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A full angle, also called a complete angle, round angle, or perigon, is an angle equal to radians corresponding to the central angle of an entire circle.Four right angles or two straightangles equal one full angle.

A right angle is an angle equal to half the angle from one end of a line segment to the other. A right angle is radians or . A triangle containing a right angle is called a right triangle. However, a triangle cannot contain more than one right angle, since the sum of the two right angles plus the third angle would exceed the total possessed by a triangle.The patterns of cracks observed in mud that has been dried by the sun form curves that often intersect in right angles (Williams 1979, p. 45; Steinhaus 1999, p. 88; Pearce 1990, p. 12).

The exterior angle bisectors (Johnson 1929, p. 149), also called the external angle bisectors (Kimberling 1998, pp. 18-19), of a triangle are the lines bisecting the angles formed by the sides of the triangles and their extensions, as illustrated above.Note that the exterior angle bisectors therefore bisect the supplementaryangles of the interior angles, not the entire exterior angles.There are therefore three pairs of oppositely oriented exterior angle bisectors. The exterior angle bisectors intersect pairwise in the so-called excenters , , and . These are the centers of the excircles, i.e., the three circles that are externally tangent to the sides of the triangle (or their extensions).The points determined on opposite sides of a triangle by an angle bisector from each vertex lie on a straight line if either (1) all or (2) one out of the three bisectors is an external angle bisector (Johnson 1929, p. 149; Honsberger..

An exterior angle of a polygon is the angle formed externally between two adjacent sides. It is therefore equal to , where is the corresponding internal angle between two adjacent sides (Zwillinger 1995, p. 270).Consider the angles formed between a side of a polygon and the extension of an adjacent side. Since there are two directions in which a side can be extended, there are two such angles at each vertex. However, since corresponding angles are opposite, they are also equal.Confusingly, a bisector of an angle is known as an exterior angle bisector, while a bisector of an angle (which is simply a line oriented in the opposite direction as the interior angle bisector) is not given any special name.The sum of the angles in a convex polygon is equal to radians (), since this corresponds to one complete rotation of the polygon...

The (interior) bisector of an angle, also called the internal angle bisector (Kimberling 1998, pp. 11-12), is the line or line segment that divides the angle into two equal parts.The angle bisectors meet at the incenter , which has trilinear coordinates 1:1:1.The length of the bisector of angle in the above triangle is given bywhere and .The points , , and have trilinear coordinates , , and , respectively, and form the vertices of the incentral triangle.

Given two intersecting lines or line segments, the amount of rotation about the point of intersection (the vertex) required to bring one into correspondence with the other is called the angle between them. The term "plane angle" is sometimes used to distinguish angles in a plane from solid angles measured in space (International Standards Organization 1982, p. 5).The term "angle" can also be applied to the rotational offset between intersecting planes about their common line of intersection, in which case the angle is called the dihedral angle of the planes.Angles are usually measured in degrees (denoted ), radians (denoted rad, or without a unit), or sometimes gradians (denoted grad).The concept of an angle can be generalized from the circle to the sphere, in which case it is known as solid angle. The fraction of a sphere subtended by an object (its solid angle) is measured in steradians, with the entire sphere..

The radian is a unit of angular measure defined such that an angle of one radian subtended from the center of a unit circle produces an arc with arc length 1.A full angle is therefore radians, so there are per radians, equal to or 57./radian. Similarly, a right angle is radians and a straight angle is radians.Radians are the most useful angular measure in calculus because they allow derivative and integral identities to be written in simple terms, e.g.,for measured in radians.Unless stated otherwise, all angular quantities considered in this work are assumed to be specified in radians.

A cone with elliptical cross section. The parametric equations for an elliptic cone of height , semimajor axis , and semiminor axis are(1)(2)(3)where and .The elliptic cone is a quadratic ruledsurface, and has volume(4)The coefficients of the first fundamental form(5)(6)(7)second fundamental form coefficients(8)(9)(10)The lateral surface area can then be calculated as(11)(12)(13)where is a complete elliptic integral of the second kind and assuming .The Gaussian curvature is(14)and the mean curvature is(15)

A conical frustum is a frustum created by slicing the top off a cone (with the cut made parallel to the base). For a right circular cone, let be the slant height and and the base and top radii. Then(1)The surface area, not including the top and bottomcircles, is(2)(3)The volume of the frustum is given by(4)But(5)so(6)(7)(8)This formula can be generalized to any pyramid by letting be the base areas of the top and bottom of the frustum. Then the volume can be written as(9)The area-weighted integral of over the frustum is(10)(11)so the geometric centroid is located alongthe z-axis at a height(12)(13)(Eshbach 1975, p. 453; Beyer 1987, p. 133; Harris and Stocker 1998, p. 105). The special case of the cone is given by taking , yielding .

The mapping of a grid of regularly ruled squares onto a cone with no overlap or misalignment. Cone nets are possible for vertex angles of , , and , where the dark edges in the upper diagrams above are joined. Beautiful photographs of cone net models (lower diagrams above) are presented in Steinhaus (1999). The transformation from a point in the grid plane to a point on the cone is given by(1)(2)(3)where , 1/2, or 3/4 is the fraction of a circle forming the base, and(4)(5)(6)

Two cones placed base-to-base.The bicone with base radius and half-height has surface area and volume(1)(2)The centroid is at the origin, and the inertia tensor about the centroid is given by(3)

For a logarithmic spiral given parametricallyas(1)(2)evolute is given by(3)(4)As first shown by Johann Bernoulli, the evolute of a logarithmic spiral is therefore another logarithmic spiral, having and ,In some cases, the evolute is identical to the original,as can be demonstrated by making the substitution to the new variable(5)Then the above equations become(6)(7)(8)(9)which are equivalent to the form of the original equation if(10)(11)(12)where only solutions with the minus sign in exist. Solving gives the values summarized in the following table.10.2744106319...20.1642700512...30.1218322508...40.0984064967...50.0832810611...60.0725974881...70.0645958183...80.0583494073...90.0533203211...100.0491732529...

The catacaustic of a logarithmic spiral, where the origin is taken as the radiant point, is another logarithmic spiral. For an original spiral with parametric equations(1)(2)the catacaustic with radiant point at the originis(3)(4)

The logarithmic spiral is a spiral whose polarequation is given by(1)where is the distance from the origin, is the angle from the x-axis, and and are arbitrary constants. The logarithmic spiral is also known as the growth spiral, equiangular spiral, and spira mirabilis. It can be expressed parametrically as(2)(3)This spiral is related to Fibonacci numbers, the golden ratio, and the golden rectangle, and is sometimes called the golden spiral.The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3).The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw..

Let the opposite sides of a convex cyclic hexagon be , , , , , and , and let the polygon diagonals , , and be so chosen that , , and have no common polygon vertex (and likewise for , , and ), thenThis is an extension of Ptolemy's theorem tothe hexagon.

The dual of Brianchon's theorem (Casey 1888, p. 146), discovered by B. Pascal in 1640 when he was just 16 years old (Leibniz 1640; Wells 1986, p. 69). It states that, given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line.In 1847, Möbius (1885) published the following generalization of Pascal's theorem: if all intersection points (except possibly one) of the lines prolonging two opposite sides of a -gon inscribed in a conic section are collinear, then the same is true for the remaining point.

The circle through the cusp of the arbelos and the tangent points of the first Pappus circle, which is congruent to the two Archimedes' circles. If and , then the radius of the Bankoff circle is

Draw the perpendicular line from the intersection of the two small semicircles in the arbelos. The two circles and tangent to this line, the large semicircle, and each of the two semicircles are then congruent and known as Archimedes' circles.For an arbelos with outer semicircle of unit radius and parameter , Archimedes' circles have radii(1)and centers(2)(3)Circles that are constructed in a natural way using an arbelos and are congruent to Archimedes' circles are known as Archimedean circles.

