The Equation That Couldn't Be Solved Quotes

“The key point to keep in mind, however, is that symmetry is one of the most important tools in deciphering nature's design.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Even today, I am in total awe of the following wondrous chain of ideas and interconnections. Guided throughout by principles of symmetry, Einstein first showed that acceleration and gravity are really two sides of the same coin. He then expanded the concept to demonstrate that gravity merely reflects the geometry of spacetime. The instruments he used to develop the theory were Riemann's non-Euclidean geometries-precisely the same geometries used by Felix Klein to show that geometry is in fact a manifestation of group theory (because every geometry is defined by its symmetries-the objects it leaves unchanged). Isn't this amazing?”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Surprisingly, palindromes appear not just in witty word games but also in the structure of the male-defining Y chromosome. The Y's full genome sequencing was completed only in 2003. This was the crowning achievement of a heroic effort, and it revealed that the powers of preservation of this sex chromosome have been grossly underestimated. Other human chromosome pairs fight damaging mutations by swapping genes. Because the Y lacks a partner, genome biologists had previously estimated that its cargo was about to dwindle away in perhaps as little as five million years. To their amazement, however, the researchers on the sequencing team discovered that the chromosome fights withering with palindromes. About six million of its fifty million DNA letters form palindromic sequences-sequences that read the same forward and backward on the two strands of the double helix. These copies not only provide backups in case of bad mutations, but also allow the chromosome, to some extent, to have sex with itself-arms can swap position and genes are shuffled. As team leader David Page of MIT has put it, "The Y chromosome is a hall of mirrors.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 through the discovery of the most suitable Lie group that contains the product U(1) X SU(2) x SU(3).”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 "charges" of the three quarks that make up a proton or a neutron are all different (red, blue, green), and they add up to give zero color charge or "white" (equivalent to being electrically neutral in electromagnetism). Since color symmetry is at the base of the gluon-mediated force between quarks, the theory of these forces has become known as quantum chromodynamics. The marriage of the electroweak theory (which describes the electromagnetic and weak forces) with quantum chromodynamics (which describes the strong force) produced the standard model-the basic theory of elementary particles and the physical laws that govern them.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 mathematics. This word is followed, however, by the revolutionary slogans "unite; indivisibilite de la republic") and "Liberte, egalite, fraternite ou la mort" ("Liberty, equality, brotherhood, or death"). After these republican proclamations, as if this is all part of one continuous thought, the mathematical analysis resumes. Clearly, in Galois's mind, the concepts of unity and indivisibility applied equally well to mathematics and to the spirit of the revolution. Indeed, group theory achieved precisely that-a unity and indivisibility of the patterns underlying a wide range of seemingly unrelated disciplines.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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. Fields as far apart as the laws of nature and music suddenly became mysteriously connected. The Tower of Babel of symmetries miraculously fused into a single language.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 fantasy and imagination.

