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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...)

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a smooth surface, vaguely conical in shape and embedded in a basket-like mesh of points, rotates in empty space
Non-uniform rational B-splines (NURBS) are commonly used in computer graphics for generating and representing curves and surfaces for both analytic shapes (described by mathematical formulas) and modeled shapes. Here the shape of the surface is determined by control points, shown as small spheres surrounding the surface itself. The square at the bottom sets the maximum width and length of the surface. Based on early work by Pierre Bézier and Paul de Casteljau, NURBS are generalizations of both B-splines (basis splines) and Bézier curves and surfaces. Unlike simple Bézier curves and surfaces, which are non-rational, NURBS can represent exactly certain analytic shapes such as conic sections and spherical sections. They are widely used in computer-aided design (CAD), manufacturing (CAM), and engineering (CAE), although T-splines and subdivision surfaces may be more suitable for more complex organic shapes.

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  • Image 1 Alfred North Whitehead OM FRS FBA (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found application to a wide variety of disciplines, including ecology, theology, education, physics, biology, economics, and psychology, among other areas. In his early career Whitehead wrote primarily on mathematics, logic, and physics. His most notable work in these fields is the three-volume Principia Mathematica (1910–1913), which he wrote with former student Bertrand Russell. Principia Mathematica is considered one of the twentieth century
  • The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.
    The boundary of a Reuleaux triangle is a constant width curve based on an equilateral triangle. All points on a side are equidistant from the opposite vertex.
  • The complete graph K4 has ten matchings, corresponding to the value T(4) = 10 of the fourth telephone number.
    The complete graph K4 has ten matchings, corresponding to the value T(4) = 10 of the fourth telephone number.
  • Image 4In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ, ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two. Unless θ is a rational multiple of π, there will also be at least two distinct distances.This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi [hu], and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square. (Full article...)
    In mathematics, the three-gap theorem, three-distance theorem, or Steinhaus conjecture states that if one places n points on a circle, at angles of θ, 2θ, 3θ, ... from the starting point, then there will be at most three distinct distances between pairs of points in adjacent positions around the circle. When there are three distances, the largest of the three always equals the sum of the other two. Unless θ is a rational multiple of π, there will also be at least two distinct distances.

    This result was conjectured by Hugo Steinhaus, and proved in the 1950s by Vera T. Sós, János Surányi [hu], and Stanisław Świerczkowski; more proofs were added by others later. Applications of the three-gap theorem include the study of plant growth and musical tuning systems, and the theory of light reflection within a mirrored square. (Full article...)
  • As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.
    As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows sin x and its Taylor approximations by polynomials of degree 1, 3, 5, 7, 9, 11, and 13 at x = 0.
  • Schwarz lantern on display in the German Museum of Technology, Berlin
    Schwarz lantern on display in the German Museum of Technology, Berlin
  • Image 7Bernt Michael HolmboeBernt Michael Holmboe (23 March 1795 – 28 March 1850) was a Norwegian mathematician. He was home-tutored from an early age, and was not enrolled in school until 1810. Following a short period at the Royal Frederick University, which included a stint as assistant to Christopher Hansteen, Holmboe was hired as a mathematics teacher at the Christiania Cathedral School in 1818, where he met the future renowned mathematician Niels Henrik Abel. Holmboe's lasting impact on mathematics worldwide has been said to be his tutoring of Abel, both in school and privately. The two became friends and remained so until Abel's early death. Holmboe moved to the Royal Frederick University in 1826, where he worked until his own death in 1850.Holmboe's significant impact on mathematics in the fledgling Norway was his textbook in two volumes for secondary schools. It was widely used, but faced competition from Christopher Hansteen's alternative offering, sparking what may have been Norway's first debate about school textbooks. (Full article...)
    HolmboeBM2.jpg
    Bernt Michael Holmboe

    Bernt Michael Holmboe (23 March 1795 – 28 March 1850) was a Norwegian mathematician. He was home-tutored from an early age, and was not enrolled in school until 1810. Following a short period at the Royal Frederick University, which included a stint as assistant to Christopher Hansteen, Holmboe was hired as a mathematics teacher at the Christiania Cathedral School in 1818, where he met the future renowned mathematician Niels Henrik Abel. Holmboe's lasting impact on mathematics worldwide has been said to be his tutoring of Abel, both in school and privately. The two became friends and remained so until Abel's early death. Holmboe moved to the Royal Frederick University in 1826, where he worked until his own death in 1850.

    Holmboe's significant impact on mathematics in the fledgling Norway was his textbook in two volumes for secondary schools. It was widely used, but faced competition from Christopher Hansteen's alternative offering, sparking what may have been Norway's first debate about school textbooks. (Full article...)
  • Image 8In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:
  
    
      
        
          ∑
          
            n
            =
            1
          
          
            ∞
          
        
        
          
            1
            n
          
        
        =
        1
        +
        
          
            1
            2
          
        
        +
        
          
            1
            3
          
        
        +
        
          
            1
            4
          
        
        +
        
          
            1
            5
          
        
        +
        ⋯
        .
      
    
    {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n))=1+{\frac {1}{2))+{\frac {1}{3))+{\frac {1}{4))+{\frac {1}{5))+\cdots .}
  
The first 
  
    
      
        n
      
    
    {\displaystyle n}
  
 terms of the series sum to approximately 
  
    
      
        ln
        ⁡
        n
        +
        γ
      
    
    {\displaystyle \ln n+\gamma }
  
, where 
  
    
      
        ln
      
    
    {\displaystyle \ln }
  
 is the natural logarithm and 
  
    
      
        γ
        ≈
        0.577
      
    
    {\displaystyle \gamma \approx 0.577}
  
 is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. (Full article...)
    In mathematics, the harmonic series is the infinite series formed by summing all positive unit fractions:


    The first terms of the series sum to approximately , where is the natural logarithm and is the Euler–Mascheroni constant. Because the logarithm has arbitrarily large values, the harmonic series does not have a finite limit: it is a divergent series. Its divergence was proven in the 14th century by Nicole Oresme using a precursor to the Cauchy condensation test for the convergence of infinite series. It can also be proven to diverge by comparing the sum to an integral, according to the integral test for convergence. (Full article...)
  • A unit cube with a hole cut through it, large enough to allow Prince Rupert's cube to pass
    A unit cube with a hole cut through it, large enough to allow Prince Rupert's cube to pass
  • Composite numbers can be arranged into rectangles but prime numbers cannot.
    Composite numbers can be arranged into rectangles but prime numbers cannot.
  • The red graph is the dual graph of the blue graph, and vice versa.
    The red graph is the dual graph of the blue graph, and vice versa.
  • Convergence of a convex curve to a circle under the curve-shortening flow. Inner curves (lighter color) are flowed versions of the outer curves. Time steps between curves are not uniform.
    Convergence of a convex curve to a circle under the curve-shortening flow. Inner curves (lighter color) are flowed versions of the outer curves. Time steps between curves are not uniform.

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Dodecahedron.svg
A dodecahedron, one of the five Platonic solids
Image credit: User:DTR

A regular polytope is a geometric figure with a high degree of symmetry. Examples in two dimensions include the square, the regular pentagon and hexagon, and so on. In three dimensions the regular polytopes include the cube, 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. Circles and 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. (Full article...)

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