The term "arbelos" means shoemaker's knife in Greek, and this term is applied to the shaded area in the above figure which resembles the blade of a knife used by ancient cobblers (Gardner 1979). Archimedes himself is believed to have been the first mathematician to study the mathematical properties of this figure. The position of the central notch is arbitrary and can be located anywhere along the diameter.The arbelos satisfies a number of unexpected identities (Gardner 1979, Schoch). 1. Call the diameters of the left and right semicircles and , respectively, so the diameter of the enclosing semicircle is 1. Then the arc length along the bottom of the arbelos is(1)so the arc length along the enclosing semicircle is the same as the arc length along the two smaller semicircles. 2. Draw the perpendicular from the tangent of the two semicircles to the edge of the large circle. Then the area of the arbelos is the same as the area of the circle with..

Transfinite induction, like regular induction, is used to show a property holds for all numbers . The essential difference is that regular induction is restricted to the natural numbers , which are precisely the finite ordinal numbers. The normal inductive step of deriving from can fail due to limit ordinals.Let be a well ordered set and let be a proposition with domain . A proof by transfinite induction uses the following steps (Gleason 1991, Hajnal 1999): 1. Demonstrate is true. 2. Assume is true for all . 3. Prove , using the assumption in (2). 4. Then is true for all . To prove various results in point-set topology, Cantor developed the first transfinite induction methods in the 1880s. Zermelo (1904) extended Cantor's method with a "proof that every set can be well-ordered," which became the axiom of choice or Zorn's Lemma (Johnstone 1987). Transfinite induction and Zorn's lemma are often used interchangeably (Reid 1995), or are strongly..

The truth of an infinite sequence of propositions for , ..., is established if (1) is true, and (2) implies for all . This principle is sometimes also known as the method of induction.

According to G. Pólya, the method of finding geometric objects by intersection. 1. For example, the centers of all circles tangent to a straight line at a given point lie on a line that passes through and is perpendicular to . 2. In addition, the circle centered at with radius is the locus of the centers of all circles of radius passing through . The intersection of and consists of two points and which are the centers of two circles of radius tangent to at .Many constructions with straightedge and compass are based on this method, as, for example, the construction of the center of a given circle by means of the perpendicular bisector theorem.

Circumscribe a triangle about a circle, another circle around the triangle, a square outside the circle, another circle outside the square, and so on. The circumradius and inradius for an -gon are then related by(1)so an infinitely nested set of circumscribed polygons and circles has(2)(3)(4)Kasner and Newman (1989) and Haber (1964) state that , but this is incorrect, and the actual answer is(5)(OEIS A051762).By writing(6)it is possible to expand the series about infinity, change the order of summation, do the sum symbolically, and obtain the quickly converging series(7)where is the Riemann zeta function.Bouwkamp (1965) produced the following infinite productformulas for the constant,(8)(9)(10)where is the sinc function (cf. Prudnikov et al. 1986, p. 757), is the Riemann zeta function, and is the Dirichlet lambda function. Bouwkamp (1965) also produced the formula with accelerated convergence(11)where(12)(cited in Pickover..

A triangle is a 3-sided polygon sometimes (but not very commonly) called the trigon. Every triangle has three sides and three angles, some of which may be the same. The sides of a triangle are given special names in the case of a right triangle, with the side opposite the right angle being termed the hypotenuse and the other two sides being known as the legs. All triangles are convex and bicentric. That portion of the plane enclosed by the triangle is called the triangle interior, while the remainder is the exterior.The study of triangles is sometimes known as triangle geometry, and is a rich area of geometry filled with beautiful results and unexpected connections. In 1816, while studying the Brocard points of a triangle, Crelle exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?" (Wells 1991, p. 21).It is..

