In geometry, a torus (pl. tori) is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle. In most contexts it is assumed that the axis does not touch the circle - in this case the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit.

Other types of torus include the horn torus, which is generated when the axis is tangent to the circle, and the spindle torus, which is generated when the axis is a chord of the circle. A degenerate case is when the axis is a diameter of the circle, which simply generates the surface of a sphere. The ring torus bounds a solid known as a toroid. The adjective toroidal can be applied to tori, toroids or, more generally, any ring shape as in toroidal inductors and transformers. Real world examples of (approximately) toroidal objects include doughnuts, inner tubes, many lifebuoys, O-rings and vortex rings.

In topology, a ring torus is homeomorphic to the Cartesian product of two circles: ''S''¹ × ''S''¹, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into Euclidean space, but another way to do this is the Cartesian product of the embedding of ''S''¹ in the plane. This produces a geometric object called the Clifford torus, surface in 4-space.

The word ''torus'' comes from the Latin word meaning cushion.

Geometry

A torus can be defined parametrically by: :$x(u,\; v)\; =\; (R\; +\; r\; \backslash cos\{v\})\; \backslash cos\{u\}\; \backslash ,$ :$y(u,\; v)\; =\; (R\; +\; r\; \backslash cos\{v\})\; \backslash sin\{u\}\; \backslash ,$ :$z(u,\; v)\; =\; r\; \backslash sin\{v\}\; \backslash ,$

where :''u'',''v'' are in the interval [0, 2π), :''R'' (or ''A'') is the distance from the center of the tube to the center of the torus, :''r'' (or ''a'') is the radius of the tube.

''R'' and ''r'' are also known as the "major radius" and "minor radius", respectively. The ratio of the two is known as the "aspect ratio". A doughnut has an aspect ratio of 2 to 3.

An implicit equation in Cartesian coordinates for a torus radially symmetric about the ''z-''axis is :$\backslash left(R\; -\; \backslash sqrt\{x^2\; +\; y^2\}\backslash right)^2\; +\; z^2\; =\; r^2,\; \backslash ,\backslash !$ or the solution of $f(x,y,z)\; =\; 0$, where :$f(x,y,z)\; =\; \backslash left(R\; -\; \backslash sqrt\{x^2\; +\; y^2\}\backslash right)^2\; +\; z^2\; -\; r^2.\backslash ,\backslash !$

Algebraically eliminating the square root gives a quartic equation, :$(x^2+y^2+z^2\; +\; R^2\; -\; r^2)^2\; =\; 4R^2(x^2+y^2).\; \backslash ,\backslash !$

The three different classes of standard tori correspond to the three possible relative sizes of ''r'' and ''R''. When ''R'' > ''r'', the surface will be the familiar ring torus. The case ''R'' = ''r'' corresponds to the horn torus, which in effect is a torus with no "hole". The case ''R'' < ''r'' describes the self-intersecting spindle torus. When ''R'' = 0, the torus degenerates to the sphere.

The surface area and interior volume of this torus are easily computed using Pappus's centroid theorem giving

:$A\; =\; 4\; \backslash pi^2\; R\; r\; =\; \backslash left(\; 2\backslash pi\; r\; \backslash right)\; \backslash left(2\; \backslash pi\; R\; \backslash right)\; \backslash ,$ :$V\; =\; 2\; \backslash pi^2\; R\; r^2\; =\; \backslash left\; (\; \backslash pi\; r\; ^2\; \backslash right\; )\; \backslash left(\; 2\; \backslash pi\; R\; \backslash right).\; \backslash ,$

These formulas are the same as for a cylinder of length 2π''R'' and radius ''r'', created by cutting the tube and unrolling it by straightening out the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.

As a torus is the product of two circles, a modified version of the spherical coordinate system is sometimes used. In traditional spherical coordinates there are three measures, ''R'', the distance from the center of the coordinate system, and $\backslash theta$ and $\backslash phi$, angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; $\backslash phi$ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of $\backslash theta$ is moved to the center of ''r'', and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles". In modern use these terms are more commonly used to discuss magnetic confinement fusion devices.

Topology

Topologically, a torus is a closed surface defined as the product of two circles: ''S''¹ × ''S''¹. This can be viewed as lying in C² and is a subset of the 3-sphere ''S''³ of radius √2. This topological torus is also often called the Clifford torus. In fact, ''S''³ is filled out by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of ''S''³ as a fiber bundle over ''S''² (the Hopf bundle).

