Geometry and Cult of Personality (song): Difference between pages

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{{Refimprove|date=September 2007}}
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{{Infobox Single <!-- See Wikipedia:WikiProject_Songs -->
{{otheruses}}
| Name = Cult of Personality
| Cover = Living Colour Cult of Personality.jpg
| Cover size =
| Border =
| Caption =
| Artist = [[Living Colour]]
| Album = [[Vivid (album)|Vivid]]
| A-side =
| B-side =
| Released = [[1988]]
| Format = [[CD Single|CD]], [[Gramophone record|Vinyl]], [[Compact Cassette|Cassette]]
| Recorded = 1987–1988, 2007
| Genre = [[Hard rock]]<br>[[Heavy metal music|Heavy metal]]<br>[[Funk metal]]
| Length = 4:54
| Label = [[Epic Records|Epic]]/[[Sony Music Entertainment|CBS]]
| Writer = [[Living Colour]]
| Producer = [[Ed Stasium]]
| Audio sample? =
| Certification =
| Last single =
| This single = "'''Cult of Personality'''"<br />(1988)
| Next single = "[[Glamour Boys]]"<br />(1988)
| Misc ={{Extra track listing
| Album = [[Vivid (album)|Vivid]]
| Type = studio
| prev_track =
| prev_no =
| this_track = "'''Cult of Personality'''"
| track_no = Track 1
| next_track = "I Want to Know"
| next_no = Track 2
}}
}}


"'''Cult of Personality'''" is a song by the [[heavy metal music|heavy metal]] band [[Living Colour]] and the [[lead single]] from their [[debut album]], ''[[Vivid (album)|Vivid]]''. Its [[music video]] earned two [[MTV Video Music Award]]s for [[MTV Video Music Award for Best Group Video|Best Group Video]] and [[MTV Video Music Award for Best New Artist|Best New Artist]]. Released in 1989, "Cult of Personality" reached #13 on the [[Billboard Hot 100]] and #9 on the Billboard Album Rock Tracks chart.
[[Image:Calabi-Yau.png|thumb|right|150px|[[Calabi-Yau manifold]]]]
'''Geometry''' ([[Greek language|Greek]] ''γεωμετρία''; geo = earth, metria = measure) is a part of [[mathematics]] concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning [[length]]s, [[area]]s, and [[volume]]s, in the third century B.C., geometry was put into an [[axiomatic system|axiomatic form]] by [[Euclid]], whose treatment - [[Euclidean geometry]] - set a standard for many centuries to follow. The field of [[astronomy]], especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia.


The song begins with an edited quote from the beginning of a [[speech]] by [[Malcolm X]]. As it appears in the song, the quote is:
Introduction of [[coordinates]] by [[René Descartes]] and the concurrent development of [[algebra]] marked a new stage for geometry, since geometric figures, such as [[plane curve]]s, could now be represented [[analytic geometry|analytically]], i.e., with functions and equations. This played a key role in the emergence of [[calculus]] in the seventeenth century. Furthermore, the theory of [[perspective (graphical)|perspective]] showed that there is more to geometry than just the metric properties of figures. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with [[Euler]] and [[Carl Friedrich Gauss|Gauss]] and led to the creation of [[topology]] and [[differential geometry]].
<blockquote>''". . . And during the few moments that we have left, . . . We want to talk right down to earth in a language that everybody here can easily understand."''</blockquote>
The unabridged beginning of the speech is:
<blockquote>"''...And during the few moments that we have left, we want to have just an off-the-cuff chat between you and me -- us. We want to talk right down to earth in a language that everybody here can easily understand."'' <ref>Malcolm X: "Message to the Grass Roots": http://www.americanrhetoric.com/speeches/malcolmxgrassroots.htm</ref></blockquote>
At 4:35, [[John F. Kennedy]] is quoted, saying ''"Ask not what your country can do for you,"'' and the song ends with [[Franklin D. Roosevelt]] saying ''"The only thing we have to fear is fear itself."''


