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Sphere

Sphere

:For other uses, see sphere (disambiguation). sphere (disambiguation) A sphere is a perfectly symmetrical geometrical object. In mathematics, the term refers to the surface or boundary of a ball, but in non-mathematical usage, the term is used to refer either to a three-dimensional ball or to its surface. This article deals with the mathematical concept of sphere.

Geometry

In three-dimensional Euclidean geometry, a sphere is the set of points in R³ which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. The fixed point is called the center or centre, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere.

Equations

real number In analytic geometry, a sphere with center (x₀, y₀, z₀) and radius r is the set of all points (x, y, z) such that :

(x - x_0 )^2 + (y - y_0 )^2 + ( z -  z_0 )^2 =  r^2 \,

The points on the sphere with radius r can be parametrized via :

x = x_0 + r \sin \theta \; \cos \phi

y = y_0 + r \sin \theta \; \sin \phi  \qquad (0 \leq \theta \leq \pi \mbox -\pi < \phi \leq \pi) \,

z = z_0 + r \cos \theta  \,

(see also trigonometric functions and spherical coordinates). A sphere of any radius centered at the origin is described by the following differential equation: :

x \, dx + y \, dy + z \, dz = 0.

This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other. The surface area of a sphere of radius r is: :

A = 4 \pi r^2 \,

and its enclosed volume is: :

V = \frac

The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension minimizes surface area. surface tension for the Gravity Probe B experiment which differs from a perfect sphere by no more than a mere 40 atoms of thickness as it refracts the image of Einstein in the background. It is thought that only neutron stars are smoother.]] The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere. This fact, along with the volume and surface formulas given above, was already known to Archimedes. A sphere can also be defined as the surface formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.

Terminology

Pairs of points on a sphere that lie on a straight line through its center are called antipodal points. A great circle is a circle on the sphere that has the same center as the sphere. If a particular point on a sphere is designated as its north pole, then the corresponding antipodal point is called the south pole and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the two poles is called the axis. Circles on the sphere that are parallel to the equator are lines of latitude. This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid).

Generalization to other dimensions

Spheres can be generalized to other dimensions. For any natural number n, an n-sphere is the set of points in (n+1)-dimensional Euclidean space which are at distance r from a fixed point of that space, where r is, as before, a positive real number:
- a 0-sphere is a pair of points

(-r, r)

- a 1-sphere is a circle of radius r
- a 2-sphere is an ordinary sphere
- a 3-sphere is a sphere in 4-dimensional Euclidean space. Spheres for n > 2 are sometimes called hyperspheres. The n-sphere of unit radius centred at the origin is denoted Sⁿ and is often referred to as "the" n-sphere.

Generalization to metric spaces

More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set :S(x;r) = . If the center is a distinguished point considered as origin of E, e.g. in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken equal to one, i.e. in the case of a unit sphere. In contrast to a ball, a sphere may be empty. For example, in Zⁿ with Euclidean metric, a sphere of radius r is nonempty only if r² can be written as sum of n squares of integers.

Topology

In topology, an n-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere described above under Geometry, but perhaps lacking its metric.
- a 0-sphere is a pair of points with the discrete topology
- a 1-sphere is a circle (up to homeomorphism); thus, for example, (the image of) any knot is a 1-sphere
- a 2-sphere is an ordinary sphere (up to homeomorphism); thus, for example, any spheroid is a 2-sphere The n-sphere is denoted Sⁿ. It is an example of a compact topological manifold without boundary. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere. The Heine-Borel theorem is used in a short proof that a Euclidean n-sphere is compact. The sphere is the inverse image of a one-point set under the continuous function ||x||. Therefore the sphere is closed. Sⁿ is also bounded. Therefore it is compact.