The surface described above, given the relative topology from R³, is homeomorphic to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by stereographically projecting the topological torus into R³ from the north pole of ''S''³.

The torus can also be described as a quotient of the Cartesian plane under the identifications :(''x'',''y'') ~ (''x''+1,''y'') ~ (''x'',''y''+1). Or, equivalently, as the quotient of the unit square by pasting the opposite edges together, described as a fundamental polygon $ABA^\{-1\}B^\{-1\}$.

The fundamental group of the torus is just the direct product of the fundamental group of the circle with itself: :$\backslash pi\_1(\backslash mathbb\{T\}^2)\; =\; \backslash pi\_1(S^1)\; \backslash times\; \backslash pi\_1(S^1)\; \backslash cong\; \backslash mathbb\{Z\}\; \backslash times\; \backslash mathbb\{Z\}.$ Intuitively speaking, this means that a closed path that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole. So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. This might be imagined as two shoelaces passing through each other, then unwinding, then rewinding.

If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged.

The first homology group of the torus is isomorphic to the fundamental group (this follows from Hurewicz theorem since the fundamental group is abelian).

Two-sheeted cover

The 2-torus double-covers the 2-sphere, with four ramification points. Every conformal structure on the 2-torus can be represented as a two-sheeted cover of the 2-sphere. The points on the torus corresponding to the ramification points are the Weierstrass points. In fact, the conformal type of the torus is determined by the cross-ratio of the four points.

''n''-dimensional torus

The torus has a generalization to higher dimensions, the ''n''-dimensional torus, often called the ''n''-torus for short. (This is one of two different meanings of the term "''n''-torus".) Recalling that the torus is the product space of two circles, the ''n''-dimensional torus is the product of ''n'' circles. That is: :$\backslash mathbb\{T\}^n\; =\; \backslash underbrace\{S^1\; \backslash times\; S^1\; \backslash times\; \backslash cdots\; \backslash times\; S^1\}\_n.$ The torus discussed above is the 2-dimensional torus. The 1-dimensional torus is just the circle. Just as for the 2-torus, the ''n''-torus can be described as a quotient of R^''n'' under integral shifts in any coordinate. That is, the ''n''-torus is R^''n'' modulo the action of the integer lattice Z^''n'' (with the action being taken as vector addition). Equivalently, the ''n''-torus is obtained from the ''n''-dimensional hypercube by gluing the opposite faces together.

An ''n''-torus in this sense is an example of an ''n-''dimensional compact manifold. It is also an example of a compact abelian Lie group. This follows from the fact that the unit circle is a compact abelian Lie group (when identified with the unit complex numbers with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.

Toroidal groups play an important part in the theory of compact Lie groups. This is due in part to the fact that in any compact Lie group ''G'' one can always find a maximal torus; that is, a closed subgroup which is a torus of the largest possible dimension. Such maximal tori ''T'' have a controlling role to play in theory of connected ''G''.

Automorphisms of ''T'' are easily constructed from automorphisms of the lattice Z^''n'', which are classified by integral matrices ''M'' of size ''n''×''n'' which are invertible with integral inverse; these are just the integral ''M'' of determinant +1 or −1. Making ''M'' act on R^''n'' in the usual way, one has the typical toral automorphism on the quotient.

The fundamental group of an ''n''-torus is a free abelian group of rank ''n''. The ''k''-th homology group of an ''n''-torus is a free abelian group of rank ''n'' choose ''k''. It follows that the Euler characteristic of the ''n''-torus is 0 for all ''n''. The cohomology ring ''H''^•(T^''n'',Z) can be identified with the exterior algebra over the Z-module Z^''n'' whose generators are the duals of the ''n'' nontrivial cycles.

Configuration space

As the ''n''-torus is the ''n''-fold product of the circle, the ''n''-torus is the configuration space of ''n'' ordered, not necessarily distinct points on the circle. Symbolically, $T^n\; =\; (S^1)^n.$ The configuration space of ''unordered,'' not necessarily distinct points is accordingly the orbifold $T^n/S\_n,$ which is the quotient of the torus by the symmetric group on ''n'' letters (by permuting the coordinates).

For $n=2,$ the quotient is the Möbius strip, the edge corresponding to the orbifold points where the two coordinates coincide. For $n=3$ this quotient may be described as a solid torus with cross-section an equilateral triangle, with a twist; equivalently, as a triangular prism whose top and bottom faces are connected with a ⅓ twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical.

These orbifolds have found significant applications to music theory in the work of Dmitri Tymoczko and collaborators (Felipe Posada and Michael Kolinas, et al.), being used to model musical triads.

Flat torus

The flat torus is a specific embedding of the familiar 2-torus into Euclidean 4-space or higher dimensions. Its surface has zero Gaussian curvature everywhere. Its surface is "flat" in the same sense that the surface of a cylinder is "flat". In 3 dimensions one can bend a flat sheet of paper into a cylinder without stretching the paper, but you cannot then bend this cylinder into a torus without stretching the paper. In 4 dimensions one can (mathematically).

A simple ''4''-d Euclidean embedding is as follows: <''x'',''y'',''z'',''w''> = <''R'' cos ''u'', ''R'' sin ''u'', ''P'' cos ''v'', ''P'' sin ''v''> where ''R'' and ''P'' are constants determining the aspect ratio. It is diffeomorphic to a regular torus but not isometric. It can not be isometrically embedded into Euclidean 3-space. Mapping it into ''3''-space requires you to "bend" it, in which case it looks like a regular torus, for example, the following map <''x'',''y'',''z''> = <(''R'' + ''P'' sin ''v'')cos ''u'', (''R'' + ''P'' sin ''v'')sin ''u'', ''P'' cos ''v''>.

A flat torus partitions the 3-sphere into two congruent solid tori subsets with the aforsaid flat torus surface as their common boundary. {| class=wikitable width=256 |256px
A stereographic projection of a Clifford torus performing a simple rotation through the ''xz''-plane. |}

''n''-fold torus

In the theory of surfaces the term ''n-''torus has a different meaning. Instead of the product of ''n'' circles, they use the phrase to mean the connected sum of ''n'' 2-dimensional tori. To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the disks' boundary circles. To form the connected sum of more than two surfaces, sum two of them at a time until they are all connected together. In this sense, an ''n''-torus resembles the surface of ''n'' doughnuts stuck together side by side, or a 2-dimensional sphere with ''n'' handles attached.

An ordinary torus is a 1-torus, a 2-torus is called a double torus, a 3-torus a triple torus, and so on. The ''n''-torus is said to be an "orientable surface" of "genus" ''n'', the genus being the number of handles. The 0-torus is the 2-dimensional sphere.

The classification theorem for surfaces states that every compact connected surface is either a sphere, an ''n''-torus with ''n'' > 0, or the connected sum of ''n'' projective planes (that is, projective planes over the real numbers) with ''n'' > 0.

{| class=wikitable |- align=center | | |}

Toroidal polyhedra

Polyhedra with the topological type of a torus are called toroidal polyhedra, and satisfy a modified version of the polyhedron formula, $E-F-V\; =\; 0.$

The term "toroidal polydron" is also used for higher genus polyhedra and for immersions of toroidal polyhedra.

Automorphisms

The homeomorphism group (or the subgroup of diffeomorphisms) of the torus is studied in geometric topology. Its mapping class group (the group of connected components) is isomorphic to the group GL(''n,'' Z) of invertible integer matrices, and can be realized as linear maps on the universal covering space $\backslash mathbf\{R\}^n$ that preserve the standard lattice $\backslash mathbf\{Z\}^n$ (this corresponds to integer coefficients) and thus descend to the quotient.

At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the fundamental group, as these are all naturally isomorphic; note also that the first cohomology group generates the cohomology algebra): :$\backslash mathrm\{MCG\}(T^n)\; =\; \backslash mathrm\{Aut\}(\backslash pi\_1(X))\; =\; \backslash mathrm\{Aut\}(\backslash mathbf\{Z\}^n)\; =\; \backslash mathrm\{GL\}(n,\backslash mathbf\{Z\}).$ Since the torus is an Eilenberg-MacLane space ''K''(''G,'' 1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); that this agrees with the mapping class group reflects that all homotopy equivalences can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism – and that homotopic homeomorphisms are in fact isotopic (connected through homeomorphisms, not just through homotopy equivalences). More tersely, the map $\backslash mathrm\{Homeo\}(T^n)\; \backslash to\; \backslash mathrm\{SHE\}(T^n)$ is 1-connected (isomorphic on path-components, onto fundamental group). This is a "homeomorphism reduces to homotopy reduces to algebra" result.

Thus the short exact sequence of the mapping class group splits (an identification of the torus as the quotient of $\backslash mathbf\{R\}^n$ gives a splitting, via the linear maps, as above): :$1\; \backslash rightarrow\; \{\backslash rm\; Homeo\}\_0(T^n)\; \backslash rightarrow\; \{\backslash rm\; Homeo\}(T^n)\; \backslash rightarrow\; \{\backslash rm\; MCG\}(T^n)\; \backslash rightarrow\; 1,$ so the homeomorphism group of the torus is a semidirect product, $\backslash mathrm\{Homeo\}(T^n)\; \backslash cong\; \backslash mathrm\{Homeo\}\_0(T^n)\; \backslash rtimes\; \backslash mathrm\{GL\}(n,\backslash mathbf\{Z\}).$

The mapping class group of higher genus surfaces is much more complicated, and an area of active research.

Coloring a torus

If a torus is divided into regions, then it is always possible to color the regions with no more than seven colors so that neighboring regions have different colors. (Contrast with the four color theorem for the plane.)

Cutting a torus

A standard torus (specifically, a ring torus) can be cut with ''n'' planes into at most (''n''³ + 3''n''² + 8''n'')}} parts, integer sequence . The initial terms of this sequence are 1, 2, 6, 13, for ''n'' starting from 0.

See also

Algebraic torus

Annulus (mathematics)

Clifford torus

Complex torus

Doughnut

Dupin cyclide

Elliptic curve

Irrational cable on a torus

Joint European Torus

Loewner's torus inequality

Maximal torus

Period lattice

Sphere

Spiric section

Standard torus

Surface

Toric section

Toric variety

Toroid (geometry)

Torus knot

Umbilic torus

Villarceau circles

Notes

References

NOCIONES DE GEOMETRIA ANALITICA Y ALGEBRA LINEAL, ISBN 9789701065969, Author: KOZAK ANA MARIA, POMPEYA PASTORELLI SONIA, VERDANEGA PEDRO EMILIO, Editorial: MCGRAW-HILL, ''Edition 2007'', 744 pages, language; spanish

Allen Hatcher. Algebraic topology. Cambridge University Press, 2002. ISBN 0-521-79540-0.

V.V. Nikulin, I.R.Shafarevich. Geometries and Groups. Springer, 1987. ISBN 3540152814, ISBN 9783540152811.

http://www.mathcurve.com/surfaces/tore/tore.shtml

"Tore (notion géométrique)" at Encyclopédie des Formes Mathématiques Remarquables

External links

Creation of a torus at cut-the-knot

"4D torus" Fly-through cross-sections of a four dimensional torus.

"Relational Perspective Map" Visualizing high dimensional data with flat torus.

"Torus Games" Free downloadable games for Windows and Mac OS X that highlight the topology of a torus.

Polydos

Category:Surfaces

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bgcolour	#6495ED
name	Richard Serra
birth date	November 02, 1939
birth place	San Francisco, California
nationality	American
field	minimalist sculptor
training	Yale University
movement	Process Art
influenced by	Robert Smithson
awards	}}

Richard Serra (born November 2, 1939) is an American minimalist sculptor and video artist known for working with large-scale assemblies of sheet metal. Serra was involved in the Process Art Movement.

Early life and education

His father was Spanish native of Mallorca and mother was Russian in Odessa (committed suicide in 1979). Serra was born in San Francisco and he went on to study English literature at the University of California, Berkeley and later at the University of California, Santa Barbara between 1957 and 1961. While at Santa Barbara, he studied art with Howard Warshaw and Rico Lebrun. On the West Coast, he helped support himself by working in steel mills, which was to have a strong influence on his later work. Serra discussed his early life and influences in an interview in 1993. He described the San Francisco shipyard where his father worked as a pipe-fitter as another important influence to his work, saying of his early memory: “All the raw material that I needed is contained in the reserve of this memory which has become a reoccurring dream.”

