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wallpaper group is a mathematical concept used to classify repetitive designs on two-dimensional surfaces, such as floors and walls, based on the
symmetries in the pattern. Such patterns occur frequently in
decorative art. The mathematical study of such patterns reveals that exactly 17 different types of pattern can occur.
Wallpaper groups are examples of an abstract algebraic structure known as a group. Groups are frequently used in mathematics to study the notion of symmetry. Wallpaper groups are related to the simpler frieze groups, and to the more complex three-dimensional crystallographic groups.
Catalan numbers, named for the
Eugène Charles Catalan, are a
natural numbers that are important in
combinatorial mathematics. The sequence begins:
The Catalan numbers are solutions to numerous counting problems which often have a recursive flavour. In fact, one author lists over 60 different possible interpretations of these numbers. For example, the nth Catalan number is the number of full binary trees with n internal nodes, or n+1 leaves. It is also the number of ways of associating n applications of a binary operator as well as the number of ways that a convex polygon with n + 2 sides can be cut into triangles by connecting vertices with straight lines.
|David Hilbert||David Hilbert (January 23, 1862, Wehlau, Prussia–February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He established his reputation as a great mathematician and scientist by inventing or developing a broad range of ideas, such as invariant theory, the axiomization of geometry, and the notion of Hilbert space, one of the foundations of functional analysis. Hilbert and his students supplied significant portions of the mathematic infrastructure required for quantum mechanics and general relativity. He is one of the founders of proof theory, mathematical logic, and the distinction between mathematics and metamathematics, and warmly defended Cantor's set theory and transfinite numbers. A famous example of his world leadership in mathematics is his 1900 presentation of a set of problems that set the course for much of the mathematical research of the 20th century.|
Leonhard Euler (pronounced oiler;
IPA /ˈɔɪlər/) (April 15, 1707
Switzerland - September 18, 1783
Russia) was a
physicist. He is considered to be the dominant mathematician of the
18th century and one of the greatest mathematicians of all time; he is certainly among the most prolific, with collected works filling over 70 volumes.
Euler developed many important concepts and proved numerous lasting theorems in diverse areas of mathematics, from calculus to number theory to topology. In the course of this work, he introduced many of modern mathematical terminologies, defining the concept of a function, and its notation, such as sin, cos, and tan for the trigonometric functions.
Euclid's Elements (
Greek: Στοιχεῖα) is a
geometric treatise, consisting of 13 books, written by the
Egypt during the early
3rd century BC. It comprises a collection of definitions, postulates (
axioms), propositions (
theorems) and proofs thereof. Euclid's books are in the fields of
Euclidean geometry, as well as the ancient
Greek version of
number theory. The Elements is one of the oldest extant axiomatic deductive treatments of
geometry, and has proven instrumental in the development of
logic and modern
It is considered one of the most successful textbooks ever written: the Elements was one of the very first books to go to press, and is second only to the Bible in number of editions published (well over 1000). For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century did it cease to be considered something all educated people had read. It is still (though rarely) used as a basic introduction to geometry today.
|Monty Hall problem||The
Monty Hall problem is a puzzle involving
probability similar to the American game show
Let's Make a Deal. The name comes from the show's host,
Monty Hall. A widely known, but problematic (see below) statement of the problem is from Craig F. Whitaker of
Columbia, Maryland in a letter to
Marilyn vos Savant's September 9, 1990, column in
Parade Magazine (as quoted by Bohl, Liberatore, and Nydick).
The problem is also called the Monty Hall paradox; it is a veridical paradox in the sense that the solution is counterintuitive, although the problem does not yield a logical contradiction.
|Trigonometric functions||The trigonometric functions are functions of an angle; they are most important when studying triangles and modeling periodic phenomena, among many other applications. They are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to positive and negative values and even to complex numbers.|
|Four color theorem||The
four color theorem states that given any plane separated into regions, such as a political map of the counties of a state, the regions may be colored using no more than four
colors in such a way that no two adjacent regions receive the same color. Two regions are called adjacent if they share a border segment, not just a point. "Color by Number" worksheets and exercises, which combine learning art and math for people of young ages, are a good example of the four color theorem.