External links

- [http://mathworld.wolfram.com/Hypersphere.html Mathworld website]
- [http://www.mathsisfun.com/geometry/sphere.html More Sphere Images] Math is Fun Category:Surfaces Category:Differential geometry Category:Topology Category:Elementary geometry ja:球 ja:球面 simple:Sphere

Sphere (disambiguation)

Most commonly, a sphere is a ball-shaped object; however, there are alternative uses:
- Sphere Books was a British paperback-publisher of the 1960s to the 1980s.
- Sphere is a book written by Michael Crichton, which was subsequently turned into a film of the same name.
- Sphere (jazz band)
- Sphere (rpg maker) is a cross platform, open source computer program used to make Role-playing games similar to those on the SNES console.
- A sphere of knowledge
- A sphere of influence is the area or region heavily influenced by a strong culture or political entity.
- Properties of a planet: biosphere, lithosphere, hydrosphere and atmosphere

Symmetrical

Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by one of the operations. In 2D geometry the main kinds of symmetry of interest are with respect to the basic Euclidean plane isometries: translations, rotations, reflections, and glide reflections. glide reflection

Mathematical model for symmetry

The set of all symmetry operations considered, on all objects in a set X, can be modelled as a group action g : G × X → X, where the image of g in G and x in X is written as g·x. If, for some g, g·x = y then x and y are said to be symmetric to each other. For each object x, operations g for which g·x = x form a group, the symmetry group of the object, a subgroup of G. If the symmetry group of x is the trivial group then x is said to be asymmetric, otherwise symmetric. A general example is that G is a group of bijections g: V → V acting on the set of functions x: V → W by (gx)(v)=x(g−1(v)) (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of x consists of all g for which x(v)=x(g(v)) for all v. G is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of G may not be the symmetry group of any object. For example, if the group contains for every v and w in V a g such that g(v)=w, then only the symmetry groups of constant functions x contain that group. However, the symmetry group of constant functions is G itself. In a modified version for vector fields, we have (gx)(v)=h(g,x(g−1(v))) where h rotates any vectors and pseudovectors in x, and inverts any vectors (but not pseudovectors) according to rotation and inversion in g, see symmetry in physics. The symmetry group of x consists of all g for which x(v)=h(g,x(g(v))) for all v. In this case the symmetry group of a constant function may be a proper subgroup of G: a constant vector has only rotational symmetry with respect to rotation about an axis if that axis is in the direction of the vector, and only inversion symmetry if it is zero. For a common notion of symmetry in Euclidean space, G is the Euclidean group E(n), the group of isometries, and V is the Euclidean space. The rotation group of an object is the symmetry group if G is restricted to E+(n), the group of direct isometries. (For generalizations, see the next subsection.) Objects can be modeled as functions x, of which a value may represent a selection of properties such as color, density, chemical composition, etc. Depending on the selection we consider just symmetries of sets of points (x is just a boolean function of position v), or, at the other extreme, e.g. symmetry of right and left hand with all their structure. For a given symmetry group, the properties of part of the object, fully define the whole object. Considering points equivalent which, due to the symmetry, have the same properties, the equivalence classes are the orbits of the group action on the space itself. We need the value of x at one point in every orbit to define the full object. A set of such representatives forms a fundamental domain. An object with a desired symmetry can be produced by choosing for every orbit a single function value. Starting from a given object x we can e.g.:
- take the values in a fundamental domain (i.e., add copies of the object)
- take for each orbit some kind of average or sum of the values of x at the points of the orbit (ditto, where the copies may overlap) If it is desired to have no more symmetry than that in the symmetry group, then the object to be copied should be asymmetric. As pointed out above, some groups of isometries are not the symmetry group of any object, except in the modified model for vector fields. For example, this applies in 1D for the group of all translations. The fundamental domain is only one point, so we can not make it asymmetric, so any "pattern" invariant under translation is also invariant under reflection (these are the uniform "patterns"). In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry. If there is also reflectional symmetry, the constant function value contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D example is an infinite cylinder with a current perpendicular to the axis; the magnetic field (a pseudovector) is, in the direction of the cylinder, constant, but nonzero. For vectors (in particular the current density) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. This cylindrical symmetry without mirror planes through the axis is also only possible in the vector field version of the symmetry concept. A similar example is a cylinder rotating about its axis, where magnetic field and current density are replaced by angular momentum and velocity, respectively. A symmetry group is said to act transitively on a repeated feature of an object if, for every pair of occurrences of the feature there is a symmetry operation mapping the first to the second. For example, in 1D, the symmetry group of acts transitively on all these points, while does not act transitively on all points. Equivalently, the first set is only one conjugacy class with respect to isometries, while the second has two classes.

