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Portal:Mathematics - Wikipedia

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Mathematics is the study of representing and reasoning about abstract objects (such as numbers, points, spaces, sets, structures, and games). Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. (Full article...)

• Image 2
  Richard Phillips Feynman

  (

  ; May 11, 1918 – February 15, 1988) was an American

  theoretical physicist

  . He is best known for his work in the

  path integral formulation

  of

  quantum mechanics

  , the theory of

  quantum electrodynamics

  , the physics of the

  superfluidity

  of supercooled

  liquid helium

  , and in

  particle physics

  , for which he proposed the

  parton model

  . For his contributions to the development of quantum electrodynamics, Feynman received the

  Nobel Prize in Physics

  in 1965 jointly with

  Julian Schwinger

  and

  Shin'ichirō Tomonaga

  .

  Feynman developed a pictorial representation scheme for the mathematical expressions describing the behavior of subatomic particles, which later became known as Feynman diagrams and is widely used. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World, he was ranked the seventh-greatest physicist of all time. (Full article...)

• Image 3 Plots of logarithm functions, with three commonly used bases. The special points log_b b = 1 are indicated by dotted lines, and all curves intersect in log_b 1 = 0.

  In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the 3rd power: 1000 = 10³ = 10 × 10 × 10. More generally, if x = b^y, then y is the logarithm of x to base b, written log_b x, so log₁₀ 1000 = 3. As a single-variable function, the logarithm to base b is the inverse of exponentiation with base b.

  The logarithm base 10 is called the decimal or common logarithm and is commonly used in science and engineering. The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics because of its very simple derivative. The binary logarithm uses base 2 and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written log x. (Full article...)

• Image 5
  

  One of Molyneux's celestial globes, which is displayed in

  Middle Temple

  Library – from the frontispiece of the

  Hakluyt Society

  's 1889 reprint of

  A Learned Treatise of Globes, both Cœlestiall and Terrestriall

  , one of the English editions of Robert Hues'

  Latin

  work

  Tractatus de Globis

  (1594)

  
  Emery Molyneux

  (

  EM-ər-ee MOL-in-oh

  ; died June 1598) was an

  English Elizabethan

  maker of

  globes

  ,

  mathematical instruments

  and

  ordnance

  . His terrestrial and celestial globes, first published in 1592, were the first to be made in England and the first to be made by an Englishman.

  Molyneux was known as a mathematician and maker of mathematical instruments such as compasses and hourglasses. He became acquainted with many prominent men of the day, including the writer Richard Hakluyt and the mathematicians Robert Hues and Edward Wright. He also knew the explorers Thomas Cavendish, Francis Drake, Walter Raleigh and John Davis. Davis probably introduced Molyneux to his own patron, the London merchant William Sanderson, who largely financed the construction of the globes. When completed, the globes were presented to Elizabeth I. Larger globes were acquired by royalty, noblemen and academic institutions, while smaller ones were purchased as practical navigation aids for sailors and students. The globes were the first to be made in such a way that they were unaffected by the humidity at sea, and they came into general use on ships. (Full article...)

• Image 6
  

  Damage from

  Hurricane Katrina

  in 2005. Actuaries need to estimate long-term levels of such damage in order to accurately price property insurance, set appropriate

  reserves

  , and design appropriate

  reinsurance

  and capital management strategies.

  An

  actuary

  is a professional with advanced mathematical skills who deals with the measurement and management of

  risk

  and uncertainty. These risks can affect both sides of the

  balance sheet

  and require

  asset management

  ,

  liability

  management, and valuation skills. Actuaries provide assessments of financial security systems, with a focus on their complexity, their mathematics, and their mechanisms. The name of the corresponding academic discipline is

  actuarial science

  .

  While the concept of insurance dates to antiquity, the concepts needed to scientifically measure and mitigate risks have their origins in 17th-century studies of probability and annuities. Actuaries in the 21st century require analytical skills, business knowledge, and an understanding of human behavior and information systems; actuaries use this knowledge to design programs that manage risk, by determining if the implementation of strategies proposed for mitigating potential risks does not exceed the expected cost of those risks actualized. The steps needed to become an actuary, including education and licensing, are specific to a given country, with various additional requirements applied by regional administrative units; however, almost all processes impart universal principles of risk assessment, statistical analysis, and risk mitigation, involving rigorously structured training and examination schedules, taking many years to complete. (Full article...)

