that wen pollen particles are suspended in water, they get into a state of agitated motion...
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Mario Livio |
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a84f978
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Is it odd how asymmetrical Is "symmetry"? "Symmetry" is asymmetrical. How odd it is. This stanza remains unchanged if read word by word from the end to the beginning-it is symmetrical with respect to backward reading."
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Mario Livio |
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69049e5
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Through the works of Weinberg, Glashow, and Salam on the electroweak theory and the elegant framework developed by the physicists David Gross, David Politzer, and Frank Wilczek for quantum chromodynamics, the characteristic group of the standard model has been identified with a product of three Lie groups denoted by U(1), SU(2), and SU(3). In some sense, therefore, the road toward the ultimate unification of the forces of nature has to go t..
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Mario Livio |
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41668e3
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Even though it is almost impossible to attribute with certainty any specific mathematical achievements either to Pythagoras himself or to his followers, there is no question that they have been responsible for a mingling of mathematics, philosophy of life, and religion unparalleled in history. In this respect it is perhaps interesting to note the historical coincidence that Pythagoras was a contemporary of Buddha and Confucius.
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Mario Livio |
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593fc43
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We have already seen that gauge symmetry that characterizes the electroweak force-the freedom to interchange electrons and neturinos-dictates the existence of the messenger electroweak fields (photon, W, and Z). Similarly, the gauge color symmetry requires the presence of eight gluon fields. The gluons are the messengers of the strong force that binds quarks together to form composite particles such as the proton. Incidentally, the color "c..
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Mario Livio |
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6323c69
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The properties that define a group are: 1. Closure. The offspring of any two members combined by the operation must itself be a member. In the group of integers, the sum of any two integers is also an integer (e.g., 3 + 5 = 8). 2. Associativity. The operation must be associative-when combining (by the operation) three ordered members, you may combine any two of them first, and the result is the same, unaffected by the way they are bracketed..
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Mario Livio |
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680e445
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The importance of mirror-reflection symmetry to our perception and aesthetic appreciation, to the mathematical theory of symmetries, to the laws of physics, and to science in general, cannot be overemphasized, and I will return to it several times. Other symmetries do exist, however, and they are equally relevant.
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Mario Livio |
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b003c15
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In other words, Birkhoff proposed a formula for the feeling of aesthetic value: M = O / C. The meaning of this formula is: For a given degree of complexity, the aesthetic measure is higher the more order the object possesses. Alternatively, if the amount of order is specified, the aesthetic measure is higher the less complex the object. Since for most practical purposes, the order is determined primarily by the symmetries of the object, Bir..
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Mario Livio |
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b6b676f
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An interesting question is whether symmetry with respect to translation, and indeed reflection and rotation too, is limited to the visual arts, or may be exhibited by other artistic forms, such as pieces of music. Evidently, if we refer to the sounds, rather than to the layout of the written musical score, we would have to define symmetry operations in terms other than purely geometrical, just as we did in the case of the palindromes. Once ..
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Mario Livio |
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7101a4e
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Our mathematics is a combination of invention and discoveries. The axioms of Euclidean geometry as a concept were an invention, just as the rules of chess were an invention. The axioms were also supplemented by a variety of invented concepts, such as triangles, parallelograms, ellipses, the golden ratio, and so on. The theorems of Euclidean geometry, on the other hand, were by and large discoveries; they were the paths linking the different..
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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 |