After studying painting with Josef Albers at the Yale University School of Art and Architecture between 1961 and 1964, Serra continued his training abroad, spending a year each in Florence and Paris. In 1964, he was awarded a Fulbright Scholarship for Rome. Since then, he has lived in New York, where he first used rubber in 1966 and began applying his characteristic work material lead in 1968. In New York, his circle of friends included Carl Andre, Walter De Maria, Eva Hesse, Sol LeWitt, and Robert Smithson.

He is the brother of famed San Francisco trial attorney Tony Serra. Serra lives in Tribeca, New York and on Cape Breton Island in Nova Scotia.

Work

Sculpture

In 1966, Serra made his first sculptures out of nontraditional materials such as fiberglass and rubber. Serra's earliest work was abstract and process-based made from molten lead hurled in large splashes against the wall of a studio or exhibition space. Still, he is better known for his minimalist constructions from large rolls and sheets of metal (COR-TEN-Steel). Many of these pieces are self-supporting and emphasize the weight and nature of the materials. Rolls of lead are designed to sag over time. His exterior steel sculptures go through an initial oxidation process, but after 8–10 years, the patina of the steel settles to one color that will remain relatively stable over the piece's life. Serra often constructs site-specific installations, frequently on a scale that dwarfs the observer. Serra's site-specific works often challenge viewers’ perception of their body in relation to interior spaces and landscapes, and his work often encourages movement in and around his sculptures.

In 1981, Serra installed ''Tilted Arc'', a gently curved, 3.5 meter high arc of rusting mild steel in the Federal Plaza in New York City. There was controversy over the installation from day one, largely from workers in the buildings surrounding the plaza who complained that the steel wall obstructed passage through the plaza. A public hearing in 1985 voted that the work should be moved, but Serra argued the sculpture was site specific and could not be placed anywhere else. Serra famously issued an often-quoted statement regarding the nature of site-specific art when he said, "To remove the work is to destroy it." Eventually on 15 March 1989, the sculpture was dismantled by federal workers and taken for scrap. In May 1989 the piece was cut into three parts and consigned to a New York warehouse where it has languished ever since. William Gaddis satirized these events in his 1994 novel ''A Frolic of His Own''.

In 2002, a similar installation titled ''Vectors'' was to be built at the California Institute of Technology from the bequest of Eli Broad. The piece, to be four steel plates of similar material as ''Tilted Arc'' zig-zagging across one of the few green spaces at the university, met significant opposition by the student body and professors as being a "'derivative” rehash of earlier works, or an 'arrogant' piece that [belied] Institute values." The piece was never installed.

Another famous work of Serra's is the mammoth sculpture ''Snake'', a trio of sinuous steel sheets creating a curving path, permanently located in the largest gallery of the Guggenheim Museum Bilbao. In 2005, the museum mounted an exhibition of more of Serra's work, incorporating ''Snake'' into a collection entitled ''The Matter of Time''. The whole work consists of eight sculptures measuring between 12 and 14 feet in height and weighing from 44 to 276 tons.

He has not always fared so well in Spain, however; also in 2005, the Centro de Arte Reina Sofía in Madrid announced that a 38-tonne sculpture of his had been "mislaid". In a recent development, a duplicate copy is going to be made and displayed in Madrid.

In spring 2005, Serra returned to San Francisco to install his first public work in that city (previous negotiations for a commission fell through) – two 50-foot steel blades in the main open space of the new University of California, San Francisco (UCSF) campus. Weighing 160 tons, placing the work in its Mission Bay location posed serious challenges, since it is, like many parts of San Francisco, built on landfill. In 2000 he installed ''Charlie Brown,'' a 60-foot-tall sculpture in the new Gap Inc. headquarters in San Francisco. To encourage oxidation, or rust, sprinklers were initially directed toward the four German-made slabs of steel that make up the work (see External links).

Work similar to that of his in the Netherlands (pictured) can be found in Storm King Art Center in Upstate New York.

From May 7 to June 15, 2008 Richard Serra showed his installation ''Promenade'' at the Grand Palais, Paris. "A radical, poetic landscape of steel, minimalist yet full of movement." Serra was the second artist, after Anselm Kiefer, who was invited to fill the 13,500 m² nave of the Grand Palais with a group of new works created specially for the event.

Birmingham City Council is currently considering a proposal for an outdoor installation by Richard Serra in front of their new Library of Birmingham to replace the destroyed ''Forward'' sculpture by Raymond Mason in Centenary Square.