• Image 7
  Archimedes Thoughtful

  by

  Fetti

  (1620)

  
  Archimedes of Syracuse

  (

  AR-kih-MEE-deez

  ;

  c. 287

  – c.

   212 BC

  ) was an

  Ancient Greek mathematician

  ,

  physicist

  ,

  engineer

  ,

  astronomer

  , and

  inventor

  from the ancient city of

  Syracuse

  in

  Sicily

  . Although few details of his life are known, based on his surviving work, he is considered one of the leading scientists in

  classical antiquity

  , and one of the greatest mathematicians of all time. Archimedes anticipated modern

  calculus

  and

  analysis

  by applying the concept of the

  infinitesimals

  and the

  method of exhaustion

  to derive and rigorously prove many

  geometrical theorems

  , including the

  area of a circle

  , the

  surface area

  and

  volume

  of a

  sphere

  , the area of an

  ellipse

  , the area under a

  parabola

  , the volume of a segment of a

  paraboloid of revolution

  , the volume of a segment of a

  hyperboloid of revolution

  , and the area of a

  spiral

  .

  Archimedes' other mathematical achievements include deriving an approximation of pi (π), defining and investigating the Archimedean spiral, and devising a system using exponentiation for expressing very large numbers. He was also one of the first to apply mathematics to physical phenomena, working on statics and hydrostatics. Archimedes' achievements in this area include a proof of the law of the lever, the widespread use of the concept of center of gravity, and the enunciation of the law of buoyancy known as Archimedes' principle. In astronomy, he made measurements of the apparent diameter of the Sun and the size of the universe. He is also said to have built a planetarium device that demonstrated the movements of the known celestial bodies, and may have been a precursor to the Antikythera mechanism. He is also credited with designing innovative machines, such as his screw pump, compound pulleys, and defensive war machines to protect his native Syracuse from invasion. (Full article...)

• Image 8

  The number

  π

  (

  ; spelled out as

  pi

  ) is a

  mathematical constant

  , approximately equal to 3.14159, that is the

  ratio

  of a

  circle

  's

  circumference

  to its

  diameter

  . It appears in many formulae across

  mathematics

  and

  physics

  , and some of these formulae are commonly used for defining

  π

  , to avoid relying on the definition of the

  length of a curve

  .

  The number π is an irrational number, meaning that it cannot be expressed exactly as a ratio of two integers, although fractions such as 22 7 {\displaystyle {\tfrac {22}{7}}} are commonly used to approximate it. Consequently, its decimal representation never ends, nor enters a permanently repeating pattern. It is a transcendental number, meaning that it cannot be a solution of an algebraic equation involving only finite sums, products, powers, and integers. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. The decimal digits of π appear to be randomly distributed, but no proof of this conjecture has been found. (Full article...)

• Image 10
  
  Leonhard Euler

  (

  OY-lər

  ; 15 April 1707 – 18 September 1783) was a Swiss

  polymath

  who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of

  graph theory

  and

  topology

  and made influential discoveries in many other branches of mathematics, such as

  analytic number theory

  ,

  complex analysis

  , and

  infinitesimal calculus

  . He also introduced much of modern mathematical terminology and

  notation

  , including the notion of a

  mathematical function

  . He is known for his work in

  mechanics

  ,

  fluid dynamics

  , optics, astronomy, and music theory. Euler has been called a "universal genius" who "was fully equipped with almost unlimited powers of imagination, intellectual gifts and extraordinary memory". He spent most of his adult life in

  Saint Petersburg

  , Russia, and in

  Berlin

  , then the capital of

  Prussia

  .

  Euler is credited for popularizing the Greek letter π {\displaystyle \pi } (lowercase pi) to denote the ratio of a circle's circumference to its diameter, as well as first using the notation f ( x ) {\displaystyle f(x)} for the value of a function, the letter i {\displaystyle i} to express the imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle \Sigma } (capital sigma) to express summations, the Greek letter Δ {\displaystyle \Delta } (capital delta) for finite differences, and lowercase letters to represent the sides of a triangle while representing the angles as capital letters. He gave the current definition of the constant e {\displaystyle e} , the base of the natural logarithm, now known as Euler's number. Euler made contributions to applied mathematics and engineering, such as his study of ships which helped navigation, his three volumes on optics contributed to the design of microscopes and telescopes, and he studied the bending of beams and the critical load of columns. (Full article...)