Non-isometric symmetry

As mentioned above, G (the symmetry group of the space itself) may differ from the Euclidean group, the group of isometries. Examples:
- G is the group of similarity transformations, i.e. affine transformations with a matrix A that is a scalar times an orthogonal matrix. Thus dilations are added, self-similarity is considered a symmetry
- G is the group of affine transformations with a matrix A with determinant 1 or -1, i.e. the transformation which preserve area; this adds e.g. oblique reflection symmetry.
- G is the group of all bijective affine transformations
- In inversive geometry, G includes circle reflections, etc.
- More generally, an involution defines a symmetry with respect to that involution.

Reflection symmetry

See reflection symmetry.

Rotational symmetry

See rotational symmetry.

Translational symmetry

See main article translational symmetry. Translational symmetry leaves an object invariant under a discrete or continuous group of translations Ta(p) = p + a

Glide reflection symmetry

A glide reflection symmetry (in 3D: a glide plane symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. It implies translational symmetry with twice the translation vector. The symmetry group is isomorphic with Z.

Rotoreflection symmetry

In 3D, rotoreflection or improper rotation in the strict sense is rotation about an axis, combined with reflection in a plane perpendicular to that axis. As symmetry groups with regard to a roto-reflection we can distinguish:
- the angle has no common divisor with 360°, the symmetry group is not discrete
- 2n-fold rotoreflection (angle of 180°/n) with symmetry group S2n of order 2n (not to be confused with symmetric groups, for which the same notation is used; abstract group C2n); a special case is n=1, inversion, because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
- Cnh (angle of 360°/n); for odd n this is generated by a single symmetry, and the abstract group is C2n, for even n this is not a basic symmetry but a combination. See also point groups in three dimensions.

Screw axis symmetry

In 3D, screw axis symmetry is invariance under a rotation about an axis combined with translation along that axis . We can distinguish:
- there is invariance for every angle and a proportional translation distance, this applies e.g. for an infinite helix and double helix;
- the angle has no common divisor with 360°; the symmetry group is discrete, although the set of angles is not; it does not contain pure translations
- n-fold screw axis (angle of 360°/n) See also space group.

Symmetry combinations

See symmetry combinations.

Color

symmetry combinations With a color image one can associate a greyshade or black-and-white image. One way is to associate with each color a greyshade or either black or white. Alternatively, boundaries may be represented in black, and interior areas in white. When considering symmetry "ignoring colors" this tends to mean that dark colors become black and light colors white, or that boundaries become black. Sometimes there is only one meaningful conversion, in other cases the conversion has to be specified to avoid ambiguity (see e.g. the tetrakis square tiling). The new image may have more symmetry. Also colors may provide a special kind of symmetry, e.g. with corresponding points having opposite colors (including black and white), such as in the yin and yang symbol. Compare the modified symmetry model for vector fields, above.

Similarity vs. sameness

Although two objects with great similarity appear the same, they must logically be different. For example, if one rotates an equilateral triangle around its center 120 degrees, it will appear the same as it was before the rotation to an observer. In theoretical euclidean geometry, such a rotation would be unrecognizable from its previous form. In reality however, each corner of any equilateral triangle composed of matter must be composed of separate molecules in separate locations. Therefore, symmetry in real physical objects is a matter of similarity instead of sameness. The difficulty for an intelligence to differentiate such a seemingly exact similarity is understandable.

More on symmetry in geometry

The German geometer Felix Klein enunciated a very influential Erlangen programme in 1872, suggesting symmetry as unifying and organising principle in geometry (at a time when that was read 'geometries'). This is a broad rather than deep principle. Initially it led to interest in the groups attached to geometries, and the slogan transformation geometry (an aspect of the New Math, but hardly controversial in modern mathematical practice). By now it has been applied in numerous forms, as kind of standard attack on problems. A fractal, as conceived by Mandelbrot, has symmetry involving scaling. For example an equilateral triangle can be shrunk so that each of its sides are one third the length of the original's sides. These smaller triangles can be rotated and translated until they are adjacent and in the center of each of the larger triangle's lines. The smaller triangles can repeat the process, resulting in even smaller triangles on their sides. Fascinating intricate structures can be created by repeating such scaling symmetrical operations many times. If a structure has a symmetry plane then for every part of the structure there are two possibilities:
- the part has itself a symmetry plane (the same plane)
- it has a mirror image counterpart

Symmetry in mathematics

:(main article: symmetry in mathematics) An example of a mathematical expression exhibiting symmetry is a2c + 3ab + b2c. If a and b are exchanged, the expression remains unchanged due to the commutativity of addition and multiplication. Like in geometry, for the terms there are two possibilities:
- it is itself symmetric
- it has one or more other terms symmetric with it, in accordance with the symmetry kind See also symmetric function, duality (mathematics).

Symmetry in logic

A dyadic relation R is symmetric if and only if, whenever it's true that Rab, it's true that Rba. Thus, “is the same age as” is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul. Symmetric binary logical connectives are "and" (&and;, <math>\land</math>, or &), "or" (&or;), "biconditional" (iff) (↔), NAND ("not-and"), XOR ("not-biconditional"), and NOR ("not-or").

Generalization of symmetry

If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of symmetry group to that of a groupoid. Physicists have come up with other directions of generalization, such as supersymmetry and quantum groups.

Symmetry in physics

(see main article: symmetry in physics) Symmetry in physics has been generalized to mean invariance under any kind of transformation. This has become one of the most powerful tools of theoretical physics. See Noether's theorem (which, as a gross oversimplification, states that for every symmetry law, there is a conservation law).

Symmetry in biology

See symmetry in nature, bilateral symmetry, facial symmetry, pentamerism.

Symmetry in chemistry

See Spectroscopy, Molecular orbital

Symmetry in the arts and crafts

You can find the use of symmetry across a wide variety of arts and crafts.

Architecture

Architecture Image:Monticello.PNG Symmetry has long been a predominant design element in architecture; prominent examples include the Leaning Tower of Pisa, Monticello, the Astrodome, the Sydney Opera House, Gothic church windows, and the Pantheon. Symmetry is used in the design of the overall floor plan of buildings as well as the design of individual building elements such as doors, windows, floors, frieze work, and ornamentation; many facades adhere to bilateral symmetry. Links:
- [http://members.tripod.com/vismath/kim/ Williams: Symmetry in Architecture]
- [http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml Aslaksen: Mathematics in Art and Architecture]

Pottery

Image:Persian_Pottery.jpg The ancient Chinese used symmetrical patterns in their bronze castings since the 17th century B.C. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design. Persian pottery dating from 6000 B.C. used symmetric zigzags, squares, and cross-hatchings. Links:
- [http://www.chinavoc.com/arts/handicraft/bronze.htm Chinavoc: The Art of Chinese Bronzes]
- [http://www-oi.uchicago.edu/OI/MUS/VOL/NN_SUM94/NN_Sum94.html Grant: Iranian Pottery in the Oriental Institute]
- [http://www.metmuseum.org/collections/department.asp?dep=14 The Metropolitan Museum of Art - Islamic Art]