In December 2008, after almost 20 years in storage, his steel sculpture ''Slat'' was re-anchored in La Défense, the Parisian business district. The sculpture spent five years in a nearby Paris suburb, Puteaux, but in 1989 vandalism and graffiti prompted that town’s mayor to remove it. “Slat” has five 25-ton steel plates that lean on one another to form a tall, angular tepee. because of its weight, officials chose to ground it in a traffic island behind the Grande Arche.

Memorials

In 1987, Serra created ''Berlin Junction'' as a memorial to those who lost their lives to the Nazis' genocide program. First shown at the Martin-Gropius-Bau, the sculpture was installed permanently at the Berliner Philharmonie in 1988. After initially joining with architect Peter Eisenmann to submit a design for Berlin's Memorial to the Murdered Jews of Europe, Serra abruptly pulled out of the project for "personal and professional reasons" in 1998.

Performance and Video Art

Serra was one of the four performers in the premiere of the Steve Reich piece ''Pendulum Music'' on May 27, 1969 at the Whitney Museum of American Art. The other performers were Michael Snow, James Tenney and Bruce Nauman.

''Hand Catching Lead'' (1968) was Serra's first film and features a single shot of a hand in an attempt to repeatedly catch chunks of material dropped from the top of the frame. In ''Boomerang'' (1974), Serra taped Nancy Holt as she talks and hears her words played back to her after they have been delayed electronically.

Serra has made a number of films concerning the manufacture and use of his favorite material, steel. ''Steelworks'' is shot inside a German steelworks and includes an interview with a steelworker, while ''Railroad Turnbridge'' is a series of shots taken on the Burlington and Northern bridge over the Willamette River near Portland, Oregon, as it opens to let a ship pass. These films can be viewed in a room off the Arcelor gallery in the Guggenheim museum in Bilbao.

He also produced the classic 1973 short film "Television Delivers People", a critique of the corporate mass media with elevator music as the soundtrack.

Serra appears in Matthew Barney's 2002 film ''Cremaster 3'' as Hiram Abiff ("the architect"), and later as himself in the climactic ''The Order'' section – the only part of a ''Cremaster'' film commercially available on DVD.

Prints and Drawings

Since 1971, Serra has focused not only on sculptural works, but also onlarge-scale drawings on paper using various techniques. His drawing material is the paintstick, a wax-like grease crayon. Serra melts several paintsticks to form large pigment blocks. In the mid-1970s, Serra made his first "Installation Drawings" — monumental works on canvas or linen pinned directly to the wall and thickly covered with black paintstick, such as ''Abstract Slavery'', ''Taraval Beach'', ''Pacific Judson Murphy'', and ''Blank''. The drawings Serra has executed since the 1980s continue the experiments with innovative techniques but are less monumental physically. In the late 1980s he explored how to further articulate the tension of weight and gravity by placing pairs of overlapping sheets of paper saturated with paintstick in horizontal and vertical compositions.

Major presentations of Serra’s graphic oeuvre include exhibitions at the Bonnefantenmuseum, Maastricht, in 1990; at Serpentine Gallery, London, in 1992; and at Kunsthaus Bregenz, Bregenz, in 2008. At the 2006 Whitney Biennial, Serra showed a simple litho crayon drawing of an Abu Ghraib prisoner with the caption "STOP BUSH." This image was later used by the Whitney Museum to make posters for the Biennial. The posters featured an altered version of the text that read "STOP B S ." Serra also created a variation on Goya's Saturn Devouring His Son featuring George W. Bush's head in place of Saturn's. This was featured prominently in an ad for the website pleasevote.com (now defunct) on the back cover of the July 5, 2004 issue of The Nation.

Colby College recently acquired 150 works on paper by Serra, making it the second largest collection of Serra's work outside of the Museum of Modern Art in New York.

The Metropolitan Museum of Art is currently hosting a retrospective exhibit focusing on Richard Serra's Drawings. April 13, 2011 through August 28, 2011 the exhibit "Richard Serra Drawing: A Retrospective" will be on view and presents a comprehensive overview of Richard Serra's drawings and traces the development of his drawing as an art form independent from yet linked to his sculptural practice.

In popular culture

The drone band Sunn O))) used "Out-of-Round X" (1999) as the cover of their seventh studio album released in 2009, called ''Monoliths & Dimensions''.

