Why Beauty Is Truth: A History of Symmetry

Why Beauty Is Truth: A History of Symmetry is a 2007 book by Ian Stewart

Ian Stewart (mathematician)

Ian Nicholas Stewart FRS is a professor of mathematics at the University of Warwick, England, and a widely known popular-science and science-fiction writer. He is the first recipient of the , awarded jointly by the LMS and the IMA for his work on promoting mathematics.-Biography:Stewart was born...

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Following the life and work of famous mathematicians from antiquity to the present, Stewart traces Mathematics' developing handling of the concept of Symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

. One of the very first takeaways, established in the preface of this book is that it dispels the idea of the origins of Symmetry

Symmetry

Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

in Geometry, as is often the first context in which the term is introduced. This book, through its chapters establishes its origins in Algebra

Algebra

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

, more specifically Group Theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

The topics covered are:

Chapter 1: The Scribes of Babylon

The earliest records of solving quadratic equation

Quadratic equation

In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...

Chapter 2: The Household Name

Euclid

Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...

's influence on geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

in general and on regular polygon

Regular polygon

A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...

s in particular.

Chapter 3: The Persian Poet

Omar Khayyám

Omar Khayyám was aPersian polymath: philosopher, mathematician, astronomer and poet. He also wrote treatises on mechanics, geography, mineralogy, music, climatology and theology....

's solution to the cubic equation, which makes use of conic section

Conic section

In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...

Chapter 4: The Gambling Scholar

Niccolò Fontana Tartaglia

Niccolò Fontana Tartaglia was a mathematician, an engineer , a surveyor and a bookkeeper from the then-Republic of Venice...

found the first algebraic solutions to special cubic equations.

Gerolamo Cardano

Gerolamo Cardano was an Italian Renaissance mathematician, physician, astrologer and gambler...

used algebra

Algebra

to solve the cubic and quartic equation.

Chapter 5: The Cunning Fox

Carl Friedrich Gauss

Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

proved that the regular 17-gon can be constructed

Constructible polygon

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not....

using only compass and straightedge, and extended the field

Field extension

In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...

of real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s to the complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

Chapter 6: The Frustrated Doctor and the Sickly Genius

Joseph Louis Lagrange

Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

understood that all approaches to solve algebraic equations could be understood as symmetry transformations of such equations.

Alexandre-Théophile Vandermonde

Alexandre-Théophile Vandermonde was a French musician, mathematician and chemist who worked with Bézout and Lavoisier; his name is now principally associated with determinant theory in mathematics. He was born in Paris, and died there.Vandermonde was a violinist, and became engaged with...

used symmetric functions

Symmetric function

In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...

as an ansatz to solve general algebraic equations, which would lead to the development of Galois theory

Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

Paolo Ruffini

Paolo Ruffini was an Italian mathematician and philosopher.By 1788 he had earned university degrees in philosophy, medicine/surgery, and mathematics...

developed a first (incomplete) proof that the quintic equation

Quintic equation

In mathematics, a quintic function is a function of the formg=ax^5+bx^4+cx^3+dx^2+ex+f,\,where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero...

cannot be solved analytically.

Niels Abel formalized group theory

Group theory

, the indispensable tool in describing symmetries.

Chapter 7: The Luckless Revolutionary

Évariste Galois

Évariste Galois was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem...

laid the foundations to what is today known as Galois theory

Galois theory

In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...

Chapter 8: The Mediocre Engineer and the Transcendent Professor

Pierre Laurent Wantzel proved that it is impossible to double the cube

Doubling the cube

Doubling the cube is one of the three most famous geometric problems unsolvable by compass and straightedge construction...

, trisect the angle, and constructing a regular polygon

Constructible polygon

using only compass and straightedge.

Ferdinand von Lindemann

Carl Louis Ferdinand von Lindemann was a German mathematician, noted for his proof, published in 1882, that π is a transcendental number, i.e., it is not a root of any polynomial with rational coefficients....

proved the transcendence

Transcendental number

In mathematics, a transcendental number is a number that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. The most prominent examples of transcendental numbers are π and e...

of Pi

' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...

, and by implication that it is impossible to square the circle

Squaring the circle

Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge...

using only compass and straightedge.

Chapter 9: The Drunken Vandal

William Rowan Hamilton

Sir William Rowan Hamilton was an Irish physicist, astronomer, and mathematician, who made important contributions to classical mechanics, optics, and algebra. His studies of mechanical and optical systems led him to discover new mathematical concepts and techniques...

extended the field

Field extension

of complex number

Complex number

s to the quarternions.

Chapter 10: The Would-Be Soldier and the Weakly Bookworm

Marius Sophus Lie formalized Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

s and Lie algebra

Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...

Wilhelm Killing

Wilhelm Karl Joseph Killing was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry....

classified all simple Lie algebras (in what Ian Stewart calls the "greatest mathematical paper of all time")

Chapter 11: The Clerk from the Patent Office

Albert Einstein

Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

developed in his theory of general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

a symmetry of space and time.

Chapter 12: A Quantum Quintet

Max Planck

Max Karl Ernst Ludwig Planck, ForMemRS, was a German physicist who actualized the quantum physics, initiating a revolution in natural science and philosophy. He is regarded as the founder of the quantum theory, for which he received the Nobel Prize in Physics in 1918.-Life and career:Planck came...

, Erwin Schrödinger

Erwin Schrödinger

Erwin Rudolf Josef Alexander Schrödinger was an Austrian physicist and theoretical biologist who was one of the fathers of quantum mechanics, and is famed for a number of important contributions to physics, especially the Schrödinger equation, for which he received the Nobel Prize in Physics in 1933...

, Werner Heisenberg

Werner Heisenberg

Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics and is best known for asserting the uncertainty principle of quantum theory...

, Paul Dirac

Paul Dirac

Paul Adrien Maurice Dirac, OM, FRS was an English theoretical physicist who made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics...

, Eugene Wigner were major contributers to the early development of Quantum Mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

. Especially Wigner introduced symmetries into quantum physics.

Chapter 13: The Five-Dimensional Man

An overview over attempts to unify the fundamental forces

Fundamental interaction

In particle physics, fundamental interactions are the ways that elementary particles interact with one another...

, and the role of symmetry in that endeavor.

Chapter 14: The Political Journalist

Edward Witten

Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....

and Superstring theory

Superstring theory

Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...

Chapter 15: A Muddle of Mathematicians

Here, connections between field extension

Field extension

s of real number

Real number

In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s (complex number

Complex number

s, quarternions, octonions), the exceptional simple Lie algebras detected by Killing (G₂, F₄, E₆, E₇, and E₈

E8 (mathematics)

In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8...

), and symmetries occurring in string theory are explored.

Chapter 16: Seekers after Truth and Beauty

Closes the book by contemplating the role of mathematics in physical research.

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