It is often the case that using only three colors is inadequate. This applies already to the map with one region surrounded by three other regions (even though with an even number of surrounding countries three colors are enough) and it is not at all difficult to prove that five colors are sufficient to color a map.
The four color theorem was the first major theorem to be proven using a computer, and the proof is disputed by some mathematicians because it would be infeasible for a human to verify by hand (see computer-aided proof). Ultimately, in order to believe the proof, one has to have faith in the correctness of the compiler and hardware executing the program used for the proof.
The lack of mathematical elegance was another factor, and to paraphrase comments of the time, "a good mathematical proof is like a poem — this is a telephone directory!"
algorithm is a procedure (a finite
set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. The
computational complexity and efficient
implementation of the algorithm are important in computing, and this depends on suitable
Informally, the concept of an algorithm is often illustrated by the example of a recipe, although many algorithms are much more complex; algorithms often have steps that repeat ( iterate) or require decisions (such as logic or comparison). Algorithms can be composed to create more complex algorithms.
The concept of an algorithm originated as a means of recording procedures for solving mathematical problems such as finding the common divisor of two numbers or multiplying two numbers. The concept was formalized in 1936 through Alan Turing's Turing machines and Alonzo Church's lambda calculus, which in turn formed the foundation of computer science.
Most algorithms can be directly implemented by computer programs; any other algorithms can at least in theory be simulated by computer programs. In many programming languages, algorithms are implemented as functions or procedures.
manifold is an abstract
mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of
dimension is important. For example,
lines are one-dimensional, and
In a one-dimensional manifold (or one-manifold), every point has a neighborhood that looks like a segment of a line. Examples of one-manifolds include a line, a circle, and two separate circles. In a two-manifold, every point has a neighborhood that looks like a disk. Examples include a plane, the surface of a sphere, and the surface of a torus.
Manifolds are important objects in mathematics and physics because they allow more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces.
|Carl Friedrich Gauss||
Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a
scientist of profound genius who contributed significantly to many fields, including
optics. Known as "the prince of mathematicians" and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history's most influential mathematicians.
Gauss was a child prodigy, of whom there are many anecdotes pertaining to his astounding precocity while a mere toddler, and made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his magnum opus, at the age of twenty-one (1798), though it wasn't published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day.
regular polytope is a
geometric figure with a high degree of
symmetry. Examples in two
dimensions include the
square, the regular
hexagon, and so on. In three dimensions the regular polytopes include the
dodecahedron, and all other
Platonic solids. Other Platonic solids include the tetrahedron, the octahedron, the icosahedron. Examples exist in higher dimensions also, such as the 5-dimensional hendecatope.
spheres, although highly symmetric, are not considered
polytopes because they do not have flat faces. The strong symmetry of the regular polytopes gives them an
aesthetic quality that interests both non-mathematicians and mathematicians.
Many regular polytopes, at least in two and three dimensions, exist in nature and have been known since prehistory. The earliest surviving mathematical treatment of these objects comes to us from ancient Greek mathematicians such as Euclid. Indeed, Euclid wrote a systematic study of mathematics, publishing it under the title Elements, which built up a logical theory of geometry and number theory. His work concluded with mathematical descriptions of the five Platonic solids.
Georg Ferdinand Ludwig Philipp Cantor (December 3, 1845,
St. Petersburg, Russia – January 6, 1918,
Halle, Germany) was a German mathematician who is best known as the creator of
set theory. Cantor established the importance of
one-to-one correspondence between sets, defined
well-ordered sets, and proved that the
real numbers are "more numerous" than the
natural numbers. In fact,
Cantor's theorem implies the existence of an "infinity of infinities." He defined the
ordinal numbers, and their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware.
Cantor's work encountered resistance from mathematical contemporaries such as Leopold Kronecker and Henri Poincaré, and later from Hermann Weyl and L.E.J. Brouwer. Ludwig Wittgenstein raised philosophical objections. Nowadays, the vast majority of mathematicians who are neither constructivists nor finitists accept Cantor's work on transfinite sets and arithmetic, recognizing it as a major paradigm shift.