A Heronian triangle is a triangle having rational side lengths and rational area. The triangles are so named because such triangles are related to Heron's formula(1)giving a triangle area in terms of its side lengths , , and semiperimeter . Finding a Heronian triangle is therefore equivalent to solving the Diophantine equation(2)The complete set of solutions for integer Heronian triangles (the three side lengths and area can be multiplied by their least common multiple to make them all integers) were found by Euler (Buchholz 1992; Dickson 2005, p. 193), and parametric versions were given by Brahmagupta and Carmichael (1952) as(3)(4)(5)(6)(7)This produces one member of each similarity class of Heronian triangles for any integers , , and such that , , and (Buchholz 1992).The first few integer Heronian triangles sorted by increasing maximal side lengths, are ((3, 4, 5), (5, 5, 6), (5, 5, 8), (6, 8, 10), (10, 10, 12), (5, 12, 13), (10, 13,..

(1)The first few values are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, ... (OEISA001223). Rankin has shown that(2)for infinitely many and for some constant (Guy 1994). At a March 2003 meeting on elementary and analytic number in Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof that(3)(Montgomery 2003). Unfortunately, this proof turned out to be flawed.An integer is called a jumping champion if is the most frequently occurring difference between consecutive primes for some (Odlyzko et al.).

If is a member of a set , then is said to be an element of , written . If is not an element of , this is written .The term element also refers to a particular member of a group, or entry in a matrix or unevaluated determinant .

A subset is a portion of a set. is a subset of (written ) iff every member of is a member of . If is a proper subset of (i.e., a subset other than the set itself), this is written . If is not a subset of , this is written . (The notation is generally not used, since automatically means that and cannot be the same.)The subsets (i.e., power set) of a given set can befound using Subsets[list].An efficient algorithm for obtaining the next higher number having the same number of 1 bits as a given number (which corresponds to computing the next subset) is given by Gosper (1972) in PDP-10 assembler.The set of subsets of a set is called the power set of , and a set of elements has subsets (including both the set itself and the empty set). This follows from the fact that the total number of distinct k-subsets on a set of elements is given by the binomial sumFor sets of , 2, ... elements, the numbers of subsets are therefore 2, 4, 8, 16, 32, 64, ... (OEIS A000079). For example, the set..

Newton's iteration is an algorithm for computing the square root of a number via the recurrence equation(1)where . This recurrence converges quadratically as .Newton's iteration is simply an application of Newton'smethod for solving the equation(2)For example, when applied numerically, the first few iterations to Pythagoras's constant are 1, 1.5, 1.41667, 1.41422, 1.41421, ....The first few approximants , , ... to are given by(3)These can be given by the analytic formula(4)(5)These can be derived by noting that the recurrence can be written as(6)which has the clever closed-form solution(7)Solving for then gives the solution derived above.The following table summarizes the first few convergents for small positive integer OEIS, , ...11, 1, 1, 1, 1, 1, 1, 1, ...2A001601/A0510091, 3/2, 17/12, 577/408, 665857/470832, ...3A002812/A0715791, 2, 7/4, 97/56, 18817/10864, 708158977/408855776, .....

Min Max Min Max Re Im Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as(1)where and are the invariants of the Weierstrass elliptic function with modular discriminant(2)(Klein 1877). If , where is the upper half-plane, then(3)is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).Klein's absolute invariant is implemented in the WolframLanguage as KleinInvariantJ[tau].The function is the same as the j-function, modulo a constant multiplicative factor.Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).Klein's invariant can be given explicitly by(4)(5)(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function(6) is a Jacobi theta function, the are..

Underdamped simple harmonic motion is a special case of dampedsimple harmonic motion(1)in which(2)Since we have(3)it follows that the quantity(4)(5)is positive. Plugging in the trial solution to the differential equation then gives solutions that satisfy(6)i.e., the solutions are of the form(7)Using the Euler formula(8)this can be rewritten(9)We are interested in the real solutions. Since we are dealing here with a linear homogeneous ODE, linear sums of linearly independent solutions are also solutions. Since we have a sum of such solutions in (9), it follows that the imaginary and real parts separately satisfy the ODE and are therefore the solutions we seek. The constant in front of the sine term is arbitrary, so we can identify the solutions as(10)(11)so the general solution is(12)The initial values are(13)(14)so and can be expressed in terms of the initial conditions by(15)(16)The above plot shows an underdamped simple harmonic..

A method for solving ordinary differential equations using the formulawhich advances a solution from to . Note that the method increments a solution through an interval while using derivative information from only the beginning of the interval. As a result, the step's error is . This method is called simply "the Euler method" by Press et al. (1992), although it is actually the forward version of the analogous Euler backward method.While Press et al. (1992) describe the method as neither very accurate nor very stable when compared to other methods using the same step size, the accuracy is actually not too bad and the stability turns out to be reasonable as long as the so-called Courant-Friedrichs-Lewy condition is fulfilled. This condition states that, given a space discretization, a time step bigger than some computable quantity should not be taken. In situations where this limitation is acceptable, Euler's forward method becomes..

The power of a matrix for a nonnegative integer is defined as the matrix product of copies of ,A matrix to the zeroth power is defined to be the identity matrix of the same dimensions, . The matrix inverse is commonly denoted , which should not be interpreted to mean .

The product of two matrices and is defined as(1)where is summed over for all possible values of and and the notation above uses the Einstein summation convention. The implied summation over repeated indices without the presence of an explicit sum sign is called Einstein summation, and is commonly used in both matrix and tensor analysis. Therefore, in order for matrix multiplication to be defined, the dimensions of the matrices must satisfy(2)where denotes a matrix with rows and columns. Writing out the product explicitly,(3)where(4)(5)(6)(7)(8)(9)(10)(11)(12)Matrix multiplication is associative, as can be seenby taking(13)where Einstein summation is again used. Now, since , , and are scalars, use the associativity of scalar multiplication to write(14)Since this is true for all and , it must be true that(15)That is, matrix multiplication is associative. Equation(13) can therefore be written(16)without ambiguity. Due to associativity,..

The inverse of a square matrix , sometimes called a reciprocal matrix, is a matrix such that(1)where is the identity matrix. Courant and Hilbert (1989, p. 10) use the notation to denote the inverse matrix.A square matrix has an inverse iff the determinant (Lipschutz 1991, p. 45). The so-called invertible matrix theorem is major result in linear algebra which associates the existence of a matrix inverse with a number of other equivalent properties. A matrix possessing an inverse is called nonsingular, or invertible.The matrix inverse of a square matrix may be taken in the Wolfram Language using the function Inverse[m].For a matrix(2)the matrix inverse is(3)(4)For a matrix(5)the matrix inverse is(6)A general matrix can be inverted using methods such as the Gauss-Jordan elimination, Gaussian elimination, or LU decomposition.The inverse of a product of matrices and can be expressed in terms of and . Let(7)Then(8)and(9)Therefore,(10)so(11)where..

Denote the sum of two matrices and (of the same dimensions) by . The sum is defined by adding entries with the same indicesover all and . For example,Matrix addition is therefore both commutative andassociative.

The natural norm induced by the L-infty-normis called the maximum absolute row sum norm and is defined byfor a matrix . This matrix norm is implemented as Norm[m, Infinity].

The natural norm induced by the L1-normis called the maximum absolute column sum norm and is defined byfor a matrix . This matrix norm is implemented as MatrixNorm[m, 1] in the Wolfram Language package MatrixManipulation` .