• Image 11 General relativity

  , also known as the

  general theory of relativity

  , and as

  Einstein's theory of gravity

  , is the

  geometric

  theory of

  gravitation

  published by

  Albert Einstein

  in 1915 and is the current description of gravitation in

  modern physics

  . General

  relativity

  generalizes

  special relativity

  and refines

  Newton's law of universal gravitation

  , providing a unified description of gravity as a geometric property of

  space

  and

  time

  , or four-dimensional

  spacetime

  . In particular, the

  curvature of spacetime

  is directly related to the

  energy

  and

  momentum

  of whatever is

  present, including

  matter

  and

  radiation

  . The relation is specified by the

  Einstein field equations

  , a system of second-order

  partial differential equations

  .

  Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitation in classical physics. These predictions concern the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light, and include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the Shapiro time delay and singularities/black holes. So far, all tests of general relativity have been shown to be in agreement with the theory. The time-dependent solutions of general relativity enable us to talk about the history of the universe and have provided the modern framework for cosmology, thus leading to the discovery of the Big Bang and cosmic microwave background radiation. Despite the introduction of a number of alternative theories, general relativity continues to be the simplest theory consistent with experimental data. (Full article...)

• Image 13 The weighing pans of this balance scale contain zero objects, divided into two equal groups.

  In mathematics,

  zero

  is an

  even number

  . In other words, its

  parity

  —the quality of an

  integer

  being even or odd—is even. This can be easily verified based on the definition of "even": zero is an integer

  multiple

  of

  2

  , specifically

  0 × 2

  . As a result, zero shares all the properties that characterize even numbers: for example, 0 is neighbored on both sides by odd numbers, any decimal integer has the same parity as its last digit—so, since 10 is even, 0 will be even, and if

  y

  is even then

  y + x

  has the same parity as

  x

  —indeed,

  0 + x

  and

  x

  always have the same parity.

  Zero also fits into the patterns formed by other even numbers. The parity rules of arithmetic, such as even − even = even, require 0 to be even. Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. Applications of this recursion from graph theory to computational geometry rely on zero being even. Not only is 0 divisible by 2, it is divisible by every power of 2, which is relevant to the binary numeral system used by computers. In this sense, 0 is the "most even" number of all. (Full article...)

• Image 14
  Josiah Willard Gibbs

  (

  ; February 11, 1839 – April 28, 1903) was an American mechanical engineer and scientist who made fundamental theoretical contributions to

  physics

  ,

  chemistry

  , and

  mathematics

  . His work on the applications of

  thermodynamics

  was instrumental in transforming

  physical chemistry

  into a rigorous deductive science. Together with

  James Clerk Maxwell

  and

  Ludwig Boltzmann

  , he created

  statistical mechanics

  (a term that he coined), explaining the

  laws of thermodynamics

  as consequences of the statistical properties of

  ensembles

  of the possible states of a physical system composed of many particles. Gibbs also worked on the application of

  Maxwell's equations

  to problems in

  physical optics

  . As a mathematician, he created modern

  vector calculus

  (independently of the British scientist

  Oliver Heaviside

  , who carried out similar work during the same period) and described the Gibbs phenomenon in the theory of Fourier analysis.

  In 1863, Yale University awarded Gibbs the first American doctorate in engineering. After a three-year sojourn in Europe, Gibbs spent the rest of his career at Yale, where he was a professor of mathematical physics from 1871 until his death in 1903. Working in relative isolation, he became the earliest theoretical scientist in the United States to earn an international reputation and was praised by Albert Einstein as "the greatest mind in American history". In 1901, Gibbs received what was then considered the highest honor awarded by the international scientific community, the Copley Medal of the Royal Society of London, "for his contributions to mathematical physics". (Full article...)