Cryptography (or cryptology) is derived from
Greek κρυπτός kryptós "hidden," and the verb γράφω gráfo "write". In modern times, it has become a branch of
information theory, as the
mathematical study of information and especially its transmission from place to place. The noted cryptographer
Ron Rivest has observed that "cryptography is about communication in the presence of adversaries." It is a central contributor to several fields:
information security and related issues, particularly,
access control. One of cryptography's primary purposes is hiding the meaning of messages, not usually the existence of such messages.
In modern times, cryptography also contributes to computer science. Cryptography is central to the techniques used in computer and network security for such things as access control and information confidentiality. Cryptography is also used in many applications encountered in everyday life; the security of ATM cards, computer passwords, and electronic commerce all depend on cryptography.
|Polar coordinate system||The
polar coordinate system is a
coordinate system in which
points are given by an
angle and a distance from a central point known as the pole (equivalent to the origin in the more familiar
Cartesian coordinate system). The polar coordinate system is used in many fields, including
robotics. It is especially useful in situations where the relationship between two points is most easily expressed in terms of angles and distance; in the Cartesian coordinate system, such a relationship can only be found through
trigonometric formulae. For many types of curves, a polar equation is the simplest means of representation of variables.
It is known that the Greeks used the concepts of angle and radius. The astronomer Hipparchus (190-120 BC) tabulated a table of chord functions giving the length of the chord for each angle, and there are references to his using polar coordinates in establishing stellar positions.
|Fermat's Last Theorem||
Fermat's Last Theorem is one of the most famous
theorems in the
history of mathematics. It states that:
Despite how closely the problem is related to the Pythagorean theorem, which has infinite solutions and hundreds of proofs, Fermat's subtle variation is much more difficult to prove. Still, the problem itself is easily understood even by schoolchildren, making it all the more frustrating and generating perhaps more incorrect proofs than any other problem in the history of mathematics.
The 17th-century mathematician Pierre de Fermat wrote in 1637 in his copy of Bachet's translation of the famous Arithmetica of Diophantus: "I have a truly marvelous proof of this proposition which this margin is too narrow to contain." However, no correct proof was found for 357 years, until it was finally proven using very deep methods by Andrew Wiles in 1995 (after a failed attempt a year before).
Blaise Pascal (pronounced [blez pɑskɑl]), (June 19, 1623 – August 19, 1662) was a
philosopher. He was a
child prodigy who was educated by his father. Pascal's earliest work was in the natural and applied
sciences where he made important contributions to the construction of mechanical
calculators, the study of
fluids, and clarified the concepts of
vacuum by generalizing the work of
Evangelista Torricelli. Pascal also wrote powerfully in defense of the
A mathematician of the first order, Pascal helped create two major new areas of research. He wrote a significant treatise on projective geometry at the age of sixteen and corresponded with Pierre de Fermat from 1654 on probability theory, strongly influencing the development of modern economics and social science.
Following a mystical experience in late 1654, he abandoned his scientific work and devoted himself to philosophy and theology. However, he had suffered from ill-health throughout his life and his new interests were ended by his early death two months after his 39th birthday.
Platonic solid is a
regular polyhedron. These are the three-dimensional analogs of the convex
regular polygons. There are precisely five such figures (shown on the left). The name of each figure is derived from the number of its faces: respectively 4, 6, 8, 12 and 20. They are unique in that the sides, edges and angles are all
Due to their aesthetic beauty and symmetry, the Platonic solids have been a favorite subject of geometers for thousands of years. They are named after the ancient Greek philosopher Plato who claimed the classical elements were constructed from the regular solids.
The Platonic solids have been known since antiquity. The five solids were certainly known to the ancient Greeks and there is evidence that these figures were known long before then. The neolithic people of Scotland constructed stone models of all five solids at least 1000 years before Plato.
Game theory is a branch of mathematics that is often used in the context of
economics. It studies strategic interactions between
agents. In strategic games, agents choose
strategies which will maximize their return, given the strategies the other agents choose. The essential feature is that it provides a formal modelling approach to social situations in which decision makers interact with other agents. Game theory extends the simpler optimisation approach developed in
The field of game theory came into being with the 1944 classic Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. A major center for the development of game theory was RAND Corporation where it helped to define nuclear strategies.