"Cult of Personality" was performed live during the April 1, 1989 edition of ''[[Saturday Night Live]]'' with host [[Mel Gibson]]. It was also performed on ''[[The Arsenio Hall Show]]'' that same year.
Since the nineteenth century discovery of [[non-Euclidean geometry]], the concept of [[space]] has undergone a radical transformation. Contemporary geometry considers [[manifold]]s, 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 [[Riemannian geometry]] and [[general relativity]]. One of the youngest physical theories, [[string theory]], is also very geometric in flavour.


==Lyrics==
The visual nature of geometry makes it initially more accessible than other parts of mathematics, such as [[algebra]] or [[number theory]]. However, the geometric language is also used in contexts that are far removed from its traditional, Euclidean provenance, for example, in [[fractal geometry]], and especially in [[algebraic geometry]].<ref>It is quite common in algebraic geometry to speak about ''geometry of [[algebraic variety|algebraic varieties]] over [[finite field]]s'', possibly [[singularity theory|singular]]. From a naïve perspective, these objects are just finite sets of points, but by invoking powerful geometric imagery and using well developed geometric techniques, it is possible to find structure and establish properties that make them somewhat analogous to the ordinary [[sphere]]s or [[Cone (geometry)|cone]]s.</ref>
Along with the above quotes interspersed, the lyrics in the song refer to several well-known political and social figures (thus implicitly establishing each as the leader of a [[cult of personality]]).
*[[Malcolm X]]
*[[Benito Mussolini]]
*[[John F. Kennedy]]
*[[Joseph Stalin]]
*[[Franklin D. Roosevelt]]
*[[Mohandas Karamchand Gandhi|Mahatma Gandhi]]


==Appearances in other media==
==History==
*"Cult of Personality" appears on the ''[[Grand Theft Auto: San Andreas]]'' video game soundtrack.
{{main|History of geometry}}
*"Cult of Personality" also appears on ''[[Guitar Hero III: Legends of Rock]]'', the band had to re-record it as the master tracks could not be found. During the rerecording, a new, tougher solo was added.
*"Cult of Personality" was briefly used as the theme song for [[professional wrestler]], [[CM Punk]], after he became the [[Ring of Honor]] Heavyweight Champion.
*"Cult of Personality" was featured on an episode of the game show "[[Don't Forget the Lyrics]]" hosted by [[Wayne Brady]].
*"Cult of Personality" is frequently played by the [[CBS Orchestra]] on ''[[Late Night with David Letterman]]''.
*"Cult of Personality" was performed by [[Phil Ritchie]] on the American [[reality TV]] series ''[[Rockstar: Supernova]]''.
*"Cult of Personality" was also performed by Ty Taylor on Rock Star INXS


==References==
[[Image:Woman teaching geometry.jpg|thumb|230px|''Woman teaching geometry''. Illustration at the beginning of a medieval translation of [[Euclid's Elements]], (c.[[1310]])]]
<!--[[Image:Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570 (560x900).jpg|right|200px|thumb|The [[frontispiece]] of Sir Henry Billingsley's first English version of Euclid's ''[[Element (mathematics)|Elements]]'', [[1570]]]]-->
The earliest recorded beginnings of geometry can be traced to ancient [[Mesopotamia]], [[Ancient Egypt|Egypt]], and the [[Indus Valley Civilization|Indus Valley]] from around [[3000 BCE]]. 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 mathematics|Egyptian]] ''[[Rhind Mathematical Papyrus|Rhind Papyrus]]'' and ''[[Moscow Mathematical Papyrus|Moscow Papyrus]]'', the [[Babylonian mathematics|Babylonian clay tablets]], and the [[Indian mathematics|Indian]] ''[[Shulba Sutras]]'', while the Chinese had the work of [[Mozi]], [[Zhang Heng]], and the ''[[Nine Chapters on the Mathematical Art]]'', edited by [[Liu Hui]].