• Image 15

  In

  classical mechanics

  , the

  Laplace–Runge–Lenz vector

  (

  LRL vector

  ) is a

  vector

  used chiefly to describe the shape and orientation of the

  orbit

  of one

  astronomical body

  around another, such as a

  binary star

  or a planet revolving around a star. For

  two bodies interacting

  by

  Newtonian gravity

  , the LRL vector is a

  constant of motion

  , meaning that it is the same no matter where it is calculated on the orbit; equivalently, the LRL vector is said to be

  conserved

  . More generally, the LRL vector is conserved in all problems in which two bodies interact by a

  central force

  that varies as the

  inverse square

  of the distance between them; such problems are called

  Kepler problems

  .

  Thus the hydrogen atom is a Kepler problem, since it comprises two charged particles interacting by Coulomb's law of electrostatics, another inverse-square central force. The LRL vector was essential in the first quantum mechanical derivation of the spectrum of the hydrogen atom, before the development of the Schrödinger equation. However, this approach is rarely used today. (Full article...)

The

is an iconic example of a

in

. This three-dimensional

structure, resembling a

or

, reflects the long-term behavior of solutions to the

, a set of three

used by mathematician and meteorologist

as a simple description of fluid circulation in a shallow layer (of liquid or gas) uniformly heated from below and cooled from above. To be more specific, the figure is set in a three-dimensional coordinate system whose axes measure the rate of convection in the layer (

), the horizontal temperature variation (

), and the vertical temperature variation (

). As these quantities change over time, a path is traced out within the coordinate system reflecting a particular solution to the differential equations. Lorenz's analysis revealed that while all solutions are completely

, some choices of input parameters and initial conditions result in solutions showing complex, non-repeating patterns that are highly dependent on the exact values chosen. As stated by Lorenz in his 1963 paper

: "Two states differing by imperceptible amounts may eventually evolve into two considerably different states". He later coined the term "

" to describe the phenomenon. One implication is that computing such chaotic solutions to the Lorenz system (i.e., with a computer program) to arbitrary precision is not possible, as any real-world computer will have a limitation on the precision with which it can represent numerical values. The particular solution plotted in this animation is based on the parameter values used by Lorenz (

,

, and

, constants reflecting certain physical attributes of the fluid). Note that the animation repeatedly shows one solution plotted over a specific period of time; as previously mentioned, the true solution never exactly retraces itself. Not all solutions are chaotic, however. Some choices of parameter values result in solutions that tend toward

at a fixed point (as seen, for example, in

). Initially developed to describe atmospheric convection, the Lorenz equations also arise in simplified models for

,

and motors, and

.

• Image 1 Richard Wesley Hamming

  (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for

  computer engineering

  and telecommunications. His contributions include the

  Hamming code

  (which makes use of a

  Hamming matrix

  ), the

  Hamming window

  ,

  Hamming numbers

  ,

  sphere-packing

  (or

  Hamming bound

  ),

  Hamming graph

  concepts, and the

  Hamming distance

  .

  Born in Chicago, Hamming attended University of Chicago, University of Nebraska and the University of Illinois at Urbana–Champaign, where he wrote his doctoral thesis in mathematics under the supervision of Waldemar Trjitzinsky (1901–1973). In April 1945, he joined the Manhattan Project at the Los Alamos Laboratory, where he programmed the IBM calculating machines that computed the solution to equations provided by the project's physicists. He left to join the Bell Telephone Laboratories in 1946. Over the next fifteen years, he was involved in nearly all of the laboratories' most prominent achievements. For his work, he received the Turing Award in 1968, being its third recipient. (Full article...)

• Image 2
  
  José Félix Mendieta Villarroel

  (born 15 November 1958) is a Bolivian politician and trade unionist who served as a member of the

  Chamber of Deputies

  from

  Cochabamba

  , representing circumscription 28 from 2010 to 2015.

  Though educated in pedagogy, Mendieta spent most of his career in commercial driving, climbing the ranks of the sector's trade unions to eventually become general secretary of the Sacaba Mixed Motor Transport Union. Though traditionally conservative, under the leadership of figures like Mendieta, many of the country's drivers' unions were reoriented towards the left. (Full article...)

• Image 4 A unit cube with a hole cut through it, large enough to allow Prince Rupert's cube to pass

  In

  geometry

  ,

  Prince Rupert's cube

  is the largest

  cube

  that can pass through a hole cut through a unit

  cube

  without splitting it into separate pieces. Its side length is approximately 1.06, 6% larger than the side length 1 of the unit cube through which it passes. The problem of finding the largest square that lies entirely within a unit cube is closely related, and has the same solution.