Game theory has played, and continues to play a large role in the social sciences, and is now also used in many diverse academic fields. Beginning in the 1970s, game theory has been applied to animal behaviour, including evolutionary theory. Many games, especially the prisoner's dilemma, are used to illustrate ideas in political science and ethics. Game theory has recently drawn attention from computer scientists because of its use in artificial intelligence and cybernetics.
real number denoted by the
0.999… is exactly
1. In other words, "0.999…" represents the same number as the symbol "1". Various
proofs of this identity have been formulated with varying
rigour, preferred development of the real numbers, background assumptions, historical context, and target audience.
The equality has long been taught in textbooks, and in the last few decades, researchers of mathematics education have studied the reception of this equation among students, who often reject the equality. The students' reasoning is typically based on one of a few common erroneous intuitions about the real numbers; for example, a belief that each unique decimal expansion must correspond to a unique number, an expectation that infinitesimal quantities should exist, that arithmetic may be broken, an inability to understand limits or simply the belief that 0.999… should have a last 9. These ideas are false with respect to the real numbers, which can be proven by explicitly constructing the reals from the rational numbers, and such constructions can also prove that 0.999… = 1 directly.
|Infinite monkey theorem||The
infinite monkey theorem states that a
monkey hitting keys at
random on a
typewriter keyboard for an infinite amount of time will almost surely type or create a particular chosen text, such as the complete works of
William Shakespeare. Note that "
almost surely" in this context is a mathematical term with a specific meaning, and that the "monkey" is not an actual monkey; rather, it is a vivid metaphor for an abstract device that produces an unending, random sequence of letters.
The theorem graphically illustrates the perils of reasoning about infinity by imagining a vast but finite number. If every atom in the visible universe were a monkey producing a billion keystrokes a second from the Big Bang until today, it is still very unlikely that any monkey would get as far as "slings and arrows" in Hamlet's most famous soliloquy. The infinite monkey theorem is straightforward to prove, even without appealing to more advanced results.
number is an
abstract object that represents a count or
symbol for a number is called a
arithmetical operations of numbers, such as
division, are generalized in the branch of
abstract algebra, the study of abstract number systems such as
Numbers can be classified into sets called number systems. The most familiar numbers are the natural numbers, which to some mean the non-negative integers and to others mean the positive integers. In everyday parlance the non-negative integers are commonly referred to as whole numbers, the positive integers as counting numbers, symbolised by . Mathematics is used in many classes throughout the course of one's education.
The integers consist of the natural numbers (positive whole numbers and zero) combined with the negative whole numbers, which are symbolised by (from the German Zahl, meaning "number").
A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator. Fractions can be positive, negative, or zero. The set of all fractions includes the integers, since every integer can be written as a fraction with denominator 1. The symbol for the rational numbers is a bold face (for quotient).
Alan Mathison Turing,
OBE (June 23, 1912 – June 7, 1954), was an
Turing is often considered to be the father of modern computer science. Turing provided an influential formalisation of the concept of the algorithm and computation with the Turing machine, formulating the now widely accepted "Turing" version of the Church–Turing thesis, namely that any practical computing model has either the equivalent or a subset of the capabilities of a Turing machine. With the Turing test, he made a significant and characteristically provocative contribution to the debate regarding artificial intelligence: whether it will ever be possible to say that a machine is conscious and can think. He later worked at the National Physical Laboratory, creating one of the first designs for a stored-program computer, although it was never actually built. In 1947 he moved to the University of Manchester to work, largely on software, on the Manchester Mark I then emerging as one of the world's earliest true computers.
During World War II, Turing worked at Bletchley Park, Britain's codebreaking centre, and was for a time head of Hut 8, the section responsible for German Naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine which could find settings for the Enigma machine.
|Eigenvalues and eigenvectors||In
eigenvector of a
transformation is a
vector, different from the zero vector, which that transformation simply multiplies by a constant factor, called the eigenvalue of that vector. Often, a transformation is completely described by its eigenvalues and eigenvectors. The eigenspace for a factor is the
set of eigenvectors with that factor as eigenvalue, together with the zero vector.