[[Euclid|Euclid's]] ''[[Euclid's Elements|The Elements of Geometry]]'' (c. [[300 BCE]]) was one of the most important early texts on geometry, in which he presented geometry in an ideal [[axiom]]atic form, which came to be known as [[Euclidean geometry]]. The treatise is not, as is sometimes thought, a compendium of all that [[Hellenistic]] mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;<ref>{{cite book|last=Boyer|authorlink=Carl Benjamin Boyer|title=|year=1991|chapter=Euclid of Alexandria|pages=104|quote=The ''Elements'' was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all ''elementary'' mathematics-}}</ref> Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but the others fell into disuse and were lost.{{Fact|date=July 2007}}

In the [[Middle Ages]], [[Islamic mathematics|Muslim mathematicians]] 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]]. [[Thābit ibn Qurra]] (known as Thebit in [[Latin]]) (836-901) dealt with [[arithmetic]]al operations applied to [[ratio]]s of geometrical quantities, and contributed to the development of [[analytic geometry]]. [[Omar Khayyám]] (1048-1131) found geometric solutions to [[cubic equation]]s, and his extensive studies of the [[parallel postulate]] contributed to the development of [[Non-Euclidian geometry]].{{Fact|date=July 2007}}

In the early 17th century, there were two important developments in geometry. The first, and most important, was the creation of [[analytic geometry]], or geometry with [[Coordinate system|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 the study of geometry without measurement, just the study of how points align with each other.

Two developments in geometry in the nineteenth century changed the way it had been studied previously. These were the discovery of [[non-Euclidean geometry|non-Euclidean geometries]] by [[Nikolai Ivanovich Lobachevsky|Lobachevsky]], [[János Bolyai|Bolyai]] and [[Carl Friedrich Gauss|Gauss]] 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]], 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 system]]s.

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]]. The traditional type of geometry was recognized as that of [[homogeneous space]]s, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same.

== What is geometry? ==
[[Image:Chinese pythagoras.jpg|thumb|300px|right|Visual [[proof (mathematics)|proof]] of the [[Pythagorean theorem]] for the (3, 4, 5) [[triangle]] as in the [[Chou Pei Suan Ching]] 500–200&nbsp;BC.]]

Recorded development of geometry spans more than two [[millennia]]. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages.
The geometric paradigms presented below should be viewed as '[[Pictures at an exhibition]]' of a sort: they do not exhaust the subject of geometry but rather reflect some of its defining themes.

=== Practical geometry ===
There is little doubt that geometry originated as a ''practical'' science, concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for [[length]]s, [[area]]s and [[volume]]s, such as [[Pythagorean theorem]], [[circumference]] and [[area of a disk|area]] of a circle, area of a [[triangle]], volume of a [[Cylinder (geometry)|cylinder]], [[sphere]], and a [[pyramid (geometry)|pyramid]]. Development of [[astronomy]] led to emergence of [[trigonometry]] and [[spherical trigonometry]], together with the attendant computational techniques.

=== Axiomatic geometry ===

A method of computing certain inaccessible distances or heights based on [[similarity (geometry)|similarity]] of geometric figures and attributed to [[Thales]] presaged more abstract approach to geometry taken by [[Euclid]] in his [[Euclid's Elements|Elements]], one of the most influential books ever written. Euclid introduced certain [[axiom]]s, or [[postulate]]s, 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. In the twentieth century, [[David Hilbert]] employed axiomatic reasoning in his attempt to update Euclid and provide modern foundations of geometry.

=== Geometric constructions ===

Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are the [[compass and straightedge]]. 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. The approach to geometric problems with geometric or mechanical means is known as [[synthetic geometry]].

=== Numbers in geometry ===

Already [[Pythagoreans]] considered the role of numbers in geometry. However, the discovery of [[Commensurability (mathematics)|incommensurable]] lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favour of (concrete) geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of [[coordinate]]s by [[Descartes]], who realized that the study of geometric shapes can be facilitated by their algebraic representation. [[Analytic geometry]] applies methods of algebra to geometric questions, typically by relating geometric [[curve]]s and algebraic [[equation]]s. These ideas played a key role in the development of [[calculus]] in the seventeenth 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 ===

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 [[polygon]]s, lines intersecting and tangent to [[conic section]]s, the [[Pappus configuration|Pappus]] and [[Menelaus theorem|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 [[sphere packing|packing of spheres]] of equal size in space ([[Kepler conjecture]])? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. [[Projective geometry|Projective]], [[Convex geometry|convex]] and [[discrete geometry|discrete]] geometry are three subdisciplines within present day geometry that deal with these and related questions.

A new chapter in ''Geometria situs'' was opened by [[Leonhard Euler]], who boldly cast out metric properties of geometric figures and considered their most fundamental geometrical structure based solely on shape. [[Topology]], which grew out of geometry, but turned into a large independent discipline, 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 knot]]s.