Quilts

Image:kitchen_kaleid.PNG As quilts are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry. Links:
- [http://its.guilford.k12.nc.us/webquests/quilts/quilts.htm Quate: Exploring Geometry Through Quilts]
- [http://log24.com/log04/0809.htm Quilt Geometry]

Carpets, rugs

Image:orientalrug.JPG A long tradition of the use of symmetry in rug patterns spans a variety of cultures. American Navajo Indians used bold diagonals and rectangular motifs. Many Oriental rugs have intricate reflected centers and borders that translate a pattern. Not surprisingly most rugs use quadrilateral symmetry -- a motif reflected across both the horizontal and vertical axes. Links:
- [http://www.marlamallett.com/default.htm Mallet: Tribal Oriental Rugs]
- [http://navajocentral.org/rugs.htm Dilucchio: Navajo Rugs]

Music

Form

Symmetry has been used as a formal constraint by many composers, such as the arch form (ABCBA) used by Steve Reich, Béla Bartók, and James Tenney (or swell). In classical music, Bach used the symmetry concepts of permutation and invariance; see (external link "Fugue No. 21," [http://jan.ucc.nau.edu/~tas3/wtc/ii21s.pdf pdf] or [http://jan.ucc.nau.edu/~tas3/wtc/ii21.html Shockwave]).

Pitch structures

Symmetry is also an important consideration in the formation of scales and chords, traditional or tonal music being made up of non-symmetrical groups of pitches, such as the diatonic scale or the major chord. Symmetrical scales or chords, such as the whole tone scale, augmented chord, or diminished seventh chord (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are ambiguous as to the key or tonal center, and have a less specific diatonic functionality. However, composers such as Alban Berg, Béla Bartók, and George Perle have used axes of symmetry and/or interval cycles in an analogous way to keys or non-tonal tonal centers. Perle (1992) explains "C-E, D-F#, [and] Eb-G, are different instances of the same interval...the other kind of identity...has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows:" Thus in addition to being part of the interval-4 family, C-E is also a part of the sum-4 family (with C equal to 0). Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are enharmonic with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal progressions in the works of Romantic composers such as Gustav Mahler and Richard Wagner form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, Alexander Scriabin, Edgard Varese, and the Vienna school. At the same time, these progressions signal the end of tonality. The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's Quartet, Op. 3 (1910). (Perle, 1990)

Equivalency

Tone rows or pitch class sets which are invariant under retrograde are horizontally symmetrical, under inversion vertically. See also Asymmetric rhythm.

Other arts and crafts

Chair      Bracelet      Celtic knotwork The concept of symmetry is applied to the design of objects of all shapes and sizes -- you can find it in the design of beadwork, furniture, sand paintings, knotwork, masks, and musical instruments (to name just a handful of examples).

Aesthetics

Symmetry does not by itself confer beauty to an object — many symmetrical designs are boring or overly challenging, and on the other hand preference for, or dislike of, exact symmetry is apparently dependent on cultural background. Along with texture, color, proportion, and other factors, symmetry does however play an important role in determining the aesthetic appeal of an object. See also M. C. Escher, wallpaper group, tiling.

Symmetry in games and puzzles

- See also symmetric games.
- See sudoku. Puzzles
- [http://m759.freeservers.com/puzzle.html The Diamond 16 Puzzle] Board Games
- [http://www.symmetryperfect.com/shots The Symmetrical Chess Collection]

Symmetry in literature

See palindrome.

Symmetry in telecommunications

Some telecommunications services (specifically data products) may be referred to as symmetrical or asymmetrical. This refers to the bandwidth allocated for data sent and received. Most internet services used by residential customers are asymmetrical: the data sent to the server normally is far less than that returned by the server.