  Prince Rupert's cube is named after Prince Rupert of the Rhine, who asked whether a cube could be passed through a hole made in another cube of the same size without splitting the cube into two pieces. A positive answer was given by John Wallis. Approximately 100 years later, Pieter Nieuwland found the largest possible cube that can pass through a hole in a unit cube. (Full article...)

• Image 5 The cover page of Ars Conjectandi

  Ars Conjectandi (Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Nicolaus I Bernoulli. The seminal work consolidated, apart from many combinatorial topics, many central ideas in probability theory, such as the very first version of the law of large numbers: indeed, it is widely regarded as the founding work of that subject. It also addressed problems that today are classified in the twelvefold way and added to the subjects; consequently, it has been dubbed an important historical landmark in not only probability but all combinatorics by a plethora of mathematical historians. The importance of this early work had a large impact on both contemporary and later mathematicians; for example, Abraham de Moivre.

  Bernoulli wrote the text between 1684 and 1689, including the work of mathematicians such as Christiaan Huygens, Gerolamo Cardano, Pierre de Fermat, and Blaise Pascal. He incorporated fundamental combinatorial topics such as his theory of permutations and combinations (the aforementioned problems from the twelvefold way) as well as those more distantly connected to the burgeoning subject: the derivation and properties of the eponymous Bernoulli numbers, for instance. Core topics from probability, such as expected value, were also a significant portion of this important work. (Full article...)

• Image 6
  Srinivasa Ramanujan AiyangarFRS

  (22 December 1887 – 26 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in

  pure mathematics

  , he made substantial contributions to

  mathematical analysis

  ,

  number theory

  ,

  infinite series

  , and

  continued fractions

  , including solutions to mathematical problems then considered unsolvable.

  Ramanujan initially developed his own mathematical research in isolation. According to Hans Eysenck, "he tried to interest the leading professional mathematicians in his work, but failed for the most part. What he had to show them was too novel, too unfamiliar, and additionally presented in unusual ways; they could not be bothered". Seeking mathematicians who could better understand his work, in 1913 he began a mail correspondence with the English mathematician G. H. Hardy at the University of Cambridge, England. Recognising Ramanujan's work as extraordinary, Hardy arranged for him to travel to Cambridge. In his notes, Hardy commented that Ramanujan had produced groundbreaking new theorems, including some that "defeated me completely; I had never seen anything in the least like them before", and some recently proven but highly advanced results. (Full article...)

• Image 8 Graph of log₂ x as a function of a positive real number x

  In mathematics, the binary logarithm (log₂ n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x,

  :

  x = log 2 ⁡ n ⟺ 2 x = n . {\displaystyle x=\log _{2}n\quad \Longleftrightarrow \quad 2^{x}=n.}

  For example, the binary logarithm of

  1

  is

  0

  , the binary logarithm of

  2

  is

  1

  , the binary logarithm of

  4

  is

  2

  , and the binary logarithm of

  32

  is

  5

  .

  The binary logarithm is the logarithm to the base 2 and is the inverse function of the power of two function. There are several alternatives to the log₂ notation for the binary logarithm; see the Notation section below. (Full article...)

• Image 10 Pell's equation for n = 2 and six of its integer solutions
  Pell's equation

  , also called the

  Pell–Fermat equation

  , is any

  Diophantine equation

  of the form

  x 2 − n y 2 = 1 , {\displaystyle x^{2}-ny^{2}=1,}

  where

  n

  is a given positive

  nonsquare integer

  , and integer solutions are sought for

  x

  and

  y

  . In

  Cartesian coordinates

  , the equation is represented by a

  hyperbola

  ; solutions occur wherever the curve passes through a point whose

  x

  and

  y

  coordinates are both integers, such as the

  trivial solution

  with

  x

  = 1 and

  y

  = 0.

  Joseph Louis Lagrange

  proved that, as long as

  n

  is not a

  perfect square

  , Pell's equation has infinitely many distinct integer solutions. These solutions may be used to accurately

  approximate

  the

  square root

  of

  n

  by

  rational numbers

  of the form

  x

  /

  y

  .