Given a square complex or real matrix , a matrix norm is a nonnegative number associated with having the properties 1. when and iff , 2. for any scalar , 3. , 4. . Let , ..., be the eigenvalues of , then(1)The matrix -norm is defined for a real number and a matrix by(2)where is a vector norm. The task of computing a matrix -norm is difficult for since it is a nonlinear optimization problem with constraints.Matrix norms are implemented as Norm[m, p], where may be 1, 2, Infinity, or "Frobenius".The maximum absolute column sum norm is defined as(3)The spectral norm , which is the square root of the maximum eigenvalue of (where is the conjugate transpose),(4)is often referred to as "the" matrix norm.The maximum absolute row sum norm isdefined by(5), , and satisfy the inequality(6)..

The Gershgorin circle theorem (where "Gershgorin" is sometimes also spelled "Gersgorin" or "Gerschgorin") identifies a region in the complex plane that contains all the eigenvalues of a complex square matrix. For an matrix , define(1)Then each eigenvalue of is in at least one of the disks(2)The theorem can be made stronger as follows. Let be an integer with , and let be the sum of the magnitudes of the largest off-diagonal elements in column . Then each eigenvalue of is either in one of the disks(3)or in one of the regions(4)where is any subset of such that (Brualdi and Mellendorf 1994).

A procedure for decomposing an matrix into a product of a lower triangular matrix and an upper triangular matrix ,(1)LU decomposition is implemented in the WolframLanguage as LUDecomposition[m].Written explicitly for a matrix, the decomposition is(2)(3)This gives three types of equations (4)(5)(6)This gives equations for unknowns (the decomposition is not unique), and can be solved using Crout's method. To solve the matrix equation(7)first solve for . This can be done by forward substitution(8)(9)for , ..., . Then solve for . This can be done by back substitution(10)(11)for , ..., 1.

The Jordan matrix decomposition is the decomposition of a square matrix into the form(1)where and are similar matrices, is a matrix of Jordan canonical form, and is the matrix inverse of . In other words, is a similarity transformation of a matrix in Jordan canonical form. The proof that any square matrix can be brought into Jordan canonical form is rather complicated (Turnbull and Aitken 1932; Faddeeva 1958, p. 49; Halmos 1958, p. 112).Jordan decomposition is also associated with the matrix equation and the special case .The Jordan matrix decomposition is implemented in the Wolfram Language as JordanDecomposition[m], and returns a list s, j. Note that the Wolfram Language takes the Jordan block in the Jordan canonical form to have 1s along the superdiagonal instead of the subdiagonal. For example, a Jordan decomposition of(2)is given by(3)(4)..

Eigenvectors are a special set of vectors associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic vectors, proper vectors, or latent vectors (Marcus and Minc 1988, p. 144).The determination of the eigenvectors and eigenvalues of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvector is paired with a corresponding so-called eigenvalue. Mathematically, two different kinds of eigenvectors need to be distinguished: left eigenvectors and right eigenvectors. However, for many problems in physics and engineering, it is sufficient to consider only right eigenvectors. The term "eigenvector" used without qualification in such applications..

Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it is equivalent to matrix diagonalization and arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few. Each eigenvalue is paired with a corresponding so-called eigenvector (or, in general, a corresponding right eigenvector and a corresponding left eigenvector; there is no analogous distinction between left and right for eigenvalues).The decomposition of a square matrix into eigenvalues and eigenvectors is known in this work as eigen..

The matrix decomposition of a square matrix into so-called eigenvalues and eigenvectors is an extremely important one. This decomposition generally goes under the name "matrix diagonalization." However, this moniker is less than optimal, since the process being described is really the decomposition of a matrix into a product of three other matrices, only one of which is diagonal, and also because all other standard types of matrix decomposition use the term "decomposition" in their names, e.g., Cholesky decomposition, Hessenberg decomposition, and so on. As a result, the decomposition of a matrix into matrices composed of its eigenvectors and eigenvalues is called eigen decomposition in this work.Assume has nondegenerate eigenvalues and corresponding linearly independent eigenvectors which can be denoted(1)Define the matrices composed of eigenvectors(2)(3)and eigenvalues(4)where is a diagonal matrix...