5. Alternating periods of introversion and extroversion.

6. Being simultaneously humble and proud.

7. Psychological androgyny-no clear adherence to gender role stereotyping.

8. Being rebellious and iconoclastic yet respectful to the domain of expertise and its history.

9. Being on one had passionate but on the other objective about one's own work.

10. Experiencing suffering and pain mingled with exhilaration and enjoyment.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Tolerance of ambiguity is a necessary condition for creativity.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Creators are hard-driving, focused, dominant, independent risk-takers.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 covariance in general relativity and the gauge symmetries of the standard model, became the buds from which forces and new particles bloomed. All in all, there is absolutely no doubt in my mind that symmetry principles almost always tells us something important, and they may provide the most valuable clues and insights toward unveiling and deciphering the underlying principles of the universe, whatever those may be. Symmetry, in this sense, is indeed fruitful.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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. The equivalence principle-the expectation that all observers, irrespective of their motions, would deduce the same laws-requires the existence of gravity. The gauge symmetries-the fact that the laws do not distinguish color, or electrons from neutrinos-dictate the existence of the messengers of the strong and electroweak forces. Yet supersymmetry is an output of string theory, a consequence of its structure rather than a source for its existence. What does this mean? Many string theorists believe that some underlying grander principle, which will necessitate the existence of string theory, is still to be found. If history is to repeat itself, then this principle may turn out to involve an all-encompassing and even more compelling symmetry, but at the moment no one has a clue what this principle might be. Since, however, we are only at the beginning of the twenty-first century, Amati's characterization may still turn out to be an astonishing prophecy.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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-all have "spin 1/2." Particles with half-integer spin are known collectively as fermions (after the Italian physicist Enrico Fermi). On the other hand, the force carriers-the photon, W, Z, and gluons-all have one unit of spin, or they are "spin-1" particles in the physics lingo. The carrier of gravity-the graviton-has "spin 2," and this was precisely the identifying property that one of the vibrating strings was found to possess. All the particles with integer units of spin are called bosons (after the Indian physicist Satyendra Bose). Just as ordinary spacetime is associated with a supersymmetry that is based on spin. The predictions of supersymmetry, if it is truly obeyed, are far-reaching. In a universe based on supersymmetry, every known particle in the universe must have an as-yet undiscovered partner (or "superparrtner"). The matter particles with spin 1/2, such as electrons and quarks, should have spin 0 superpartners. the photon and gluons (that are spin 1) should have spin-1/2 superpartners called photinos and gluinos respectively. Most importantly, however, already in the 1970s physicists realized that the only way for string theory to include fermionic patterns of vibration at all (and therefore to be able to explain the constituents of matter) is for the theory to be supersymmetric. In the supersymmetric version of the theory, the bosonic and fermionic vibrational patters come inevitably in pairs. Moreover, supersymmetric string theory managed to avoid another major headache that had been associated with the original (nonsupersymmetric) formulation-particles with imaginary mass. Recall that the square roots of negative numbers are called imaginary numbers. Before supersymmetry, string theory produced a strange vibration pattern (called a tachyon) whose mass was imaginary. Physicists heaved a sigh of relief when supersymmetry eliminated these undesirable beasts.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Supersymmetry is a subtle symmetry based on the quantum mechanical property spin.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 particles such as gluons or the W and Z owe their existence to yet other harmonics. Put simply, all the matter and force particles of the standard model are part of the repertoire that strings can play. Most impressively, however, a particular configuration of vibrating string was found to have properties that match precisely the graviton-the anticipated messenger of the gravitational force. This was the first time that the four basic forces of nature have been housed, if tentatively, under one roof.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 minuscule length known as the Planck length, the entire tenet of a smooth spacetime is lost. This length (equal to 0.000...4 of an inch, where the 4 is at the thirty-fourth decimal place) determines the scale at which gravity has to be treated quantum mechanically. For smaller scales, space turns into an ever-fluctuating "quantum foam." But the very basic premise of general relativity has been the existence of a gently curved spacetime. In other words, the central ideas of general relativity and quantum mechanics clash irreconcilably when it comes to extremely small scales.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 particles had to be found. The extensive hunt for the missing particle was conducted in an accelerator experiment in 1964 at Brookhaven National Lab on Long Island. Yuval Ne'eman told me some years later that, upon hearing that half of the data had already been scrutinized without discovering the anticipated particle, he was contemplating leaving physics altogether. Symmetry triumphed at the end-the missing particle (called the omega minus) was found, and it had precisely the properties predicted by the theory.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 profound truth in the phrase I have cited earlier in this book revealed itself once again: "Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." The physicists of the 1960s were thrilled to discover that mathematicians had already paved the way. Just as fifty years earlier Einstein learned about the geometry tool-kit prepared by Riemann, Gell-Mann and Ne'eman stumbled upon the impressive group-theoretical work of Sophus Lie.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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" for a blue-green mixture), the laws of nature would still take the same form. The laws are symmetric under any color transformation. Furthermore, the color symmetry is again a gauge symmetry-the laws of nature do not care if the colors or color assortments vary from position to position or from one moment to the next.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 electric charge lies at the root of electromagnetic forces, color originates the strong nuclear force. Each quark flavor comes in three different colors, conventionally called red, green, and blue. There are, therefore, eighteen different quarks.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 electrons are interchanged with neutrinos or with any mixture of the two. The same is true when photons are interchanged with the W and Z force-messengers. The symmetry persists even if the mixtures vary from place to place or from time to time. The invariance of the laws under such transformations performed locally in space and time has become known as gauge symmetry. In the professional jargon, a gauge transformation represents a freedom in formulating the theory that has no directly observable effects-in other words, a transformation to which the physical interpretation is insensitive. Just as the symmetry of the laws of nature under any change of the spacetime coordinates requires the existence of gravity, the gauge symmetry between electrons and neutrinos requires the existence of the photons and the W and Z messenger particles. Once again, when the symmetry is put first, the laws practically write themselves. A similar phenomenon, with symmetry dictating the presence of new particle fields, repeats itself with the strong nuclear force.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 exchange between them bundles of energy called photons. The photon is therefore the messenger of electromagnetism. The electroweak theory predicted the existence of close siblings to the photon, which play the messenger role for the weak force. These never-before-seen particles were prefigured to be about ninety times more massive than the proton and to come in both an electrically charged (called W) and a neutral (called Z) variety. Experiments performed at the European consortium for nuclear research in Geneva (known as CERN for Conseil Europeen pour la Recherche Nucleaire) discovered the W and Z particles in 1983 and 1984 respectively.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 produces conservation of angular momentum. Angular momentum is a quantity characterizing the amount of rotation an object or a system possesses (for a pointlike object it is the product of the distance from the axis of rotation and the momentum). A common manifestation of conservation of angular momentum can be seen in figure skating-when the ice skater pulls her hands inward she spins much faster.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry