Geometry (; ''geo-'' "earth", ''-metria'' "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a body of practical knowledge concerning lengths, areas, and volumes, with elements of a formal mathematical science emerging in the West as early as Thales (6th Century BC). By the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. Archimedes developed ingenious techniques for calculating areas and volumes, in many ways anticipating modern integral calculus. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, served as an important source of geometric problems during the next one and a half millennia. Both geometry and astronomy were considered in the classical world to be part of the Quadrivium, a subset of the seven liberal arts considered essential for a free citizen to master.

The introduction of coordinates by René Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.

In Euclid's time there was no clear distinction between physical space and geometrical space. Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation, and the question arose: which geometrical space best fits physical space? With the rise of formal mathematics in the 20th century, also 'space' (and 'point', 'line', 'plane') lost its intuitive contents, so today we have to distinguish between physical space, geometrical spaces (in which 'space', 'point' etc. still have their intuitive meaning) and abstract spaces. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between pseudo-Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavour.

While the visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory, geometric language is also used in contexts far removed from its traditional, Euclidean provenance (for example, in fractal geometry and algebraic geometry).

Overview[link]

The recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages.

Practical geometry[link]

Geometry originated as a practical science concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths, areas and volumes, such as Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. A method of computing certain inaccessible distances or heights based on similarity of geometric figures is attributed to Thales. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques.

Axiomatic geometry[link]

Euclid took a more abstract approach in his Elements, one of the most influential books ever written. Euclid introduced certain axioms, or postulates, expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as ''axiomatic'' or ''synthetic'' geometry. At the start of the 19th century the discovery of non-Euclidean geometries by Gauss, Lobachevsky, Bolyai, and others led to a revival of interest, and in the 20th century David Hilbert employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.

Geometric constructions[link]

Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.

Numbers in geometry[link]

In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon abstract numbers in favor of concrete geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation, and whom the Cartesian plane is named after. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculus in the 17th century and led to discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level.

Geometry of position[link]

Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three sub-disciplines within present day geometry that deal with these and related questions.

Leonhard Euler, in studying problems like the Seven Bridges of Königsberg, considered the most fundamental properties of geometric figures based solely on shape, independent of their metric properties. Euler called this new branch of geometry ''geometria situs'' (geometry of place), but it is now known as topology. Topology grew out of geometry, but turned into a large independent discipline. It does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots.

Geometry beyond Euclid[link]

For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, ''absolute'', geometry, which is known to be true ''a priori'' by an inner faculty of mind: Euclidean geometry was synthetic a priori. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture ''Ãœber die Hypothesen, welche der Geometrie zu Grunde liegen'' (''On the hypotheses on which geometry is based''), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.

Dimension[link]

Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number ''n'', possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses ''definitions''; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything ''a priori''.

The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Research may bring a satisfactory ''geometric'' reason for the significance of 10 and 11 dimensions.

Symmetry[link]

The theme of symmetry in geometry is nearly as old as the science of geometry itself. The circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of M. C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry had been recognized. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry ''is''. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.

A different type of symmetry is the principle of duality in projective geometry (see Duality (projective geometry)) among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange ''point'' with ''plane'', ''join'' with ''meet'', ''lies in'' with ''contains'', and you will get an equally true theorem. A similar and closely related form of duality exists between a vector space and its dual space.

History of geometry[link]

The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. The earliest known texts on geometry are the Egyptian ''Rhind Papyrus'' (2000-1800 BC) and ''Moscow Papyrus'' (c. 1890 BC), the Babylonian clay tablets such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks.

In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, which is credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history Eudoxus (408–c.355 BC) developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose ''Elements'', widely considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the ''Elements'' were already known, Euclid arranged them into a single, coherent logical framework. The ''Elements'' was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes (c.287–212 BC) of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave remarkably accurate approximations of Pi. He also studied the spiral bearing his name and obtained formulas for the volumes of surfaces of revolution.

In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry and geometric algebra. Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. ThÄ?bit ibn Qurra (known as Thebit in Latin) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry. Omar Khayyám (1048–1131) found geometric solutions to cubic equations. The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were early results in hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo (c.1230–c.1314), Gersonides (1288–1344), Alfonso, John Wallis, and Giovanni Girolamo Saccheri.

In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry is a geometry without measurement or parallel lines, just the study of how points are related to each other.

Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860) and Carl Friedrich Gauss (1777–1855) and of the formulation of symmetry as the central consideration in the Erlangen Programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis, and introducing the Riemann surface, and Henri Poincaré, the founder of algebraic topology and the geometric theory of dynamical systems. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics.

Contemporary geometry[link]

Euclidean geometry[link]

Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.

Differential geometry[link]

Differential geometry has been of increasing importance to mathematical physics due to Einstein's general relativity postulation that the universe is curved. Contemporary differential geometry is ''intrinsic'', meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point, and not ''a priori'' parts of some ambient flat Euclidean space.

Topology and geometry[link]

The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topology and general topology have gone their own ways.

Algebraic geometry[link]

The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. From late 1950s through mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck. This led to the introduction of schemes and greater emphasis on topological methods, including various cohomology theories. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry.

The study of low dimensional algebraic varieties, algebraic curves, algebraic surfaces and algebraic varieties of dimension 3 ("algebraic threefolds"), has been far advanced. Gröbner basis theory and real algebraic geometry are among more applied subfields of modern algebraic geometry. Arithmetic geometry is an active field combining algebraic geometry and number theory. Other directions of research involve moduli spaces and complex geometry. Algebro-geometric methods are commonly applied in string and brane theory.

See also[link]

Lists[link]

List of geometers

* :Category:Algebraic geometers

* :Category:Differential geometers

* :Category:Geometers

* :Category:Topologists

List of geometry topics

List of important publications in geometry

List of mathematics articles

Related topics[link]

''Flatland'', a book written by Edwin Abbott Abbott about two- and three-dimensional space, to understand the concept of four dimensions

Interactive geometry software

Why 10 dimensions?

Shulba Sutras

Trigonometry

References[link]

Notes[link]

Sources[link]

Boyer, C. B. ''A History of Mathematics'', 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).

Nikolai I. Lobachevsky, Pangeometry, Translator and Editor: A. Papadopoulos, Heritage of European Mathematics Series, Vol. 4, European Mathematical Society, 2010.

Bibliography[link]

Mlodinow, M.; ''Euclid's window (the story of geometry from parallel lines to hyperspace)'', UK edn. Allen Lane, 1992.

External links[link]

A geometry course from Wikiversity

''Unusual Geometry Problems''

''The Math Forum'' — Geometry

* ''The Math Forum'' — K–12 Geometry

* ''The Math Forum'' — College Geometry

* ''The Math Forum'' — Advanced Geometry

Nature Precedings — ''Pegs and Ropes Geometry at Stonehenge''

''The Mathematical Atlas'' — Geometric Areas of Mathematics

"4000 Years of Geometry", lecture by Robin Wilson given at Gresham College, 3 October 2007 (available for MP3 and MP4 download as well as a text file)

* Finitism in Geometry at the Stanford Encyclopedia of Philosophy

The Geometry Junkyard

Interactive Geometry Applications (Java and Cabri 3D)

Interactive geometry reference with hundreds of applets

Dynamic Geometry Sketches (with some Student Explorations)

Geometry classes at Khan Academy

* Category:Greek loanwords

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name	Philip Glass
background	non_performing_personnel
birth date	January 31, 1937
birth place	Baltimore, Maryland, United States
genre	Minimalism, Classical, Contemporary classical, Ambient
occupation	Composer
years active	1956–present
label	Virgin, CBS Records, Nonesuch/Elektra, Sony Classical/SME Records, Orange Mountain Music
notable instruments	Farfisa organ Piano }}

Philip Glass (born January 31, 1937) is an American composer. One of the highest profile composers writing "classical" music today, he is often said to be one of the most influential composers of the late 20th century. His music is also often controversially described as ''minimalist'', along with the work of the other "major minimalists" La Monte Young, Terry Riley and Steve Reich.

He has lately distanced himself from the "minimalist" label, describing himself instead as a composer of "music with repetitive structures." Though his early mature music shares much with what is normally called "minimalist", he has since evolved stylistically. Currently, he describes himself as a "Classicist", pointing out that he is trained in harmony and counterpoint and studied such composers as Franz Schubert, Johann Sebastian Bach and Wolfgang Amadeus Mozart with Nadia Boulanger.

Glass is a prolific composer: he has written works for the musical group which he founded, the Philip Glass Ensemble (with which he still performs on keyboards), as well as operas, musical theatre works, ten symphonies, eleven concertos, solo works, chamber music including string quartets and instrumental sonatas, and film scores. Three of his film scores have been nominated for Academy Awards.

Glass counts many artists among his friends and collaborators, including visual artists (Richard Serra, Chuck Close), writers (Doris Lessing, David Henry Hwang, Allen Ginsberg), film and theatre directors (including Errol Morris, Robert Wilson, JoAnne Akalaitis, Godfrey Reggio, Paul Schrader, Martin Scorsese, Christopher Hampton, Bernard Rose, and many others), choreographers (Lucinda Childs, Jerome Robbins, Twyla Tharp), and musicians and composers (Ravi Shankar, David Byrne, the conductor Dennis Russell Davies, Foday Musa Suso, Laurie Anderson, Linda Ronstadt, Paul Simon, Joan LaBarbara, Arthur Russell, David Bowie, Brian Eno, Roberto Carnevale, Patti Smith, Aphex Twin, Lisa Bielawa, Andrew Shapiro, John Moran, Bryce Dessner and Nico Muhly). Among recent collaborators are Glass's fellow New Yorker Woody Allen, Stephen Colbert, and poet and songwriter Leonard Cohen.

Life and work[link]

:''For a list of works, see List of compositions by Philip Glass''

Beginnings, early education and influences (1937–1964)[link]

Glass was born on January 31, 1937, in Baltimore, Maryland, the son of Ida (née Gouline) and Benjamin Charles Glass, and the grandson of Jewish immigrants from Lithuania. His father owned a record store, and consequently Glass's record collection consisted to a large extent of unsold records, including modern music (such as Hindemith, Bartók, Schoenberg, Shostakovich) and Western classical music (including Ludwig van Beethoven's String Quartets and Schubert's B Piano Trio, which he cites as a "big influence"), at a very early age. He then studied the flute as a child at the Peabody Conservatory of Music and entered an accelerated college program at the University of Chicago at the age of 15, where he studied Mathematics and Philosophy. In Chicago he discovered the serialism of Webern and composed a twelve-tone string trio. In 1954 Glass went to Paris for the first time, encountering the films of Jean Cocteau, which made a lasting impression on him. He visited artists' studios and saw their work; "the bohemian life you see in [Cocteau's] Orphée was the life I ... was attracted to, and those were the people I hung out with."

Glass then went on to the Juilliard School of Music where the keyboard became his main instrument. His composition teachers included Vincent Persichetti and William Bergsma, while fellow students included Steve Reich. During this time, in 1959, he was a winner in the BMI Foundation's BMI Student Composer Awards, one of the most prestigious international prizes for young composers. In the summer of 1960, he studied with Darius Milhaud at the summer school of the Aspen Music Festival and composed a Violin Concerto for a fellow student, Dorothy Pixley-Rothschild. After leaving Juilliard in 1962, Glass moved to Pittsburgh and worked as a school-based composer-in-residence in the public school system, composing various choral, chamber and orchestral music.

Paris (1964–1966)[link]

In 1964, Glass received a Fulbright Scholarship and went to Paris, where he studied with the eminent composition teacher Nadia Boulanger from autumn of 1964 to summer of 1966. Glass's years in Paris as a student made a lasting impression and influenced his work ever since, as the composer admitted in 1979: "The composers I studied with Boulanger are the people I still think about most—Bach and Mozart."

Glass later stated in his autobiography ''Music by Philip Glass'' (1987) that the new music performed at Pierre Boulez's ''Domaine Musical'' concerts in Paris lacked any excitement for him (with the notable exceptions of music by John Cage and Morton Feldman), but he was deeply impressed by new films and theatre performances. He encountered revolutionary films of the French New Wave, such as those of Jean-Luc Godard and François Truffaut, which ignored the rules set by an older generation of artists., and Glass made friends with American visual artists (the sculptor Richard Serra and his wife Nancy Graves), actors and directors (JoAnne Akalaitis, Ruth Maleczech, David Warrilow, and Lee Breuer, with whom Glass later founded the experimental theatre group Mabou Mines). Together with Akalaitis (they married in 1965), Glass in turn attended performances by theatre groups including Jean-Louis Barrault's Odéon theatre, The Living Theatre and the Berliner Ensemble in 1964 to 1965. These significant encounters resulted in a collaboration with Breuer for which Glass contributed music for a 1965 staging of Samuel Beckett's ''Comédie'' (''Play'', 1963). The resulting piece (written for two soprano saxophones) was directly influenced by the play's open-ended, repetitive and almost musical structure and was the first one of a series of four early pieces in a minimalist, yet still dissonant, idiom. After ''Play'', Glass also acted in 1966 as music director of a Breuer production of Brecht's ''Mother Courage and Her Children'', featuring the theatre score by Paul Dessau.

In parallel with his early excursions in experimental theatre, Glass worked in winter 1965 and spring 1966 as a music director and composer on a film score (''Chappaqua'', Conrad Rooks, 1966) with Ravi Shankar and Alla Rakha, which added another important influence on Glass's musical thinking. His distinctive style arose from his work with Shankar and Rakha and their perception of rhythm in Indian music as being entirely additive. He renounced all his compositions in a moderately modern style resembling Milhaud's, Aaron Copland's, and Samuel Barber's, and began writing pieces based on repetitive structures of Indian music and a sense of time influenced by Samuel Beckett: a piece for two actresses and chamber ensemble, a work for chamber ensemble and his first numbered string quartet (No.1, 1966).

Glass then left Paris for northern India in 1966, where he came in contact with Tibetan refugees and began to gravitate towards Buddhism. He met Tenzin Gyatso, the 14th Dalai Lama, in 1972, and has been a strong supporter of the Tibetan independence ever since.

Minimalism: From ''Strung Out'' to ''Music in 12 Parts'' (1967–1974)[link]

Shortly after arriving in New York City in March 1967, Glass attended a performance of works by Steve Reich (including the ground-breaking minimalist piece ''Piano Phase''), which left a deep impression on him; he simplified his style and turned to a radical "consonant vocabulary". Finding little sympathy from traditional performers and performance spaces, Glass eventually formed an ensemble with fellow ex-students Steve Reich, Jon Gibson, and others, and began performing mainly in art galleries and studio lofts of SoHo. The visual artist Richard Serra provided Glass with Gallery contacts, while both collaborated on various sculptures, films and installations; from 1971 to 1974 he became Serra's regular studio assistant.

Between summer of 1967 and the end of 1968, Glass composed nine works, including ''Strung Out '' (for amplified solo violin, composed in summer of 1967), Gradus (for solo saxophone, 1968), ''Music in the Shape of a Square'' (for two flutes, composed in May 1968, an homage to Erik Satie), ''How Now '' (for solo piano, 1968) and ''1+1'' (for amplified tabletop, November 1968) which were "clearly designed to experiment more fully with his new-found minimalist approach". The first concert of Glass's new music was at Jonas Mekas's Film-Makers Cinemathèque (Anthology Film Archives) in September 1968. This concert included the first work of this series with ''Strung Out'' (performed by the violinist Pixley-Rothschild) and ''Music in the Shape of a Square'' (performed by Glass and Gibson). The musical scores were tacked on the wall, and the performers had to move while playing. Glass's new works met with a very enthusiastic response by the audience which consisted mainly of visual and performance artists who were highly sympathetic to Glass's reductive approach.

Apart from his music career, Glass had a moving company with his cousin, the sculptor Jene Highstein, and worked as a plumber and cab driver (in 1973 to 1978). During this time he made friends with other New York based artists such as Sol LeWitt, Nancy Graves, Michael Snow, Bruce Nauman, Laurie Anderson, and Chuck Close, who created a now famous portrait of Glass. (Glass returned the favour in 2005 with ''A Musical Portrait of Chuck Close'' for piano.)

With ''1+1'' and ''Two Pages'' (composed in February 1969) Glass turned to a more "rigorous approach" to his "most basic minimalist technique, additive process", pieces which were followed in the same year by ''Music in Contrary Motion'' and ''Music in Fifths'' (a kind of homage to his composition teacher Nadia Boulanger, who pointed out "hidden fifths" in his works but regarded them as cardinal sins). Eventually Glass's music grew less austere, becoming more complex and dramatic, with pieces such as ''Music in Similar Motion'' (1969), and ''Music with Changing Parts'' (1970). These pieces were performed by The Philip Glass Ensemble in the Whitney Museum of American Art in 1969 and in the Solomon R. Guggenheim Museum in 1970, often encountering hostile reaction from critics,; but Glass's music was also met with enthusiasm from younger artists such as Brian Eno and David Bowie (at the Royal College of Art ca. 1970). Eno described this encounter with Glass's music as one of the "most extraordinary musical experiences of [his] life", as a "viscous bath of pure, thick energy", concluding "this was actually the most detailed music I'd ever heard. It was all intricacy, exotic harmonics". In 1970 Glass returned to the theatre, composing music for the theatre group Mabou Mines, resulting in his first minimalist pieces employing voices: ''Red Horse Animation'' and ''Music for Voices'' (both 1970, and premiered at the Paula Cooper Gallery).

After differences of opinion with Steve Reich in 1971, Glass formed the Philip Glass Ensemble (while Reich formed Steve Reich and Musicians), an amplified ensemble including keyboards, wind instruments (saxophones, flutes), and soprano voices.

Glass's music for his ensemble culminated in the four-hour-long ''Music in Twelve Parts'' (1971–1974), which began as a single piece with twelve instrumental parts but developed into a cycle that summed up Glass's musical achievement since 1967, and even transcended it – the last part features a twelve-tone theme, sung by the soprano voice of the ensemble. "I had broken the rules of modernism and so I thought it was time to break some of my own rules", according to Glass. Though he finds the term minimalist inaccurate to describe his later work, Glass does accept this term for pieces up to and including ''Music in 12 Parts'', excepting this last part which "was the end of minimalism" for Glass. As he pointed out: "I had worked for eight or nine years inventing a system, and now I'd written through it and come out the other end."

Another Look at Harmony: The Portrait Trilogy (1975–1979)[link]

Glass continued his work with a series of instrumental works, called '' Another Look at Harmony '' (1975–1977). For Glass this series demonstrated a new start, hence the title: "What I was looking for was a way of combining harmonic progression with the rhythmic structure I had been developing, to produce a new overall structure. (...) I'd taken everything out with my early works and it was now time to decide just what I wanted to put in—a process that would occupy me for several years to come." Parts 1 and 2 of "Another Look at Harmony" were included in a collaboration with Robert Wilson, a piece of musical theater that was later designated by Glass as the first opera of his portrait opera trilogy: ''Einstein on the Beach''. Composed in Spring to Fall of 1975 in close collaboration with Wilson, Glass's first opera was first premiered in summer 1976 at the Festival d'Avignon, and in November of the same year to a mixed and partly enthusiastic reaction from the audience at the Metropolitan Opera in New York City. Scored for the Philip Glass Ensemble, solo violin, chorus, and featuring actors (reciting texts by Christopher Knowles, Lucinda Childs and Samuel M. Johnson), Glass's and Wilson's essentially plotless opera was conceived as a "metaphorical look at Albert Einstein: scientist, humanist, amateur musician—and the man whose theories (...) led to the splitting of the atom", evoking nuclear holocaust in the climactic scene, as critic Tim Page pointed out. As with ''Another Look at Harmony'', "''Einstein'' added a new functional harmony that set it apart from the early conceptual works". Composer Tom Johnson came to the same conclusion, comparing the solo violin music to Johann Sebastian Bach, and the "organ figures (...) to those Alberti basses Mozart loved so much". The piece was praised by the Washington Post as "one of the seminal artworks of the century."

''Einstein on the Beach'' was followed by further music for projects by the theatre group Mabou Mines such as ''Dressed like an Egg'' (1975), and again music for plays and adaptations from prose by Samuel Beckett, such as ''The Lost Ones'' (1975), ''Cascando'' (1975), ''Mercier and Camier'' (1979). Glass also turned to other media; two multi-movement instrumental works for the Philip Glass Ensemble originated as music for film and TV: ''North Star'' (1977 for the Documentary "Mark di Suvero, sculptor" by Francois de Menil and Barbara Rose) and four short cues for Jim Henson's TV-series for children, ''Sesame Street'', named ''Geometry of Circles'' (1979).

Another series, ''Fourth Series'' (1977–79), included music for chorus and organ ("Part One", 1977), organ and piano ("Part Two" and "Part Four", 1979), and music for a radio adaption of Constance DeJong's novel ''Modern Love'' ("Part Three", 1978). Part Two and Part Four were used (and hence renamed) in two dance productions by choreographer Lucinda Childs (who had already contributed to and performed in ''Einstein on the Beach''). "Part Two" was included in ''Dance'' (a collaboration with visual artist Sol LeWitt, 1979), and "Part Four" was renamed as ''Mad Rush'', and performed by Glass on several occasions such as the first public appearance of the 14th Dalai Lama in New York City in Fall 1981. The piece demonstrates Glass's turn to more traditional models: the composer added a conclusion to an open-structured piece which "can be interpreted as a sign that he [had] abandoned the radical non-narrative, undramatic approaches of his early period", as the pianist Steffen Schleiermacher points out.

In Spring 1978, Glass received a commission from the Netherlands Opera (as well as a Rockefeller Foundation grant) which "marked the end of his need to earn money from non-musical employment." With the commission Glass continued his work in music theater, composing his opera ''Satyagraha'' (composed in 1978–1979, premiered in 1980 at Rotterdam), based on the early life of Mahatma Gandhi in South Africa, Rabindranath Tagore, and Martin Luther King Jr.. For ''Satyagraha '', Glass worked in close collaboration with two "SoHo friends": the writer Constance deJong, who provided the libretto, and the set designer Robert Israel. This piece was in other ways a turning point for Glass, as it was his first work since 1963 scored for symphony orchestra, even if the most prominent parts were still reserved for solo voices and chorus. Shortly after completing the score in August 1979, Glass met the conductor Dennis Russell Davies, who he helped prepare for performances in Germany (using a piano-four-hands version of the score); together they started to plan another opera, to be premiered at the Stuttgart State Opera.

Completing the Portrait Trilogy: Akhnaten and beyond (1980–1986)[link]

While planning a third part of his "Portrait Trilogy", Glass turned to smaller music theatre projects such as the non-narrative ''Madrigal Opera'' (for six voices and violin and viola, 1980), and ''The Photographer'', a biographic study on the photographer Eadweard Muybridge (1982). Glass also continued to write for the orchestra with his most famous film score to date, ''Koyaanisqatsi'' (Godfrey Reggio, 1981–1982). Some pieces which were not used in the film (such as ''Facades'') eventually appeared on the album ''Glassworks'' (1982, CBS Records), which brought Glass's music to a wider public.

The "Portrait Trilogy" was completed with ''Akhnaten'' (1982–1983, premiered in 1984), a vocal and orchestral composition sung in Akkadian, Biblical Hebrew, and Ancient Egyptian. In addition, this opera featured an actor reciting ancient Egyptian texts in the language of the audience. ''Akhnaten'' was commissioned by the Stuttgart Opera in a production designed by Achim Freyer. It premiered simultaneously at the Houston Opera in a production directed by David Freeman and designed by Peter Sellars. At the time of the commission, the Stuttgart Opera House was undergoing renovation, necessitating the use of a nearby playhouse with a smaller orchestra pit. Upon learning this, Glass and conductor Dennis Russell Davies visited the playhouse, placing music stands around the pit to determine how many players the pit could accommodate. The two found that they could not fit a full orchestra in the pit. Glass decided to eliminate the violins, which had the effect of "giving the orchestra a low, dark sound that came to characterize the piece and suited the subject very well." As Glass remarked in 1992, ''Akhnaten'' is significant in his work since it represents a "first extension out of a triadic harmonic language", an experiment with the polytonality of his teachers Persichetti and Milhaud, a musical technique which Glass compares to "an optical illusion, such as in the paintings of Josef Albers".

Glass again collaborated with Robert Wilson on another opera, ''the CIVIL warS'' (1983, premiered in 1984), which also functioned as the final part ("the Rome section) of Wilson's epic work by the same name, originally planned for an "international arts festival that would accompany the Olympic Games in Los Angeles". (Glass also composed a highly prestigious work for chorus and orchestra for the opening of the Games, ''The Olympian: Lighting of the Torch and Closing ''). The premiere of ''The CIVIL warS'' in Los Angeles never materialized and the opera was in the end premiered at the Opera of Rome. Glass's and Wilson's opera includes musical settings of Latin texts by the 1st-century-Roman playwright Seneca and allusions to the music of Giuseppe Verdi and from the American Civil War, featuring the 19th century figures Giuseppe Garibaldi and Robert E. Lee as characters.

In the mid-1980s, Glass produced "works in different media at an extraordinarily rapid pace". Projects from that period include music for dance (''Dance Pieces'', Jerome Robbins, 1983, and ''In the Upper Room'', Twyla Tharp, 1986), and music for theatre productions ''Endgame'' (1984), and ''Company'' (1983). Beckett vehemently disapproved of the production of ''Endgame'' at the American Repertory Theater (Cambridge, Massachusetts), which featured Joanne Akalaitis's direction and Glass's ''Prelude'' for timpani and double bass, but in the end, he authorized the music for ''Company'', four short, intimate pieces for string quartet that were played in the intervals of the dramatization. This composition was initially regarded by the composer as a piece of Gebrauchsmusik ('music for use')—"like salt and pepper (...) just something for the table�, as he noted. Eventually ''Company'' was published as Glass's String Quartet No.2 and in a version for string orchestra, being performed by ensembles ranging from student orchestras to renowned ones such as the Kronos Quartet and the Kremerata Baltica.

This interest in writing for the string quartet and the string orchestra led to a chamber and orchestral film score for ''Mishima: A Life in Four Chapters'' (Paul Schrader, 1984–85), which Glass recently described as his "musical turning point" that developed his "technique of film scoring in a very special way".

Glass also dedicated himself to vocal works with two sets of songs, ''Three Songs for chorus'' (1984, settings of poems by Leonard Cohen, Octavio Paz and Raymond Levesque), and a song cycle initiated by CBS Masterworks Records: ''Songs from Liquid Days'' (1985), with texts by songwriters such as David Byrne, Paul Simon, in which the Kronos Quartet is featured (as it is in ''Mishima'') in a prominent role. Glass also continued his series of operas with adaptations from literary texts such as ''The Juniper Tree'' (an opera collaboration with composer Robert Moran, 1984), Edgar Allan Poe's ''The Fall of the House of Usher'' (1987), and also worked with novelist Doris Lessing on the opera ''The Making Of The Representative For Planet 8'' (1985–86, and performed by the Houston Grand Opera and English National Opera in 1988).

Operas and the turn to symphonic music (1987–1991)[link]

Compositions such as ''Company'', ''Facades'' and String Quartet No.3 (the last two extracted from the scores to ''Koyaanisqatsi'' and ''Mishima'') gave way to a series of works more accessible to ensembles such as the string quartet and symphony orchestra, in this returning to the structural roots of his student days. In taking this direction his chamber and orchestral works were also written in a more and more traditional and lyrical style. In these works, Glass often employs old musical forms such as the chaconne and the passacaglia – for instance in ''Satyagraha'', the Violin Concerto No. 1 (1987), Symphony No. 3 (1995), ''Echorus'' (1995) and also recent works such as Symphony No. 8 (2005), and ''Songs and Poems for Solo Cello'' (2006).

A series of orchestral works that were originally composed for the concert hall commenced with the 3-movement Violin Concerto No. 1 (1987). This work was commissioned by the American Composers Orchestra and written for and in close collaboration with the violinist Paul Zukofsky and the conductor Dennis Russel Davies, who since then has encouraged the composer to write numerous orchestral pieces. The Concerto is dedicated to the memory of Glass's father: "His favorite form was the violin concerto, and so I grew up listening to the Mendelssohn, the Paganini, the Brahms concertos. (...) So when I decided to write a violin concerto, I wanted to write one that my father would have liked." Among its multiple recordings, in 1992, the Concerto was performed and recorded by Gidon Kremer and the Vienna Philharmonic Orchestra. This turn to orchestral music was continued with a symphonic trilogy of "portraits of nature", commissioned by the Cleveland Orchestra, the Rotterdam Philharmonic Orchestra, and the Atlanta Symphony Orchestra: ''The Light'' (1987), ''The Canyon'' (1988), and ''Itaipu'' (1989).

While composing for symphonic ensembles, Glass also composed music for piano, with the cycle of five movements titled ''Metamorphosis'' (adapted from music for a theatrical adaptation of Franz Kafka's ''The Metamorphosis'', and for the Errol Morris film ''The Thin Blue Line'', 1988]). In the same year Glass met the poet Allen Ginsberg by chance in a book store in the East Village of New York City, and they immediately "decided on the spot to do something together, reached for one of Allen's books and chose ''Wichita Vortex Sutra''", a piece for reciter and piano which in turn developed into a music theatre piece for singers and ensemble, ''Hydrogen Jukebox'' (1990).

Glass also turned to chamber music; he composed two String Quartets (No.4 and No.5, for the Kronos Quartet, 1989 and 1991), and chamber works which originated as incidental music for plays, such as ''Music from "The Screens"'' (1989/1990). This work originated in one of many theater music collaborations with the director Joanne Akalaitis, who originally asked the Gambian musician Foday Musa Suso "to do the score [for Jean Genet's "The Screens"] in collaboration with a western composer". Glass had already collaborated with Suso in the film score to Powaqqatsi (Godfrey Reggio, 1986). ''Music from "The Screens"'' is on occasion a touring piece for Glass and Suso, and individual pieces found its way to the repertoire of Glass and the cellist Wendy Sutter. Another collaboration was a collaborative recording project with Ravi Shankar, initiated by Peter Baumann (a member of the band Tangerine Dream), which resulted in the album ''Passages'' (1990).

In the late 1980s and early 1990s, Glass's projects also included two highly prestigious opera commissions, based on the life of two explorers, Christopher Columbus (''The Voyage'' [1990], commissioned by the Metropolitan Opera, with a libretto by David Henry Hwang), and Vasco da Gama (''White Raven'') [1991], a collaboration with Robert Wilson and composed for the opening of the Expo '98). Especially in ''The Voyage'', the composer "explore[d] new territory", with its "newly arching lyricism", "Sibelian starkness and sweep", and "dark, brooding tone (...) a reflection of its increasingly chromatic (and dissonant) palette", as one commentator put it.

The Cocteau Trilogy and Symphonies (1991–1996)[link]

After these operas, Glass began working on a symphonic cycle, commissioned by the conductor Dennis Russell Davies, who told Glass at the time: "I'm not going to let you . . . be one of those opera composers who never write a symphony". Glass responded with two 3-movement symphonies (''"Low"'' [1992], and Symphony No. 2 [1994]); his first in an ongoing series of symphonies is a combination of the composer's own musical material with themes featured in prominent tracks of the David Bowie/ Brian Eno album ''Low'' (1977), whereas Symphony No.2 is described by Glass as a study in polytonality. He referred to the music of Honegger, Milhaud, and Villa-Lobos as possible models for his symphony. With the Concerto Grosso (1992), Symphony No. 3 (1995), a Concerto for Saxophone Quartet and Orchestra (1995), written for the Rascher Quartet (all commissioned by conductor Dennis Russel Davies), and ''Echorus'' (1994/95), a more transparent, refined, and intimate chamber-orchestral style paralleled the excursions of his large-scale symphonic pieces. In the four movements of his Third Symphony, Glass treats a 19-piece string orchestra as an extended chamber ensemble. In the third movement, Glass re-uses the chaconne as a formal device; one commentator characterized Glass's symphony as one of the composer's "most tautly unified works" The third Symphony was closely followed by a fourth, subtitled ''Heroes'' (1996), commissioned the American Composers Orchestra. Its six movements are symphonic reworkings of themes by Glass, David Bowie, and Brian Eno (from their album ''"Heroes"'', 1977); as in other works by the composer, it is also a hybrid work and exists in two versions: one for the concert hall, and another, shorter one for dance, choreographed by Twyla Tharp. Another commission by Dennis Russell Davies was a second series for piano, the ''Etudes'' for Piano (dedicated to Davies as well as the production designer Achim Freyer); the complete first set of ten Etudes has been recorded and performed by Glass himself. Bruce Brubaker and Dennis Russell Davies have each recorded the original set of six. Most of the Etudes are composed in the post-minimalist and increasingly lyrical style of the times: "Within the framework of a concise form, Glass explores possible sonorities ranging from typically Baroque passagework to Romantically tinged moods". Some of the pieces also appeared in different versions such as in the theatre music to Robert Wilson's ''Persephone'' (1994, commissioned by the Relache Ensemble) or ''Echorus'' (a version of Etude No.2 for two violins and string orchestra, written for Edna Mitchell and Yehudi Menuhin 1995).

Glass's prolific output in the 1990s continued to include operas with an opera triptych (1991–1996), which the composer described as an "homage" to writer and film director Jean Cocteau, based on his prose and cinematic work: ''Orphée'' (1949), ''La Belle et la Bête'' (1946), and the novel ''Les Enfants Terribles'' (1929, later made into a film by Cocteau and Jean-Pierre Melville, 1950). In the same way the triptych is also a musical homage to the work of the group of French composers associated with Cocteau, Les Six (and especially to Glass's teacher Darius Milhaud), as well as to various 18th century composers such as Gluck and Bach whose music featured as an essential part of the films by Cocteau.

The inspiration of the first part of the trilogy, ''Orphée'' (composed in 1991, and premiered in 1993 at the American Repertory Theatre) can be conceptually and musically traced to Gluck's opera ''Orfeo ed Euridice'' (''Orphée et Euridyce'', 1762/1774), which had a prominent part in Cocteau's 1949 film ''Orphee''. One theme of the opera, the death of Eurydice, has some similarity to the composer's personal life: the opera was composed after the unexpected death in 1991 of Glass's wife, artist Candy Jernigan: "(...) One can only suspect that Orpheus' grief must have resembled the composer's own", K. Robert Schwartz suggests. The opera's "transparency of texture, a subtlety of instrumental color, (...) a newly expressive and unfettered vocal writing" was praised, and ''The Guardian's'' critic remarked "Glass has a real affinity for the French text and sets the words eloquently, underpinning them with delicately patterned instrumental textures".

For the second opera, ''La Belle et la Bête'' (1994, scored for either the Philip Glass Ensemble or a more conventional chamber orchestra), Glass replaced the soundtrack (including Georges Auric's film music) of Cocteau's film, wrote "a new fully operatic score and synchronize[d] it with the film". The final part of the triptych returned again to a more traditional setting with the "Dance Opera" ''Les Enfants Terribles'' (1996), scored for voices, three pianos and dancers, with choreography by Susan Marshall. The characters are depicted by both singers and dancers. The scoring of the opera evokes Bach's Concerto for Four Harpsichords, but in another way also "the snow, which falls relentlessly throughout the opera (...) bearing witness to the unfolding events. Here time stands still. There is only music, and the movement of children through space" (Glass).

Symphonies, opera, and concertos (1997–2004)[link]

In the late 1990s and early 2000s, Glass's lyrical and romantic styles peaked with numerous projects: operas, theatre and film scores (Martin Scorsese's ''Kundun'', 1997, Godfrey Reggio's ''Naqoyqatsi'', 2002, and Stephen Daldry's ''The Hours'', 2002), a series of five concerti, and three symphonies centered on orchestra-singer and orchestra-chorus interplay. Two symphonies, Symphony No.5 "Choral" (1999) and Symphony No.7 "Toltec" (2004), and the song cycle ''Songs of Milarepa'' (1997) are thematically meditative. The operatic Symphony No.6 ''Plutonian Ode'' (2002) for soprano and orchestra was commissioned by the Brucknerhaus, Linz, and Carnegie Hall in celebration of Glass's sixty-fifth birthday, and originated as Glass's collaboration with Allen Ginsberg (poet, piano – Ginsberg, Glass), based on his eponymous poem.

Besides writing for the concert hall, Glass continued his ongoing operatic series with adaptions from literary texts: ''The Marriages of Zones 3, 4 and 5'' ([1997] story-libretto by Doris Lessing), ''In the Penal Colony'' (2000, after the novella by Franz Kafka), and the chamber opera ''The Sound of a Voice'' (2003, with David Henry Hwang), which features the Pipa, performed by Wu Man at its premiere. Glass also collaborated again with the co-author of ''Einstein on the Beach'', Robert Wilson, on ''Monsters of Grace'' (1998), and created a biographic opera on the life of astronomer Galileo Galilei (2001).

In the early 2000s, Glass started a series of five concerti with the ''Tirol Concerto for Piano and Orchestra'' (2000, premiered by Dennis Russell Davies as conductor and soloist), and the ''Concerto Fantasy for Two Timpanists and Orchestra'' (2000, for the timpanist Jonathan Haas). The ''Concerto for Cello and Orchestra'' (2001) had its premiere performance in Beijing, featuring cellist Julian Lloyd Webber; it was composed in celebration of his fiftieth birthday. These concertos were followed by the concise and rigorously neo-baroque ''Concerto for Harpsichord and Orchestra'' (2002), demonstrating in its transparent, chamber orchestral textures Glass's classical technique, evocative in the "improvisatory chords" of its beginning a toccata of Froberger or Frescobaldi, and 18th century music. Two years later, the concerti series continued with ''Piano Concerto No. 2: After Lewis and Clark'' (2004), composed for the pianist Paul Barnes. The concerto celebrates the pioneers' trek across North America, and the second movement features a duet for piano and Native American flute. With the chamber opera ''The Sound of a Voice'', Glass's Piano Concerto No. 2 might be regarded as bridging his traditional compositions and his more popular excursions to World Music, also found in ''Orion'' (also composed in 2004).

''Songs and Poems'' (2005–2007)[link]

''Waiting for the Barbarians'', an opera from J. M. Coetzee's novel (with the libretto by Christopher Hampton), had its premiere performance in September 2005. Glass defined the work as a "social/political opera", as a critique on the Bush administration's war in Iraq, a "dialogue about political crisis", and an illustration of the "power of art to turn our attention toward the human dimension of history". While the opera's themes are Imperialism, apartheid, and torture, the composer chose an understated approach by using "very simple means, and the orchestration is very clear and very traditional; it's almost classical in sound", as the conductor D. Russell Davies notes.

Two months after the premiere of this opera, in November 2005, Glass's Symphony No.8, commissioned by the Bruckner Orchester Linz, was premiered at the Brooklyn Academy of Music in New York City. After three symphonies for voices and orchestra, this piece was a return to purely orchestral and abstract composition; like previous works written for the conductor Dennis Russell Davies (the 1992 Concerto Grosso and the 1995 Symphony No.3), it features extended solo writing. Critic Allan Kozinn described the symphony's chromaticism as more extreme, more fluid, and its themes and textures as continually changing, morphing without repetition, and praised the symphony's "unpredictable orchestration", pointing out the "beautiful flute and harp variation in the melancholy second movement". Alex Ross, remarked that "against all odds, this work succeeds in adding something certifiably new to the overstuffed annals of the classical symphony. (...) The musical material is cut from familiar fabric, but it’s striking that the composer forgoes the expected bustling conclusion and instead delves into a mood of deepening twilight and unending night."

''The Passion of Ramakrishna '' (2006), was composed for the Pacific Symphony Orchestra, the Pacific Chorale and the conductor Carl St. Clair. The 45 minutes choral work is based on the writings of Indian Spiritual leader Sri Ramakrishna, which seem "to have genuinely inspired and revived the composer out of his old formulas to write something fresh", as one critic remarked, whereas another noted that "The musical style breaks little new ground for Glass, except for the glorious Handelian ending (...) the "composer’s style ideally fits the devotional text".

A Cello Suite, composed for the cellist Wendy Sutter, "Songs and Poems for Solo Cello" (2005–2007), was equally lauded by critics. It was described by Lisa Hirsch as "a major work, (...) a major addition to the cello repertory" and "deeply Romantic in spirit, and at the same time deeply Baroque". Another critic, Anne Midgette of the Washington Post, noted that the suite "maintains an unusual degree of directness and warmth"; she also noted a kinship to a major work by Johann Sebastian Bach: "Digging into the lower registers of the instrument, it takes flight in handfuls of notes, now gentle, now impassioned, variously evoking the minor-mode keening of klezmer music and the interior meditations of Bach's cello suites". Glass himself pointed out that "in many ways it owes more to Schubert than to Bach".

In 2007, Glass also worked alongside Leonard Cohen on an adaptation of Cohen's poetry collection ''Book of Longing''. The work, which premiered in June 2007 in Toronto, is a piece for seven instruments and a vocal quartet, and contains recorded spoken word performances by Cohen and imagery from his collection.

''Appomattox'', an opera surrounding the events at the end of the American Civil War, was commissioned by the San Francisco Opera and premiered on October 5, 2007. As in ''Waiting for the Barbarians'', Glass collaborated with the writer Christopher Hampton, and as with the preceding opera and Symphony No.8, the piece was conducted by Glass's long-time collaborator Dennis Russell Davies, who noted that "in his recent operas the bass line has taken on an increasing prominence,(...) (an) increasing use of melodic elements in the deep register, in the contrabass, the contrabassoon—he's increasingly using these sounds and these textures can be derived from using these instruments in different combinations. (...) He's definitely developed more skill as an orchestrator, in his ability to conceive melodies and harmonic structures for specific instrumental groups. (...) what he gives them to play is very organic and idiomatic."

Apart from this large-scale opera, Glass added a work to his catalogue of theater music in 2007, and continuing—after a gap of twenty years—to write music for the dramatic work of Samuel Beckett. He provided a "hypnotic" original score for a compilation of Beckett's short plays ''Act Without Words I'', ''Act Without Words II'', ''Rough for Theatre I'' and ''Eh Joe'', directed by JoAnne Akalaitis and premiered in December 2007. Glass's work for this production was described by ''The New York Times'' as "icy, repetitive music that comes closest to piercing the heart".

Chamber music, concertos and symphonies (2008–present)[link]

2008 to 2010 Glass continued to work on a series of chamber music pieces which started with "Songs and Poems": the ''Four Movements for Two Pianos'' (2008, premiered by Dennis Davies and Maki Namekawa in July 2008), a ''Sonata for Violin and Piano'' composed in "the Brahms tradition" (completed in 2008, premiered by violinist Maria Bachman and pianist Jon Klibonoff in February 2009); a ''String sextet'' (an adaption of the Symphony No.3 of 1995 made by Glass's musical director Michael Riesman) followed in 2009. ''Pendulum'' (2010, a one-movement piece for violin and piano), a second Suite of cello pieces for Wendy Sutter (2011), and ''Partita for solo violin'' for violinist Tim Fain (2010, first performance of the complete work 2011), are recent entries in the series.

Other works for the theater were a score for Euripides' ''The Bacchae'' (2009, directed by JoAnne Akalaitis), and ''Kepler'' (2009), yet another operatic biography of a scientist or explorer. The opera is based on the life of 17th century astronomer Johannes Kepler, against the background of the Thirty Years' War, with a libretto compiled from Kepler's texts and poems by his contemporary Andreas Gryphius. It is Glass's first opera in German, and was premiered by the Bruckner Orchester Linz and Dennis Russell Davies in September 2009. LA Times critic Mark Swed and others described the work as "oratorio-like"; Swed pointed out that the work is Glass's "most chromatic, complex, psychological score" and that the "The orchestra dominates (...) I was struck by the muted, glowing colors, the character of many orchestral solos and the poignant emphasis on bass instruments".

In 2009 and 2010, Glass returned to the concerto genre. Violin Concerto No. 2 in four movements was commissioned by violinist Robert McDuffie, and subtitled "The American Four Seasons" (2009), as an homage to Vivaldi's set of concertos "Le quattro stagioni". It premiered in December 2009 by the Toronto Symphony Orchestra, and was subsequently performed by the London Philharmonic Orchestra in April 2010. The Double Concerto for Violin and Cello and Orchestra (2010) was composed for soloists Maria Bachmann and Wendy Sutter and also as a ballet score for the Nederlands Dans Theater. Other orchestral projects of 2010 are short orchestral scores for films; to a multimedia presentation based on the novel ''Icarus at the Edge of Time'' by theoretical physicist Brian Greene, which premiered on June 6, 2010, and the score for the Brazilian film ''Nosso Lar'' (released in Brazil on September 3, 2010). Glass also donated a short work, ''Brazil'', to the video game ''Chime'', which was released on February 3, 2010.

In January 2011, Glass performed at the MONA FOMA festival in Hobart, Tasmania. The festival promotes a broad range of art forms, including experimental sound, noise, dance, theatre, visual art, performance and new media.

In August 2011, Glass presented a series of music, dance, and theater performances as part of the Days and Nights Festival. Along with the Philip Glass Ensemble, scheduled performers include Molissa Fenley and Dancers, John Moran with Saori Tsukada, as well as a screening of ''Dracula'' with Glass's score. Glass hopes to present this festival annually, with a focus on art, science, and conservation.

Glass's recently completed and projected works include Symphony No.9 (2010–2011), Symphony No.10 (2011), Cello Concerto No.2 (2012, based on the film score to Naqoyqatsi), Symphony No.11, String Quartet No.6, and the operas ''The Lost'' (2011–2012) and ''The Perfect American'' (2011). Glass's Ninth Symphony was co-commissioned by the Bruckner Orchester Linz, the American Composers Orchestra and the Los Angeles Philharmonic Orchestra. The symphony's first performance took place on January 1, 2012 at the Brucknerhaus in Linz, Austria (Dennis Russell Davies conducting the Bruckner Orchester Linz); the American premiere was on January 31, 2012 (Glass's 75th birthday), at Carnegie Hall (Dennis Russell Davies conducting the American Composers Orchestra), and the West Coast premiere with the Los Angeles Philharmonic under the baton of John Adams on April 5. ''The Lost'' is based on a play by Austrian playwright and novelist Peter Handke, ''Die Spuren der Verirrten'' (2006). It will be premiered in 2013 in Linz (Austria), conducted by Dennis Russell Davies and directed by David Pountney, who pointed out that the English translation of the original German title means "traces of those who lost their way": "[So] not knowing where you are going, let alone where you came from, seems to be a pre-condition. It is also, perhaps, the way we are? Philip’s music is perfect for this kind of abstraction". ''The Perfect American'' was composed in 2011 to a commission from Teatro Real Madrid. The libretto is based on a book of the same name by Peter Stephan Jungk and covers the final months of the life of Walt Disney. The world premiere is scheduled for Teatro Real Madrid on 22 January 2013 with British baritone Christopher Purves taking the role of Disney. The UK premiere is scheduled for the English National Opera at the London Coliseum later in 2013.

On December 1, 2011 Glass, together with Lou Reed and Laurie Anderson, joined the Occupy Wall Street protests at the Lincoln Center, reciting from the libretto of his opera Satyagraha (1978–79), which had the final performance on the same evening. A Website dedicated to Occupy Wall Street points out that "it is no doubt timely that Philip Glass's opera 'Satyagraha'—which depicts Gandhi's early struggle against colonial oppression in India—should be revived by the Metropolitan Opera in 2011, a year which has seen popular revolutions in North Africa, mass uprisings in Europe, and the emergence of Occupy Wall Street protests in the United States. (...) we see a glaring contradiction in ‘Satyagraha’ being performed at the Lincoln Center where in recent weeks protestors from Occupy Wall Street have been arrested and forcibly removed for exercising their First Amendment rights to peaceful public assembly." And as the composer states: "Unfortunately, there's always a good time to do this opera. It's just the way it is. But never has it been so close at hand (...) My position is that this is free speech and free assembly. (...) And what these people do by acting in this way, they bring back to us the importance of our basic freedoms."

Influences and connections[link]

Aside from composing in the Western classical tradition, his music has ties to rock, ambient music, electronic music, and world music. Early admirers of his minimalism include musicians Brian Eno and David Bowie. In the 1990s, Glass composed the aforementioned symphonies ''Low'' (1992) and ''Heroes'' (1996), thematically derived from the Bowie-Eno collaboration albums ''Low'' and ''"Heroes"'' (composed in late 1970s Berlin).

Glass has collaborated with recording artists such as Paul Simon, Suzanne Vega, Mick Jagger, Leonard Cohen, David Byrne, Uakti, Natalie Merchant, and Aphex Twin (yielding an orchestration of ''Icct Hedral'' in 1995 on the Donkey Rhubarb EP). Glass's compositional influence extends to musicians such as Mike Oldfield (who included parts from Glass's ''North Star'' in ''Platinum''), and bands such as Tangerine Dream and Talking Heads. Glass and his sound designer Kurt Munkacsi produced the American post-punk/new wave band Polyrock (1978 to the mid-1980s), as well as the recording of John Moran's ''The Manson Family (An Opera)'' in 1991, which featured punk legend Iggy Pop, and a second (unreleased) recording of Moran's work featuring poet Allen Ginsberg.

In 1970, Glass and Klaus Kertess (owner of the Bykert Gallery) formed a record label named ''Chatham Square Productions'' (named after the location of the studio of a Philip Glass Ensemble member Dick Landry). In 1993 Glass formed another record label, Point Music; in 1997, Point Music released ''Music for Airports'', a live, instrumental version of Eno's composition of the same name, by Bang on a Can All-Stars. In 2002, Glass and his producer Kurt Munkacsi and artist Don Christensen founded the Orange Mountain Music company, dedicated to "establishing the recording legacy of Philip Glass" and, to date, have released sixty albums of Glass's music.

Music for film[link]

Glass has composed many film scores, starting with the orchestral score for ''Koyaanisqatsi'' (1982), and continuing with two biopics, ''Mishima: A Life in Four Chapters'' (1985, resulting in the String Quartet No.3) and ''Kundun'' (1997) about the Dalai Lama, for which he received his first Academy Award nomination.

The year after scoring ''Hamburger Hill'' (1987), Glass began a collaboration with the filmmaker Errol Morris with his music for Morris's celebrated documentary ''The Thin Blue Line''. He continued composing for the Qatsi trilogy with the scores for ''Powaqqatsi'' (1988) and ''Naqoyqatsi'' (2002). In 1995 he composed the theme for Reggio's short independent film ''Evidence''. He even made a cameo appearance in Peter Weir's ''The Truman Show'' (1998), which uses music from ''Powaqqatsi'', ''Anima Mundi'' and ''Mishima'', as well as three original tracks by Glass (who is actually briefly visible performing at the piano in the film itself). In the 1990s, he also composed scores for ''Bent' (1997) and the thriller ''Candyman'' (1992) and its sequel, ''Candyman: Farewell to the Flesh'' (1995), plus a film adaptation of Joseph Conrad's ''The Secret Agent'' (1996).

In 1999, he finished a new soundtrack for the 1931 film ''Dracula''. ''The Hours'' (2002) earned him a second Academy Award nomination, and was followed by another Morris documentary, ''The Fog of War'' (2003). In the mid-2000s Glass provided the scores to films such as ''Secret Window'' (2004), ''Neverwas'' (2005), ''The Illusionist'' and ''Notes on a Scandal'', garnering his third Academy Award nomination for the latter. Glass's most recent film scores include ''No Reservations'' (Glass makes a brief cameo in the film sitting at an outdoor cafe), ''Cassandra's Dream'' (2007), ''Les Regrets'' (2009), ''Mr Nice'' (2010) and the Brazilian film ''Nosso Lar'' (2010). In 2009 Glass composed original theme music for Transcendent Man about the life and ideas of Ray Kurzweil by filmmaker Barry Ptolemy.

In the 2000s Glass's work from the 1980s again became known to wider public through various media. In 2005 his Concerto for Violin and Orchestra (1987) was featured in the surreal French thriller, ''La Moustache'', providing a tone intentionally incongruous to the banality of the movie's plot. ''Metamorphosis for Piano'' (1988) was featured in the reimagined ''Battlestar Galactica'' in the episode "Valley of Darkness", and in 2008, Rockstar Games released ''Grand Theft Auto IV'' featuring Glass's "Pruit Igoe" (from ''Koyaanisqatsi''). "Pruit Igoe" and "Prophecies" (also from ''Koyaanisqatsi'') were used both in a trailer for ''Watchmen'' and in the film itself. ''Watchmen'' also included two other Glass pieces in the score: "Something She Has To Do" (from ''The Hours'') and "Protest (Act II Scene 3)" (from ''Satyagraha'').

In 2011, it was confirmed that Glass is scoring for South Korean director Park Chan-wook's Hollywood debut film ''Stoker''.

Personal life[link]

Glass describes himself as "a Jewish-Taoist-Hindu-Toltec-Buddhist", and he is a supporter of Tibetan independence. In 1987 he co-founded the Tibet House with Columbia University professor Robert Thurman and the actor Richard Gere. Glass is a vegetarian.

Glass has four children and one granddaughter. Juliet (b. 1968) and Zachary (b. 1971) are his children from his first marriage, to theater director JoAnne Akalaitis (married 1965, divorced 1980). Granddaughter Zuri (b.1989) is Zachary's daughter. His second marriage to Luba Burtyk was dissolved. His third wife, the artist Candy Jernigan, died of liver cancer in 1991, aged 39. Marlowe and Cameron are Glass's sons with his fourth wife, Holly Critchlow (from whom Glass is divorced). Glass lives in New York and in Cape Breton, Nova Scotia. He has been romantically involved with cellist Wendy Sutter since 2005.

Glass is the first cousin once removed of Ira Glass, host of the nationally syndicated radio show ''This American Life''. Ira interviewed Glass onstage at Chicago's Field Museum; this interview was broadcast on NPR's Fresh Air. Ira interviewed Glass a second time at a fundraiser for St. Ann's Warehouse; this interview was given away to public radio listeners as a pledge drive thank you gift in 2010. Ira and Glass recorded a version of the composition Glass wrote to accompany his friend Allen Ginsberg's poem "Wichita Vortex Sutra." Glass's cousin is Ira Glass's father.

According to an interview, Franz Schubert is Glass's favorite composer.

Films about Philip Glass[link]

1976–''Music With Roots in the Aether: Opera for Television''. Tape 2: Philip Glass. Produced and directed by Robert Ashley.

1983–''Philip Glass''. From ''Four American Composers''. Directed by Peter Greenaway.

1985–''A Composer's Notes: Philip Glass and the Making of an Opera''. Directed by Michael Blackwood.

1986–''Einstein on the Beach: The Changing Image of Opera''. Directed by Mark Obenhaus.

2005–''Looking Glass''. Directed by Éric Darmon.

2007–''glass: a portrait of Philip in twelve parts''. Directed by Scott Hicks.

Awards and nominations[link]

Golden Globe Awards[link]

Best Original Score

Nominated: ''Kundun'' (1998)

Won: ''The Truman Show'' (1999)

Nominated: ''The Hours'' (2002)

BAFTA Awards[link]

Anthony Asquith Award for Film Music

Won: ''The Hours'' (2002)

Academy Awards[link]

Best Original Score

Nominated: ''Kundun'' (1997)

Nominated: ''The Hours'' (2002)

Nominated: ''Notes on a Scandal'' (2006)

Other[link]

2010 NEA Opera Honors Award

18th International Palm Springs Film Festival Award

References[link]

Further reading[link]

William Duckworth (1995, 1999). ''Talking Music: Conversations With John Cage, Philip Glass, Laurie Anderson, and Five Generations of American Experimental Composers''. New York, New York: Da Capo Press.

Philip Glass, Robert T. Jones (ed.) (1987, 1995). ''Music by Philip Glass''. New York, New York: DaCapo Press.

Richard Kostelanetz (ed.) (1997). ''Writings on Glass. Essays, Interviews, Criticism''. Berkeley, Los Angeles, London: University of California Press.

Robert Maycock (2002). ''Glass: A Biography of Philip Glass''. Sanctuary Publishing.

Potter, Keith (2000). ''Four Musical Minimalists: La Monte Young, Terry Riley, Steve Reich, Philip Glass''. Music in the Twentieth Century series. Cambridge, UK; New York, New York: Cambridge University Press.

John Richardson (1999). ''Singing Archaeology: Philip Glass's "Akhnaten"''. Wesleyan University Press.

K. Robert Schwarz (1996). ''Minimalists''. 20th-Century Composers Series. London: Phaidon Press.

Bartman, William and Kesten, Joanne (editors). ''The Portraits Speak: Chuck Close in Conversation with 27 of his subjects'', New York: A.R.T. Press, 1997

Knowlson, James (2004). ''Damned to Fame: The Life of Samuel Beckett'', New York: Grove Press.

External links[link]

Philip Glass Q&A;

NewMusicBox cover: Philip Glass (includes video)

Art of the States: Philip Glass ''Piece in the Shape of a Square'' (1967)

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