=== Geometry beyond Euclid ===

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]].<ref>Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) ''possibility'' of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact ''predicted'' the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164.</ref> This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of [[Carl Friedrich Gauss|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 inaugurational lecture ''Über 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.

=== Symmetry ===
[[Image:Order-3 heptakis heptagonal tiling.png|right|thumb|120px|A uniform [[tessellation|tiling]] of the [[hyperbolic plane]]]]
The theme of [[symmetry]] in geometry is nearly as old as the science of geometry itself. The [[circle]], [[regular polygon]]s and [[platonic solid]]s 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 nineteenth 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 (mathematics)|group]], determines what geometry ''is''. Symmetry in classical [[Euclidean geometry]] is represented by [[Congruence (geometry)|congruence]]s and rigid motions, whereas in [[projective geometry]] an analogous role is played by [[collineation]]s, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, [[William Kingdon Clifford|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]].

=== Modern geometry ===

''Modern geometry'' is the title of a popular textbook by Dubrovin, [[Sergei Petrovich Novikov|Novikov]], and Fomenko first published in 1979 (in Russian). At close to 1000 pages,
the book has one major thread: geometric structures of various types on [[manifold]]s and their applications in contemporary [[theoretical physics]]. A quarter century after its publication, [[differential geometry]], [[algebraic geometry]], [[symplectic geometry]], and [[Lie theory]] presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics.

==Contemporary geometers==

Some of the representative leading figures in modern geometry are [[Michael Atiyah]], [[Mikhail Gromov]], and [[William Thurston]]. The common feature in their work is the use of [[smooth manifold]]s as the basic idea of ''space''; they otherwise have rather different directions and interests. Geometry now is, in large part, the study of ''structures'' on manifolds that have a geometric meaning, in the sense of the [[principle of covariance]] that lies at the root of [[general relativity]] theory in theoretical physics. (See [[:Category:Structures on manifolds]] for a survey.)

Much of this theory relates to the theory of ''continuous symmetry'', or in other words [[Lie group]]s. From the foundational point of view, on manifolds and their geometrical structures, important is the concept of [[pseudogroup]], defined formally by [[Shiing-shen Chern]] in pursuing ideas introduced by [[Élie Cartan]]. A pseudogroup can play the role of a Lie group of ''infinite'' dimension.

==Dimension==

Where the traditional geometry allowed dimensions 1 (a [[line]]), 2 (a [[Plane (mathematics)|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 manifold]]s 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]]. Exactly why is something to which research may bring a satisfactory ''geometric'' answer.

==Contemporary Euclidean geometry==
{{main|Euclidean geometry}}

The study of traditional [[Euclidean geometry]] is by no means dead. It is now typically presented as the geometry of [[Euclidean space]]s of any dimension, and of the [[Euclidean group]] of [[rigid motion]]s. The fundamental formulae of geometry, such as the [[Pythagorean theorem]], can be presented in this way for a general [[inner product space]].

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 group]]s and [[polytope]]s. [[Geometric group theory]] is an expanding area of the theory of more general [[discrete group]]s, drawing on geometric models and algebraic techniques.

==Algebraic geometry==

The field of [[algebraic geometry]] is the modern incarnation of the [[Cartesian geometry]] of [[co-ordinates]]. After a turbulent period of [[axiomatization]], its foundations are in the twenty-first century on a stable basis. Either one studies the 'classical' case where the spaces are [[complex manifold]]s that can be described by [[algebraic equation]]s; or the [[scheme theory]] provides a technically sophisticated theory based on general [[commutative ring]]s.

The geometric style which was traditionally called the [[Italian school of algebraic geometry|Italian school]] is now known as [[birational geometry]]. It has made progress in the fields of [[threefold (mathematics)|threefold]]s, [[singularity theory]] and [[moduli space]]s, as well as recovering and correcting the bulk of the older results. Objects from algebraic geometry are now commonly applied in [[string theory]], as well as [[diophantine geometry]].

Methods of algebraic geometry rely heavily on [[sheaf theory]] and other parts of [[homological algebra]]. The [[Hodge conjecture]] is an open problem that has gradually taken its place as one of the major questions for mathematicians. For practical applications, [[Gröbner basis]] theory and [[real algebraic geometry]] are major subfields.

==Differential geometry==
[[Differential geometry]], which in simple terms is the geometry of [[curvature]], has been of increasing importance to [[mathematical physics]] since the suggestion that space is not [[flat space]]. Contemporary differential geometry is ''intrinsic'', meaning that space is a manifold and structure is given by a [[Riemannian metric]], or analogue, locally determining a geometry that is variable from point to point.

This approach contrasts with the ''extrinsic'' point of view, where curvature means the way a space ''bends'' within a larger space. The idea of 'larger' spaces is discarded, and instead manifolds carry [[vector bundle]]s. Fundamental to this approach is the connection between curvature and [[characteristic class]]es, as exemplified by the [[generalized Gauss-Bonnet theorem]].

==Topology and geometry==
[[Image:Trefoil knot arb.png|thumb|right|120 px|A thickening of the [[trefoil knot]]]]
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 [[homeomorphism]]s. 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.

==Axiomatic and open development==

The model of Euclid's ''Elements'', a connected development of geometry as an [[axiomatic system]], is in a tension with [[René Descartes]]'s reduction of geometry to algebra by means of a [[coordinate system]]. There were many champions of [[synthetic geometry]], Euclid-style development of projective geometry, in the nineteenth century, [[Jakob Steiner]] being a particularly brilliant figure. In contrast to such approaches to geometry as a closed system, culminating in [[Hilbert's axioms]] and regarded as of important pedagogic value, most contemporary geometry is a matter of style. [[Computational synthetic geometry]] is now a branch of [[computer algebra]].

The Cartesian approach currently predominates, with geometric questions being tackled by tools from other parts of mathematics, and geometric theories being quite open and integrated. This is to be seen in the context of the axiomatization of the whole of [[pure mathematics]], which went on in the period c.1900–c.1950: in principle all methods are on a common axiomatic footing. This reductive approach has had several effects. There is a taxonomic trend, which following Klein and his Erlangen program (a taxonomy based on the [[subgroup]] concept) arranges theories according to generalization and specialization. For example [[affine geometry]] is more general than Euclidean geometry, and more special than projective geometry. The whole theory of [[classical group]]s thereby becomes an aspect of geometry. Their [[invariant theory]], at one point in the nineteenth century taken to be the prospective master geometric theory, is just one aspect of the general [[representation theory]] of [[algebraic group]]s and [[Lie group]]s. Using [[finite field]]s, the classical groups give rise to [[finite group]]s, intensively studied in relation to the [[finite simple group]]s; and associated [[finite geometry]], which has both combinatorial (synthetic) and algebro-geometric (Cartesian) sides.

An example from recent decades is the [[twistor theory]] of [[Roger Penrose]], initially an intuitive and synthetic theory, then subsequently shown to be an aspect of [[sheaf theory]] on [[complex manifold]]s. In contrast, the [[non-commutative geometry]] of [[Alain Connes]] is a conscious use of geometric language to express phenomena of the theory of [[von Neumann algebra]]s, and to extend geometry into the domain of [[ring theory]] where the [[commutative law]] of multiplication is not assumed.

Another consequence of the contemporary approach, attributable in large measure to the Procrustean bed represented by [[Bourbaki]]ste axiomatization trying to complete the work of [[David Hilbert]], is to create winners and losers. The ''[[Ausdehnungslehre]]'' (calculus of extension) of [[Hermann Grassmann]] was for many years a mathematical backwater, competing in three dimensions against other popular theories in the area of [[mathematical physics]] such as those derived from [[quaternion]]s. In the shape of general [[exterior algebra]], it became a beneficiary of the Bourbaki presentation of [[multilinear algebra]], and from 1950 onwards has been ubiquitous. In much the same way, [[Clifford algebra]] became popular, helped by a 1957 book ''Geometric Algebra'' by [[Emil Artin]]. The history of 'lost' geometric methods, for example ''[[infinitely near point]]s'', which were dropped since they did not well fit into the pure mathematical world post-''[[Principia Mathematica]]'', is yet unwritten. The situation is analogous to the expulsion of [[infinitesimal]]s from [[differential calculus]]. As in that case, the concepts may be recovered by fresh approaches and definitions. Those may not be unique: [[synthetic differential geometry]] is an approach to infinitesimals from the side of [[categorical logic]], as [[non-standard analysis]] is by means of [[model theory]].

==See also==
{{sisterlinks|Geometry}}
=== Lists ===
* [[List of basic geometry topics]]
* [[List of geometry topics]]
* [[List of geometers]]
** [[:Category:Geometers]]
** [[:Category:Algebraic geometers]]
** [[:Category:Differential geometers]]
** [[:Category:Topologists]]
* [[List of important publications in mathematics#Geometry|Important publications in geometry]]
* [[List of mathematics articles]]

=== Related topics ===
* [[Interactive geometry software]]
{{wikisource|Flatland}}
* ''[[Flatland]]'', a book written by [[Edwin Abbott Abbott]] about two and [[three-dimensional space]], to understand the concept of four dimensions
* [[Why 10 dimensions?]]

== References ==
{{reflist}}
{{reflist}}
{{Refimprove|date=December 2007}}

==External links==
{{portal}}
{{Wikibooks}}
* [http://www.mathforum.org/library/topics/geometry/ ''The Math Forum'' — Geometry]
** [http://www.mathforum.org/geometry/k12.geometry.html ''The Math Forum'' — K–12 Geometry]
** [http://www.mathforum.org/geometry/coll.geometry.html ''The Math Forum'' — College Geometry]
** [http://www.mathforum.org/advanced/geom.html ''The Math Forum'' — Advanced Geometry]
* [http://www.math.niu.edu/~rusin/known-math/index/tour_geo.html ''The Mathematical Atlas'' — Geometric Areas of Mathematics]
* [http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618 "4000 Years of Geometry"], lecture by Robin Wilson given at [[Gresham College]], 3rd October 2007 (available for MP3 and MP4 download as well as a text file)
* [http://www.cut-the-knot.org/WhatIs/WhatIsGeometry.shtml What Is Geometry?] at [[cut-the-knot]]
* [http://www.cut-the-knot.org/geometry.shtml Geometry] at [[cut-the-knot]]
* [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
* [http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry]
* Stanford Encyclopedia of Philosophy:
** [http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry]
** [http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century]
* [http://www.egwald.ca/geometry/index.php Online Interactive Geometric Objects] by Elmer G. Wiens
* [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html Arabic mathematics : forgotten brilliance?]
* [http://www.ics.uci.edu/~eppstein/junkyard/topic.html The Geometry Junkyard]
* [http://mrperezonlinemathtutor.com/A_Geometry.html Geometry lessons in PowerPoint]
{{Mathematics-footer}}

[[Category:Geometry| ]]

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Revision as of 06:32, 13 October 2008

"Cult of Personality"
Song

"Cult of Personality" is a song by the heavy metal band Living Colour and the lead single from their debut album, Vivid. Its music video earned two MTV Video Music Awards for Best Group Video and Best New Artist. Released in 1989, "Cult of Personality" reached #13 on the Billboard Hot 100 and #9 on the Billboard Album Rock Tracks chart.

The song begins with an edited quote from the beginning of a speech by Malcolm X. As it appears in the song, the quote is:

". . . And during the few moments that we have left, . . . We want to talk right down to earth in a language that everybody here can easily understand."

The unabridged beginning of the speech is:

"...And during the few moments that we have left, we want to have just an off-the-cuff chat between you and me -- us. We want to talk right down to earth in a language that everybody here can easily understand." [1]

At 4:35, John F. Kennedy is quoted, saying "Ask not what your country can do for you," and the song ends with Franklin D. Roosevelt saying "The only thing we have to fear is fear itself."

"Cult of Personality" was performed live during the April 1, 1989 edition of Saturday Night Live with host Mel Gibson. It was also performed on The Arsenio Hall Show that same year.

Lyrics

Along with the above quotes interspersed, the lyrics in the song refer to several well-known political and social figures (thus implicitly establishing each as the leader of a cult of personality).

Appearances in other media

References

  1. ^ Malcolm X: "Message to the Grass Roots": http://www.americanrhetoric.com/speeches/malcolmxgrassroots.htm
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