  This equation was first studied extensively in India starting with Brahmagupta, who found an integer solution to 92 x 2 + 1 = y 2 {\displaystyle 92x^{2}+1=y^{2}} in his Brāhmasphuṭasiddhānta circa 628. Bhaskara II in the 12th century and Narayana Pandit in the 14th century both found general solutions to Pell's equation and other quadratic indeterminate equations. Bhaskara II is generally credited with developing the chakravala method, building on the work of Jayadeva and Brahmagupta. Solutions to specific examples of Pell's equation, such as the Pell numbers arising from the equation with n = 2, had been known for much longer, since the time of Pythagoras in Greece and a similar date in India. William Brouncker was the first European to solve Pell's equation. The name of Pell's equation arose from Leonhard Euler mistakenly attributing Brouncker's solution of the equation to John Pell. (Full article...)

• Image 11
  
  Albert Einstein

  (14 March 1879 – 18 April 1955) was a German-born

  theoretical physicist

  who is best known for developing the

  theory of relativity

  . Einstein also made important contributions to

  quantum mechanics

  . His

  mass–energy equivalence

  formula

  E = mc²

  , which arises from

  special relativity

  , has been called "the world's most famous equation". He received the 1921

  Nobel Prize in Physics

  for

  his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect

  .

  Born in the German Empire, Einstein moved to Switzerland in 1895, forsaking his German citizenship (as a subject of the Kingdom of Württemberg) the following year. In 1897, at the age of seventeen, he enrolled in the mathematics and physics teaching diploma program at the Swiss federal polytechnic school in Zurich, graduating in 1900. He acquired Swiss citizenship a year later, which he kept for the rest of his life, and afterwards secured a permanent position at the Swiss Patent Office in Bern. In 1905, he submitted a successful PhD dissertation to the University of Zurich. In 1914, he moved to Berlin to join the Prussian Academy of Sciences and the Humboldt University of Berlin, becoming director of the Kaiser Wilhelm Institute for Physics in 1917; he also became a German citizen again, this time as a subject of the Kingdom of Prussia. In 1933, while Einstein was visiting the United States, Adolf Hitler came to power in Germany. Horrified by the Nazi persecution of his fellow Jews, he decided to remain in the US, and was granted American citizenship in 1940. On the eve of World War II, he endorsed a letter to President Franklin D. Roosevelt alerting him to the potential German nuclear weapons program and recommending that the US begin similar research. (Full article...)

• ... that the role of the British Mobile Defence Corps was to carry out rescue work in the aftermath of a nuclear attack?
• ... that Carmel Naughton, having been told that girls were "stupid and couldn't do maths", sponsored a STEM scholarship fund?
• ... that Ewa Ligocka cooked another mathematician's goose?
• ... that Fairleigh Dickinson's upset victory over Purdue was the biggest upset in terms of point spread in NCAA tournament history, with Purdue being a 23+1⁄2-point favorite?
• ... that in the aftermath of the American Civil War, the only Black-led organization providing teachers to formerly enslaved people was the African Civilization Society?
• ... that in 1940 Xu Ruiyun became the first Chinese woman to receive a PhD in mathematics?
• ... that despite a mathematical model deeming the ice cream bar flavour Goody Goody Gum Drops impossible, it was still created?
• ... that people in Madagascar perform algebra on tree seeds in order to tell the future?

• ...that pi can be computed using only the number 2 by the work of Viète?
• … that the Riemann Hypothesis, one of the Millennium Problems, depends on the asymptotic growth of the Mertens Function?
• … that every positive integer can be written as the sum of three palindromic numbers in every number system with base 5 or greater?
• … that the best known lower bound for the length of the smallest superpermutation was first posted anonymously to the internet imageboard 4chan?
• ...that the mathematician Grigori Perelman was offered a Fields Medal in 2006, in part for his proof of the Poincaré conjecture, which he declined?
• ...that a regular heptagon is the regular polygon with the fewest sides which is not constructible with a compass and straightedge?
• ...that the regular trigonometric functions and the hyperbolic trigonometric functions can be related without using complex numbers through the Gudermannian function?

Showing 7 items out of 75

A 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 planes two-dimensional.

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. (Full article...)

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