Moral symmetry

- Tit for tat
- Reciprocity
- Golden Rule
- Empathy & Sympathy
- Reflective equilibrium

External links

- [http://0waldo.com (A)symmetrical wallpaper tiles] by Walter Muncaster (user:0waldo)
- [http://home.earthlink.net/~akuster/music/bartok/quartet4.htm An Analysis of the first movement of the Fourth String Quartet (1928)] by Andrew Kuster
- [http://www.usask.ca/education/coursework/skaalid/theory/theory.htm Skaalid: Design Theory]
- [http://mathforum.org/library/topics/sym_tess/ Mathforum: Symmetry/Tesselations]
- [http://www.teachersnetwork.org/teachnet/westchester/symmetry.htm Calotta: A World of Symmetry]
- [http://www.uwgb.edu/dutchs/SYMMETRY/2DPTGRP.HTM Dutch: Symmetry Around a Point in the Plane]
- [http://www.punahou.edu/acad/sanders/MathArt/MACch2sym.html Sanders: Transformations and Symmetry]
- [http://daphne.palomar.edu/design/conclude.html Saw: Design Notes]
- [http://home.earthlink.net/~jdc24/symmetry.htm Chapman: Aesthetics of Symmetry]
- [http://www.bangor.ac.uk/~mas009/psym.htm Abas: The Wonder Of Symmetry]
- [http://www.mi.sanu.ac.yu/~jablans/isis0.htm ISIS Symmetry]
- [http://etext.lib.virginia.edu/cgi-local/DHI/dhi.cgi?id=dv4-46 Symmetry and Asymmetry at The Dictionary of the History of Ideas]

References

- Perle, George (1990). The Listening Composer, p. 112. California: University of California Press. ISBN 0520069919.
- Perle, George (1992). Symmetry, the Twelve-Tone Scale, and Tonality. Contemporary Music Review 6 (2), pp. 81-96
- Weyl, Hermann (1952). Symmetry. Princeton University Press. ISBN 0-691-02374-3.
- Category:Arts ja:対称性

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Boundary

The word boundary has a variety of meanings.
- In general, the term boundary is something that encloses a region space, a territory, and/or an area. Something enclosed by a boundary, is "bounded" (by the boundary). See also border.
- For boundary in the sport of cricket, see: boundary (cricket).
- For the boundary of a set in a topology, see: boundary (topology).
- For the boundary of a topological manifold, see: manifold,
- As a psychological term, boundary can mean a mental separation which we keep between ourselves and others. Sometimes this can be referred to as personal space. Addicts use boundaries to keep themselves sober.

Solid geometry

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was studied as a sequel to plane geometry. Stereometry deals with the measurements of volumes of various solid figures: cylinder, circular cone, truncated cone, sphere, prisms, blades, wine casks. The Pythagoreans had dealt with the sphere and regular solids, but the pyramid, prism, cone and cylinder were but little known until the Platonists took them in hand. Eudoxus established their mensuration, proving the pyramid and cone to have one-third the content of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volumes of spheres are as the cubes of their radii. See also: Archimedes, Demiurge, Johannes Kepler, planimetry, Plato, Timaeus (dialogue) ...partly from the 1911 Encyclopædia Britannica

Basic topics of solid geometry

Basic topics are:
- incidence of planes and lines
- dihedral angle and solid angle
- the cube, cuboid, parallelepiped
- the tetrahedron and other pyramids
- prisms
- octahedron, dodecahedron, icosahedron
- cones and cylinders
- the sphere
- other quadrics: spheroid, ellipsoid, paraboloid and hyperboloids.

Euclidean geometry

:For a topic other than geometry whose name includes the word "Euclidean", see Euclidean algorithm. In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. Mathematicians sometimes use the term to encompass higher dimensional geometries with similar properties. It is important to remember that, in the original and correct conception, geometry is, first of all, a physical science ("the noblest of the physical sciences"); that is, the logical definition of geometry (its fundamental assumptions or axioms) arises directly out of observation. Abstract geometries may be exclusively mathematical, but, whenever so, they are different from physical geometries. Euclidean geometry sometimes means geometry in the plane which is also called plane geometry. Plane geometry is the topic of this article. Euclidean geometry is also based on the Point-Line-Plane postulate. Euclidean geometry in three dimensions is traditionally called solid geometry. For information on higher dimensions see Euclidean space. Plane geometry is the kind of geometry usually taught in secondary school. Euclidean geometry is named after the Greek mathematician Euclid. Euclid's text Elements is an early systematic treatment of this kind of geometry.

