|
344e400
|
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..
|
|
|
Mario Livio |
|
ff5b846
|
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..
|
|
|
Mario Livio |
|
7e20552
|
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..
|
|
|
Mario Livio |
|
1d21842
|
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 ..
|
|
|
Mario Livio |
|
947cb28
|
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 ..
|
|
|
Mario Livio |
|
ec8f06f
|
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."
|
|
|
Mario Livio |
|
15a240f
|
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..
|
|
|
Mario Livio |
|
026d1a7
|
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..
|
|
|
Mario Livio |
|
41080c2
|
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...
|
|
|
Mario Livio |
|
a84f978
|
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."
|
|
|
Mario Livio |
|
69049e5
|
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..
|
|
|
Mario Livio |
|
41668e3
|
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.
|
|
|
Mario Livio |
|
593fc43
|
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..
|
|
|
Mario Livio |
|
6323c69
|
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..
|
|
|
Mario Livio |
|
680e445
|
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.
|
|
|
Mario Livio |
|
b003c15
|
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..
|
|
|
Mario Livio |
|
b6b676f
|
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 ..
|
|
|
Mario Livio |
|
7101a4e
|
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..
|
|
|
Mario Livio |
|
960d939
|
Either because of mate selection, cognition, predator avoidance, or a combination of all three, our minds are attracted to and are finely tuned to the detection of symmetry. The question of whether symmetry is truly fundamental to the universe itself, or merely to the universe as perceived by humans, thus becomes particularly acute.
|
|
|
Mario Livio |
|
a60f988
|
Computer scientist and author Douglas R. Hofstadter phrased this succinctly in his fantastic book Godel, Escher, Bach: An Eternal Golden Braid: "Provability is a weaker notion than truth." In this sense, there will never be a formal method of determining for every mathematical proposition whether it is absolutely true, any more than there is a way to determine whether a theory in physics is absolutely true. Oxford's mathematical physicist R..
|
|
|
Mario Livio |
|
9920395
|
The legendary inscription above the Academy's door speaks loudly about Plato's attitude toward mathematics. In fact, most of the significant mathematical research of the fourth century BC was carried out by people associated in one way or another with the Academy. Yet Plato himself was not a mathematician of great technical dexterity, and his direct contributions to mathematical knowledge were probably minimal. Rather, he was an enthusiasti..
|
|
|
Mario Livio |
|
6f4914b
|
The Greek excellence in mathematics was largely a direct consequence of their passion for knowledge for its own sake, rather than merely for practical purposes. A story has it that when a student who learned one geometrical proposition with Euclid asked, "But what do I gain from this?" Euclid told his slave to give the boy a coin, so that the student would see an actual profit."
|
|
|
Mario Livio |
|
6715109
|
Pythagoras apparently wrote nothing, and yet his influence was so great that the more attentive of his followers formed a secretive society, or brotherhood, and were known as the Pythagoreans. Aristippus of Cyrene tells us in his Account of Natural Philosphers that Pythagoras derived his name from the fact that he was speaking (agoreuein) truth like the God at Delphi (tou Pythiou).
|
|
|
Mario Livio |
|
36c0fb0
|
Mind-boggling, isn't it? Centuries before the question of why mathematics was so effective in explaining nature was even asked, Galileo thought he already knew the answer! To him, mathematics was simply the language of the universe. To understand the universe, he argued, one must speak this language. God is indeed a mathematician.
|
|
|
Mario Livio |
|
f976d4a
|
You may begin to realize that groups will pop up wherever symmetries exist. In fact, the collection of all the symmetry transformations of any system always from a group.
|
|
|
Mario Livio |
|
536c8de
|
To claim that mathematics is purely a human invention and is successful in explaining nature only because of evolution and natural selection ignores some important facts in the nature of mathematics and in the history of theoretical models of the universe. First, while the mathematical rules (e.g., the axioms of geometry or of set theory) are indeed creations of the human mind, once those rules are specified, we lose our freedom. The defini..
|
|
|
Mario Livio |
|
e5da224
|
populations typically have such high reproduction potential that if unchecked they would increase exponentially.
|
|
|
Mario Livio |
|
5ae269f
|
The environment is the agent that does the shaking of the sieve.
|
|
|
Mario Livio |
|
e186fcd
|
Unlike most mathematical discoveries, however, no one was looking for a theory of groups or even a theory of symmetries when the concept was discovered. Quite the contrary; group theory appeared somewhat serendipitously, out of a millenia-long search for a solution to an algebraic equation. Befitting its description as a concept that crystallized simplicity out of chaos, group theory was itself born out of one of the most tumultuous stories..