In the specific case of linear algebra, the eigenvalue problem is this: given an n by n matrix A, what nonzero vectors x in exist, such that Ax is a scalar multiple of x?
The scalar multiple is denoted by the Greek letter λ and is called an eigenvalue of the matrix A, while x is called the eigenvector of A corresponding to λ. These concepts play a major role in several branches of both pure and applied mathematics — appearing prominently in linear algebra, functional analysis, and to a lesser extent in nonlinear situations.
It is common to prefix any natural name for the vector with eigen instead of saying eigenvector. For example, eigenfunction if the eigenvector is a function, eigenmode if the eigenvector is a harmonic mode, eigenstate if the eigenvector is a quantum state, and so on. Similarly for the eigenvalue, e.g. eigenfrequency if the eigenvalue is (or determines) a frequency.
Riemann hypothesis, first formulated by
Bernhard Riemann in 1859, is one of the most famous unsolved problems. It has been an open question for well over a century, despite attracting concentrated efforts from many outstanding mathematicians.
The Riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta-function ζ(s). The Riemann zeta-function is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers (i.e. at s=-2, s=-4, s=-6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with the non-trivial zeros, and states that:
Thus the non-trivial zeros should lie on the so-called critical line ½ + it with t a real number and i the imaginary unit. The Riemann zeta-function along the critical line is sometimes studied in terms of the Z-function, whose real zeros correspond to the zeros of the zeta-function on the critical line.
The Riemann hypothesis is one of the most important open problems in contemporary mathematics; a $1,000,000 prize has been offered by the Clay Mathematics Institute for a proof. Most mathematicians believe the Riemann hypothesis to be true. ( J. E. Littlewood and Atle Selberg have been reported as skeptical. Selberg's skepticism, if any, waned, from his young days. In a 1989 paper, he suggested that an analogue should hold for a much wider class of functions, the Selberg class.)
Johannes Kepler (1571 – 1630) was an
astronomer and a key figure in the 17th century astronomical revolution. He is best known for his
laws of planetary motion, based on his works
Astronomia nova and
Harmonice Mundi; Kepler's laws provided one of the foundations of
Isaac Newton's theory of
universal gravitation. Before Kepler, planets' paths were computed by combinations of the circular motions of the
celestial orbs; after Kepler astronomers shifted their attention from orbs to
orbits—paths that could be represented mathematically as an
During his career Kepler was a mathematics teacher at a Graz seminary school (later the University of Graz, Austria), an assistant to Tycho Brahe, court mathematician to Emperor Rudolf II, mathematics teacher in Linz, Austria, and adviser to General Wallenstein. He also did fundamental work in the field of optics and helped to legitimize the telescopic discoveries of his contemporary Galileo Galilei.
Kepler lived in an era when there was no clear distinction between astronomy and astrology, while there was a strong division between astronomy (a branch of mathematics within the liberal arts) and physics (a branch of the more prestigious discipline of philosophy).
Hilbert space is a
vector space with a
Hermitian form, that is
complete under its
norm. Thus it is an
inner product space, which means that it has notions of
distance and of
angle (especially the notion of
orthogonality or perpendicularity). The completeness requirement ensures that for infinite dimensional Hilbert spaces the
limits exist when expected, which facilitates various definitions from
calculus. A typical example of a Hilbert space is the space of square summable sequences.
Hilbert spaces allow simple geometric concepts, like projection and change of basis to be applied to infinite dimensional spaces, such as function spaces. They provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics.
|e (mathematical constant)||
The mathematical constant e is occasionally called Euler's number after the
Leonhard Euler, or Napier's constant in honor of the
John Napier who introduced
logarithms. It is one of the most important numbers in mathematics, alongside the additive and multiplicative identities
imaginary unit i, and
π, the circumference to diameter ratio for any circle. It has a number of equivalent definitions. One is given in the caption of the image to the right, and three more are:
The number e is also the base of the natural logarithm. Since e is transcendental, and therefore irrational, its value can not be given exactly. The numerical value of e truncated to 20 decimal places is 2.71828 18284 59045 23536.
|Graph (discrete mathematics)||Informally speaking, a
graph is a set of objects called points, nodes, or vertices connected by links called lines or edges. In a proper graph, which is by default undirected, a line from point A to point B is considered to be the same thing as a line from point B to point A. In a digraph, short for directed graph, the two directions are counted as being distinct arcs or directed edges. Typically, a graph is depicted in diagrammatic form as a set of dots (for the points, vertices, or nodes), joined by curves (for the lines or edges). Graphs have applications in both
computer science, and form the basic object of study in
Applications of graph theory are generally concerned with labeled graphs and various specializations of these. Many problems of practical interest can be represented by graphs. The link structure of a website could be represented by a directed graph: the vertices are the web pages available at the website and a directed edge from page A to page B exists if and only if A contains a link to B. A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or weighted graphs, are used to represent structures in which pairwise connections have some numerical values. For example if a graph represents a road network, the weights could represent the length of each road. A digraph with weighted edges in the context of graph theory is called a network. Networks have many uses in the practical side of graph theory, network analysis (for example, to model and analyze traffic networks).
|Homotopy groups of spheres||The
homotopy groups of spheres describe the different ways
spheres of various dimensions can be wrapped around each other. They are studied as part of
algebraic topology. The topic can be hard to understand because the most interesting and surprising results involve spheres in higher dimensions. These are defined as follows: an n-dimensional sphere, n-sphere, consists of all the points in a space of n+1 dimensions that are a fixed distance from a center point. This definition is a generalization of the familiar
circle (1-sphere) and
The goal of algebraic topology is to categorize or classify topological spaces. Homotopy groups were invented in the late 19th century as a tool for such classification, in effect using the set of mappings from a c-sphere into a space as a way to probe the structure of that space. An obvious question was how this new tool would work on n-spheres themselves. No general solution to this question has been found to date, but many homotopy groups of spheres have been computed and the results are surprisingly rich and complicated. The study of the homotopy groups of spheres has led to the development of many powerful tools used in algebraic topology.
quadratic equation is a
degree two. The general form is
where a ≠ 0 (if a = 0, then the equation becomes a linear equation). The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term.
If the discriminant , then the quadratic equation has two distinct real solutions; if , the equation has two real solutions which are equal; if , the equation has two complex solutions.
continuum hypothesis is a
hypothesis, advanced by
Georg Cantor, about the possible sizes of
infinite sets. Cantor introduced the concept of
cardinality to compare the sizes of infinite sets, and he showed that the set of
integers is strictly smaller than the set of
real numbers. The continuum hypothesis states the following:
This is equivalent to:
The real numbers have also been called the continuum, hence the name.
Euclidean geometry is a mathematical system attributed to the
Alexandria. Euclid's text
Elements was the first systematic discussion of
geometry. It has been one of the most influential books in history, as much for its method as for its mathematical content. The method consists of assuming a small set of intuitively appealing
axioms, and then proving many other
theorems) from those axioms. Although many of Euclid's results had been stated by earlier Greek mathematicians, Euclid was the first to show how these propositions could fit together into a comprehensive deductive and logical system.
The Elements begin with plane geometry, still often taught in secondary school as the first axiomatic system and the first examples of formal proof. The Elements goes on to the solid geometry of three dimensions, and Euclidean geometry was subsequently extended to any finite number of dimensions. Much of the Elements states results of what is now called number theory, proved using geometrical methods.
For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute sense. Today, however, many other self-consistent geometries are known, the first ones having been discovered in the early 19th century. It also is no longer taken for granted that Euclidean geometry describes physical space. An implication of Einstein's theory of general relativity is that Euclidean geometry is only a good approximation to the properties of physical space if the gravitational field is not too strong.
Knot theory is the branch of
topology that studies
mathematical knots, which are defined as
embeddings of a
circle S1 in 3-dimensional
Euclidean space, R3. This is basically equivalent to a conventional
knotted string with the ends of the string joined together to prevent it from becoming undone. Two mathematical knots are considered equivalent if one can be transformed into the other via continuous deformations (known as
ambient isotopies); these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.