Axiomatic approach

The traditional presentation of Euclidean geometry is as an axiomatic system, setting out to prove all the "true statements" as theorems in geometry from a set of finite number of axioms. The five postulates of the Elements are as follows: # Any two points can be joined by a straight line. # Any straight line segment can be extended indefinitely in a straight line. # Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. # All right angles are congruent. # If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The fifth postulate is called the parallel postulate, which leads to the same geometry as the following statement (note that it is formulated for two-dimensional geometry): :Through a point not on a given straight line, one and only one line can be drawn that never meets the given line. The parallel postulate seems less obvious than the others and many geometers tried in vain to prove it from them. By 1763 at least 28 different proofs of the fifth postulate had been published, but all were found to be incorrect. In the 19th century it was shown that this could not be done, by constructing hyperbolic geometry where the parallel postulate is false, while the other axioms hold. (If one simply drops the parallel postulate from the list of axioms then the result is the more general geometry called absolute geometry). Interestingly one of the results of the fifth Euclidean axiom being a non-logical axiom is that the three angles of a triangle do not by definition add to 180°. In Hyperbolic geometry the sum of the three angles are always less than 180 and can approach zero. Only under the umbrella of Euclidean geometry are then angles always portions of 180°. Another thing that was observed was that Euclid's five axioms are actually somewhat incomplete. For instance, one of his theorems is that any line segment is part of a triangle; he constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as third vertex. His axioms, however, do not guarantee that the circles actually intersect. Many revised systems of axioms were constructed, the most standard ones are Hilbert's axioms, Birkhoff's axioms, and Tarski's axioms. Tarski used his axioms to show Euclidean geometry is a complete decidable theory; that is, every proposition of Euclidean geometry can be shown to be either true or false. Euclid also had five "common notions" which can also be taken to be axioms, though he later used other properties of magnitudes. # Things which equal the same thing also equal one another. # If equals are added to equals, then the wholes are equal. # If equals are subtracted from equals, then the remainders are equal. # Things which coincide with one another equal one another. # The whole is greater than the part.

Modern introduction to Euclidean geometry

Today, Euclidean geometry is usually constructed rather than axiomatized, by means of analytic geometry. If one introduces geometry this way, one can then prove the Euclidean (or any other) axioms as theorems in this particular model. This does not have the beauty of the axiomatic approach, but it is extremely concise.

The construction

First let us define the set of points as set of pairs of real numbers

(x,y)

. Then given two points

P=(x,y)

and

Q=(z,t)

one can define distances using the following formula: :

|PQ|=\sqrt

. This is known as the Euclidean metric. All other notions as a straight line, angle, circle can be defined in terms of points as pairs of real numbers and the distances between them. For example straight line through

P

and

Q

can be defined as a set of points

A

such that the triangle

APQ

is degenerate, i.e. :

|PQ| =|PA|+|AQ| \mbox |PQ| =\pm(|PA|-|AQ|)

References

- Gödel, Escher, Bach: an Eternal Golden Braid, by Douglas Hofstadter, ISBN 0465026567 (page 91)

Classical theorems

- Ceva's theorem
- Heron's formula
- Nine-point circle
- Pythagorean theorem
- Tartaglia's formula

External links

- [http://agutie.homestead.com Geometry Step by Step from the Land of the Incas] by Antonio Gutierrez.
- [http://aleph0.clarku.edu/~djoyce/java/elements/toc.html Euclid's elements]
- [http://www.cut-the-knot.org/geometry.shtml Geometry] at cut-the-knot
-
- ko:유클리드 기하학 ja:ユークリッド幾何学

Geometry

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. It was one of the two fields of pre-modern mathematics, the other being the study of numbers. In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry.)

