The Equation That Couldn't Be Solved Quotes

“Lagrange was born in Turin (now Italy), but his family was partly French ancestry on his father's side, who was originally wealthy, managed to squander all the family's fortune in speculations, leaving his son with no inheritance. Later in life, Lagrange described this economic catastrophe as the best thing that had ever happened to him: "Had I inherited a fortune I would probably not have cast my lot with mathematics.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 into the outlet directly. The mathematician, on the other hand, says that he would take the iron to the room without the outlet, since that problem has already been solved.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 in the history of mathematics. Almost four thousand years of intellectual curiosity and struggle, spiced with intrigue, misery, and persecution, culminated in the creation of the theory in the nineteenth century. This amazing story, chronicled in the next three chapters, began with the dawn of mathematics on the banks of the Nile and Euphrates rivers.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

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

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

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

“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. Addition, for instance, is associative: (5 + 7) + 13 = 25 and 5 + (7 + 13) = 25, where the parentheses, the "punctuation marks" of mathematics, indicate which pair you add first.

3. Identity element. The group has to contain an identity element such that when combined with any member, it leaves the member unchanged. In the group of integers, the identity element is the number zero. For example, 0 + 3 = 3 + 0 = 3.

4. Inverse. For every member in the group there must exist an inverse. When a member is combined with its inverse, it gives the identity element. For the integers, the inverse of any number is the number of the same absolute value, but with the opposite sign: e.g., the inverse of 4 is -4 and the inverse of -4 is 4; 4 + (-4) = 0 and (-4) + 4 = 0.

The fact that this simple definition can lead to a theory that embraces and unifies all the symmetries of our world continues to amaze even mathematicians.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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 we do that, however, the answer to the question, Can we find translation-symmetric music? is a resounding yes. As Russian crystal physicist G. V. Wulff wrote in 1908: "The spirit of music is rhythm. It consists of the regular, periodic repetition of parts of the musical composition...the regular repetition of identical parts in the whole constitutes the essence of symmetry." Indeed, the recurring themes that are so common in musical composition are the temporal equivalents of Morris's designs and symmetry under translation. Even more generally, compositions are often based on a fundamental motif introduced at the beginning and then undergoing various metamorphoses.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“Note that a rotation by 360 degrees is equivalent to doing nothing at all, or rotating by zero degrees. This is known as the identity transformation. Why bother to define such a transformation at all? As we shall see later in the book, the identity transformation plays a similar role to that of the number zero in the arithmetic operation of addition or the number one in multiplication-when you add zero to a number or multiply a number by one, the number remains unchanged.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“No one, especially not Birkhoff himself, would claim that the intricacies of aesthetic pleasure could be reduced entirely to a mere formula. However, in Birkhoff's words, "In the inevitable analytic accompaniment of the creative process, the theory of aesthetic measure is capable of performing a double service: it gives a simple, unified account of the aesthetic experience, and it provides means for the systematic analysis of typical aesthetic fields.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

“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, Birkhoff's theory heralds symmetry as a crucial aesthetic element.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

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

“One of Lindon's amusing word-unit palindromes reads: "Girl, bathing on Bikini, eyeing boy, finds boy eyeing bikini on bathing girl." Other palindromes are symmetric with respect to back-to-front reading letter by letter-"Able was I ere I saw Elba" (attributed jokingly to Napoleon), or the title of a famous NOVA program: "A Man, a Plan, a Canal, Panama.”
― Mario Livio, The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry

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

“Uno de los primeros estudios documentados sobre las permutaciones no se desarrolla en un libro de matemáticas, sino en uno de mística judía que se remonta a alguna fecha entre los siglos III y VI. El Sefer Yetzira (Libro de la Creación) es un breve y enigmático libro que propone resolver el misterio de la creación analizando las combinaciones de las letras del alfabeto hebreo. La premisa general del libro (que la leyenda cabalística atribuye al antepasado judío Abraham) es que las diferentes categorías de letras forman bloques de construcción divinos con los cuales se pueden construir todas las cosas. En este contexto, el libro afirma que: «Dos letras forman dos palabras, tres forman seis, cuatro forman 24, cinco forman 120, seis forman 720, siete forman 5.040.»”
― Mario Livio, La ecuación jamás resuelta: Como dos genios matemáticos descubrieron el lenguaje de la simetría (Popular Science)

“El primer libro que incluyó la solución completa a la ecuación de segundo grado más general no apareció en Europa hasta el siglo XII. El autor fue el ecléctico matemático judeo-español Abraham bar Hiyya Ha-nasi (1070-1136; «Ha-nasi» significa «el jefe»). Como un recordatorio de los tempranos orígenes de las ecuaciones de segundo grado, el libro se tituló: Hibbur ha-meshihah ve-ha-tishboret (Tratado de Medidas y Cálculos). Abraham bar Hiyya explica: Aquel que desee aprender correctamente la forma de medir áreas y dividirlas, necesariamente tiene que comprender a fondo los teoremas generales de la geometría y la aritmética, sobre los que se cimenta la enseñanza de las medidas. Si domina completamente estas ideas, no podrá desviarse nunca de la verdad. Esto puso fin a una larga era durante la cual los matemáticos árabes actuaron como los seguros guardianes de las matemáticas. Durante los tres mil años que siguieron al período mesopotámico, el progreso sólo ha sido gradual.”
― Mario Livio, La ecuación jamás resuelta: Como dos genios matemáticos descubrieron el lenguaje de la simetría (Popular Science)

“En el código judío de leyes civiles y cánones (el Talmud) encontramos la historia de un exilarca, al cual se había impuesto una enorme multa. Tenía que llenar de trigo un granero que medía de base 40 × 40. El afligido hombre se dirigió al rabino Huna (aprox. 212-97 d.C.), el jefe de la Academia de Sura en Babilonia, en busca de consejo. El sabio le dijo: «Convénceles para que te cojan [dos plazos]: ahora una superficie de 20 × 20 y dentro de un tiempo otra entrega de 20 × 20, y así podrás aprovechar la mitad.» Naturalmente el área de un cuadrado con un lado de 40 unidades es 40 × 40 = 1.600 unidades cuadradas, mientras que el área combinada de dos 20 × 20 cuadrados tan sólo es de 800 unidades cuadradas.”
― Mario Livio, La ecuación jamás resuelta: Como dos genios matemáticos descubrieron el lenguaje de la simetría (Popular Science)

“El psicólogo Christopher W. Tyler del Smith-Kettlewell Eye Research Institute de San Francisco presentó sujetos con una variedad de patrones simétricos traslacionales (y de reflejo). Descubrió que esos estímulos producían la activación de una región del lóbulo occipital, cuya función por lo demás se desconoce. Sorprendentemente, en otras áreas con funciones visuales conocidas se detectó muy poca o ninguna activación. Tyler llegó a la conclusión de que esta región especializada probablemente codifica la presencia de la simetría en el campo visual.”
― Mario Livio, La ecuación jamás resuelta: Como dos genios matemáticos descubrieron el lenguaje de la simetría (Popular Science)