Exhibitions

Serra had his first solo exhibitions at the Galleria La Salita, Rome, 1966, and in the United States at the Leo Castelli Warehouse, New York. The Pasadena Art Museum organized a solo exhibition of Serra’s work in 1970. Serra has since participated in Documentas 5 (1972), 6 (1977), 7 (1982), and 8 (1987), in Kassel, the Venice Biennales of 1984 and 2001, and the Whitney Museum of American Art's Annual and Biennial exhibitions of 1968, 1970, 1973, 1977, 1979, 1981, and 1995. Serra was honored with further solo exhibitions at the Kunsthalle Tübingen, Germany, in 1978; the Musée National d'Art Moderne, Paris, in 1984; the Museum Haus Lange, Krefeld, Germany, in 1985; and the Museum of Modern Art, New York, in 1986. From 1997 to 1998 his ''Torqued Ellipses'' (1997) were exhibited at and acquired by the Dia Center for the Arts, New York. In 2005 eight major works by Serra were installed permanently at Guggenheim Museum Bilbao.

In the summer of 2007 the Museum of Modern Art presented a retrospective of Serra's work in New York. Intersection II (1992–1993) and Torqued Ellipse IV (1998) were included in this show along with three new works. The retrospective consisted of 27 of Serra's works, including three large new sculptures made specifically for the second floor of the museum, two works in the garden, and earlier pieces from the 1960s through the 1980s.

Serra continues to produce large-scale steel structures for sites throughout the world, and has become particularly renowned for his monumental arcs, spirals, and ellipses, which engage the viewer in an altered experience of space. He was invited to create a number of artworks in France: ''Philibert et Marguerite'' in the cloister of the Musée de Brou at Bourg-en-Bresse (1985), ''Octagon for Saint Eloi'' (1991) in the village of Chagny in Burgundy, and ''Threats of Hell'' at the CAPC (Musée d'Art Contemporain) in Bordeaux.

Recognition

His work was featured on BBC One in "Imagine...Richard Serra: Man of Steel" on Tuesday 25 November 2008 which described him as "Sculptor and giant of modern art Richard Serra discusses his extraordinary life and work. A creator of enormous, immediately identifiable steel sculptures that both terrify and mesmerise, Serra believes that each viewer creates the sculpture for themselves by being within it." Contributors include Chuck Close, Philip Glass and Glenn D Lowry, Director of MoMA. He was interviewed at length by the BBC's Alan Yentob.

Serra was awarded an honorary Doctor of Fine Arts by Williams College in 2008 and by Harvard University in 2010. In 1994, he was honored with the Praemium Imperiale.

Art market

The record auction price for a Serra sculpture was paid at Sotheby’s in New York in 2008, where ''12-4-8,'' a 1983 work consisting of three steel plates, sold for $1.65 million.

See also

Robert Smithson

Site-specific art

Environmental sculpture

Neo-minimalism

References

External links

Richard Serra in Literal, Latin American Voices

A short documentary by KQED-TV's Spark on a Serra's piece for UCSF.

Richard Serra geolocation finder for sculpture sites

Moving Serra from MoMA to LACMA

Biography, interviews, essays, artwork images and video clips from PBS series ''Art:21 -- Art in the Twenty-First Century'' - Season 1 (2001).

MoMA: Richard Serra Sculpture: Forty Years

PBS: Richard Serra

Richard Serra interviewed by Klaus Ottmann

Richard Serra at Gagosian Gallery

Robert Hughes: Richard Serra (22/06/05)

Television Delivers People (1973)

1992 Richard Serra monograph by Adrian Searle

Richard Serra in the Video Data Bank

BBC TV Imagine documentary on Richard Serra.

Charlie Brown, sculpture at The Gap HQ.

Richard Serra sculpture in Berlin, Germany

Richard Serra interviewed about Tilted Arc controversy, excerpt from "Public Sculpture" (ART/New York #14), video, 1982/83.

Category:1939 births Category:Living people Category:American artists Category:American sculptors Category:American printmakers Category:Video artists Category:Members of the American Academy of Arts and Letters Category:Recipients of the Pour le Mérite (civil class) Category:Minimalist artists Category:People from San Francisco, California Category:Guggenheim Fellows Category:Orden de las Artes y las Letras de España recipients Category:American people of Spanish descent

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