The earliest geometry

The earliest recorded beginnings of geometry may be traced to Ancient Egypt (see geometry in Egypt) and Ancient Babylon (see Babylonian mathematics) around 3000 B.C. 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. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras; the Egyptians had a correct formula for the volume of a frustum of a square pyramid; the Babylonians had a trigonometry table. Chinese culture at this same time period was equally advanced, so it is likely that they had an equally advanced mathematics, but no artifacts have survived from which we could learn about it. This may be partly due to their early use of paper, rather than clay tablets or stone, to record their achievements.

The Greek period (c. 600 B.C. – 600 A.D.)

The Greek Period must be considered in detail, since geometry, for most of its history, was what the Greeks made it. For the Ancient Greeks, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies “eternal forms”, or abstractions, of which physical objects are only approximations; and they developed the idea of an “axiomatic theory”, which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories.

Thales and Pythagoras

Thales (635-543 B.C.) of Ionia (now southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 B.C.) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and probably traveled to Babylon and Egypt. The theorem that bears his name was not his discovery, but he was the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers.

Plato

Plato (427-347 B.C.), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, “Let none enter here who are ignorant of geometry.” Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but a compass and straight edge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of the possible ruler and compass constructions, and three classic ruler-and-compass problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 B.C.), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century.

Euclid

Euclid (365?-275? B.C.), probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in the ideal axiomatic form. The treatise is not a compendium of all that the Greeks knew at the time about geometry; 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 it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt. The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read. # Any two points can be joined by a straight line. # Any finite straight line can be extended in a straight line. # A circle can be drawn with any center and any radius. # All right angles are equal to each other. # If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate). It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement “Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line.” This is called Playfair’s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks. The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid’s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.

Archimedes

Archimedes (287-212 B.C.), of Syracuse, Sicily, when it was a Greek city-state, was the greatest of the Greek mathematicians, and often named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

After Archimedes

After Archimedes, Greek mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Greek geometry. He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The Middle Ages, Renaissance, and Reformation

The great library of Alexandria was burned. There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later. Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign. The Islamic ascendency in the Middle East, north Africa, and Spain began about 640 A.D. Original Arab mathematics during this period was primarily algebraic rather than geometric, though there were important commentaries on geometry. Omar Khayyám, for example, was a geometer as well as a poet. Scholarship in Europe declined until even the great works of antiquity were lost to them, and survived only in the Islamic centers of learning. When Europe started to emerge from the intellectual darkness of the Middle Ages, the writers of Ancient Greece and Rome were rediscovered in Islamic libraries and translated from Arabic into Latin. Euclid’s Elements of Geometry was recovered, and the rigorous deductive methods of geometry were relearned. Development of geometry in the style of Euclid resumed, resulting in an abundance of new theorems and concepts, many of them very profound and elegant.

The 17th and early 18th centuries

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 coordinates and equations, by Rene 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. There had been some early work in this area by Greek geometers, notably Pappus (c. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788-1867). In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642-1727) and Gottfried Wilhelm von Leibniz (1646-1716). This was the beginning of a new field of mathematics now called analysis. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The late 18th and 19th centuries

Non-Euclidean geometry

The old problem of proving Euclid’s Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity. It remained to prove mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry. While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

Introduction of mathematical rigor

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by David Hilbert in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry). Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.

Analysis situs, or topology

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Eu

Sphere

Geometry

Equations

Terminology

Generalization to other dimensions

Generalization to metric spaces

Topology

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External links

Sphere (disambiguation)

Symmetrical

Mathematical model for symmetry

Non-isometric symmetry

Reflection symmetry

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Screw axis symmetry

Symmetry combinations

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Similarity vs. sameness

More on symmetry in geometry

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Generalization of symmetry

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Boundary

See also

Solid geometry

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Euclidean geometry

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Modern introduction to Euclidean geometry

The construction

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Classical theorems