Knots can be described in various ways, but the most common method is by planar diagrams (known as knot projections or knot diagrams). Given a method of description, a knot will have many descriptions, e.g., many diagrams, representing it. A fundamental problem in knot theory is determining when two descriptions represent the same knot. One way of distinguishing knots is by using a knot invariant, a "quantity" which remains the same even with different descriptions of a knot.
Research in knot theory began with the creation of knot tables and the systematic tabulation of knots. While tabulation remains an important task, today's researchers have a wide variety of backgrounds and goals. Classical knot theory, as initiated by Max Dehn, J. W. Alexander, and others, is primarily concerned with the knot group and invariants from homology theory such as the Alexander polynomial.
The discovery of the Jones polynomial by Vaughan Jones in 1984, and subsequent contributions from Edward Witten, Maxim Kontsevich, and others, revealed deep connections between knot theory and mathematical methods in statistical mechanics and quantum field theory. A plethora of knot invariants have been invented since then, utilizing sophisticated tools as quantum groups and Floer homology.
Riemann sphere is a way of extending the
complex numbers with one additional
point at infinity, in a way that makes expressions such as
On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field. However, the Riemann sphere is geometrically and analytically well-behaved, even near infinity; it is a one- dimensional complex manifold, also called a Riemann surface.
In complex analysis, the Riemann sphere facilitates an elegant theory of meromorphic functions. The Riemann sphere is ubiquitous in projective geometry and algebraic geometry as a fundamental example of a complex manifold, projective space, and algebraic variety. It also finds utility in other disciplines that depend on analysis and geometry, such as quantum mechanics and other branches of physics.
fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole". The term was coined by
Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured".
A fractal as a geometric object generally has the following features:
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics. Fractals, when zoomed in, will keep showing more and more of itself, and it keeps going for infinity.
map projection is any method used in
cartography (mapmaking) to represent the
surface of the
earth or other bodies. The term "projection" here refers to any
function defined on the earth's surface and with values on the plane, and not necessarily a
Flat maps could not exist without map projections, because a sphere cannot be laid flat over a plane without distortions. One can see this mathematically as a consequence of Gauss's Theorema Egregium. Flat maps can be more useful than globes in many situations: they are more compact and easier to store; they readily accommodate an enormous range of scales; they are viewed easily on computer displays; they can facilitate measuring properties of the terrain being mapped; they can show larger portions of the earth's surface at once; and they are cheaper to produce and transport. These useful traits of flat maps motivate the development of map projections.
mathematics and the
arts, two quantities are in the
golden ratio if the
ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The
golden ratio is a
mathematical constant, usually denoted by the
Greek letter φ (
Expressed algebraically, two quantities a and b (assuming ) are therefore in the golden ratio if
which is approximately equal to 1.6180339887.
At least since the Renaissance, many artists and architects have proportioned their works to approximate the golden ratio—especially in the form of the golden rectangle, in which the ratio of the longer side to the shorter is the golden ratio—believing this proportion to be aesthetically pleasing. Mathematicians have studied the golden ratio because of its unique and interesting properties.
Other names frequently used for or closely related to the golden ratio are golden section (Latin: sectio aurea), golden mean, golden number, divine proportion (Italian: proporzionedivina), divine section (Latin: sectio divina), golden proportion, golden cut, and mean of Phidias.
stereographic projection is a particular mapping (
function) that projects a
sphere onto a
plane. The projection is defined on the entire sphere, except at one point: the projection point. Where it is defined, the mapping is
bijective. It is
conformal, meaning that it preserves
angles. It is neither
isometric nor area-preserving: that is, it preserves neither distances nor the areas of figures.
Intuitively, then, the stereographic projection is a way of picturing the sphere as the plane, with some inevitable compromises. Because the sphere and the plane appear in many areas of mathematics and its applications, so does the stereographic projection; it finds use in diverse fields including complex analysis, cartography, geology, and photography.
|Banach-Tarski Paradox||The Banach–Tarski paradox is a theorem in set-theoretic geometry which states that a solid ball in 3-dimensional space can be split into a finite number of non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball. The reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are complicated: they are not usual solids but infinite scatterings of points. A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball) — solid in the sense of the continuum — either one can be reassembled into the other. This is often stated colloquially as "a pea can be chopped up and reassembled into the Sun".|