|
|
|
Mario Livio |
|
c456681
|
Leonardo had considerable interest in geometry, especially for its practical applications in mathematics. In his words: "Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics."
|
|
|
Mario Livio |
|
6834dca
|
Here, however, is where his genius truly took off. Galois managed to associate with each equation a sort of "genetic code" of that equation-the Galois group of the equation-and to demonstrate that the properties of the Galois group determine whether the equation is solvable by a formula or not. Symmetry became the key concept, and the Galois group was a direct measure of the symmetry properties of an equation."
|
|
|
Mario Livio |
|
5938434
|
In it, Porphyry says about Pythagoras: "He himself could hear the harmony of the Universe, and understood the music of the spheres, and the stars which move in concert with them, and which we cannot hear because of the limitations of our weak nature."
|
|
|
Mario Livio |
|
d2f9a73
|
The beauty of the principle idea of string theory is that all the known elementary particles are supposed to represent merely different vibration modes of the same basic string. Just as a violin or a guitar string can be plucked to produce different harmonics, different vibrational patterns of a basic string correspond to distinct matter particles, such as electrons and quarks. The same applies to the force carriers as well. Messenger parti..
|
|
|
Mario Livio |
|
9dcc170
|
Having witnessed in his own life much agony and the horrors of war, Kepler concluded that Earth really created two notes, mi for misery ("miseria" in Latin) and fa for famine ("fames" in Latin). In Kepler's words: "the Earth sings MI FA MI, so that even from the syllable you may guess that in this home of ours Misery and Famine hold sway."
|
|
|
Mario Livio |
|
22b7364
|
Wolfram, one of the most innovative thinkers in scientific computing and in the theory of complex systems, has been best known for the development of Mathematica, a computer program/system that allows a range of calculations not accessible before. After ten years of virtual silence, Wolfram is about to emerge with a provocative book that makes the bold claim that he can replace the basic infrastructure of science. In a world used to more th..
|
|
|
Mario Livio |
|
6d0a744
|
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 d..
|
|
|
Mario Livio |
|
8aa43c4
|
Pythagoras is in fact credited with having coined the words "philosophy" ("love of wisdom") and "mathematics" ("that which is learned"). To him, a "philosopher" was someone who "gives himself up to discovering the meaning and purpose of life itself...to uncover the secrets of nature." Pythagoras emphasized the importance of learning above all other activities, because, in his words, "most men and women, by birth or nature, lack the means to..
|
|
|
Mario Livio |
|
87ecc6a
|
The Pythagoreans were probably the first to recognize the concept that the basic forces in the universe may be expressed through the language of mathematics.
|
|
|
Mario Livio |
|
52e3345
|
Group theory has been called by the noted mathematics scholar James R. Newman "the supreme art of mathematical abstraction." It derives its incredible power from the intellectual flexibility afforded by its definition."
|
|
|
Mario Livio |
|
8268685
|
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. In particular, the familiar symmetry of the laws under translations corresponds to conservation of momentum, the symmetry with respect to the passing of time (the fact that the laws do not change with time) gives us conservation of energy, and the symmetry under rotations pro..
|
|
|
Mario Livio |
|
d59a5fd
|
In mathematics, if you are of quick mind, you can get to the "frontline" of cutting-edge research very quickly. In some other domains you may have to read entire thick volumes first. Moreover, if you have been for too long in a certain domain, you get conditioned to think like everybody else. When you are new, you are not compelled to the ideas of the people around you. The younger you are, the more likely you are to be truly original."
|
|
|
Mario Livio |
|
124c8af
|
So mathematics is indeed extraordinarily effective for some descriptions, especially those dealing with fundamental science, but it cannot describe our universe in all its dimensions. To some extent, scientists have selected what problems to work on based on those problems being amenable to a mathematical treatment.
|
|
|
Mario Livio |
|
79708db
|
The number 6 was the first perfect number, and the number of creation. The adjective "perfect" was attached that are precisely equal to the sum of all the smaller numbers that divide into them, as 6=1+2+3. The next such number, incidentally, is 28=1+2+4+7+14, followed by 496=1+2+4+8+16+31+62+124+248; by the time we reach the ninth perfect number, it contains thirty-seven digits. Six is also the product of the first female number, 2, and the..
|
|
|
Mario Livio |
|
34625a7
|
In the Middle Ages, the Elements was translated into Arabic three times. The first of these translations was carried out by al-Hajjaj ibn Yusuf ibn Matar, at the request of Caliph Harun ar-Rashid (ruled 786 - 809), who is familiar to us through the stories in The Arabian Nights. The Elements was first made known in Western Europe through Latin translations of Arabic versions. English Benedictine monk Adelard of Bath (ca. 1070 - 1145), who a..
|
|
|
Mario Livio |