A theorem sometimes called "Euclid's first theorem" or Euclid's principle states that if is a prime and , then or (where means divides). A corollary is that (Conway and Guy 1996). The fundamental theorem of arithmetic is another corollary (Hardy and Wright 1979).Euclid's second theorem states that the number of primes is infinite. This theorem, also called the infinitude of primes theorem, was proved by Euclid in Proposition IX.20 of the Elements (Tietze 1965, pp. 7-9). Ribenboim (1989) gives nine (and a half) proofs of this theorem. Euclid's elegant proof proceeds as follows. Given a finite sequence of consecutive primes 2, 3, 5, ..., , the number(1)known as the th Euclid number when is the th prime, is either a new prime or the product of primes. If is a prime, then it must be greater than the previous primes, since one plus the product of primes must be greater than each prime composing the product. Now, if is a product of primes,..

The distinct prime factors of a positive integer are defined as the numbers , ..., in the prime factorization(1)(Hardy and Wright 1979, p. 354).A list of distinct prime factors of a number can be computed in the Wolfram Language using FactorInteger[n][[All, 1]], and the number of distinct prime factors is implemented as PrimeNu[n].The first few values of for , 2, ... are 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, ... (OEIS A001221; Abramowitz and Stegun 1972, Kac 1959). This sequence is given by the inverse Möbius transform of , where is the characteristic function of the prime numbers (Sloane and Plouffe 1995, p. 22). The prime factorizations and distinct prime factors of the first few positive integers are listed in the table below.prime factorizationdistinct prime factors (A027748)1--0--221233134125515622, 377178129131022, 511111111222, 313131131422, 71523, 51612The numbers consisting only of distinct..

The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright 1979, pp. 2-3).This theorem is also called the unique factorization theorem. The fundamental theorem of arithmetic is a corollary of the first of Euclid's theorems (Hardy and Wright 1979).For rings more general than the complex polynomials , there does not necessarily exist a unique factorization. However, a principal ideal domain is a structure for which the proof of the unique factorization property is sufficiently easy while being quite general and common.

A prime factor is a factor that is prime, i.e., one that cannot itself be factored. In general, a prime factorization takes the form(1)where are prime factors and are their orders. Prime factorization can be performed in the Wolfram Language using the command FactorInteger[n], which returns a list of pairs.The following table gives the prime factorization for the positive integers .1111112131314141221222324233131323233343434142434445515253545616263646771717273737474781828384891919292939491020304050The number of not necessarily distinct prime factors of a number is denoted (Hardy and Wright 1979, p. 354) or . Conway et al. (2008) coined the term "multiprimality of " to describe , with semiprimes then being termed biprimes, numbers with three factors terms triprimes, etc. The number of prime factors is given in terms of the prime factorization above by(2)The first few values for , 2, ... are 0, 1, 1, 2, 1, 2,..

A factorization of the form(1)The factorization for was discovered by Aurifeuille, and the general form was subsequently discovered by Lucas. The large factors are sometimes written as and as follows:(2)(3)which can be written(4)(5)(6)where and(7)(8)(9)

The Pratt certificate is a primality certificate based on Fermat's little theorem converse. Prior to the work of Pratt (1975), the Lucas-Lehmer test had been regarded purely as a heuristic that worked a lot of the time (Knuth 1969). Pratt (1975) showed that Lehmer's primality heuristic could be made a nondeterministic procedure by applying it recursively to the factors of . As a consequence of this result, Pratt (1975) became the first to demonstrate that the resulting tree implies that prime factorization lies in the complexity class NP.To generate a Pratt certificate, assume that is a positive integer and is the set of prime factors of . Suppose there exists an integer (called a "witness") such that but (mod ) whenever is one of . Then Fermat's little theorem converse states that is prime (Wagon 1991, pp. 278-279).By applying Fermat's little theorem converse to and recursively to each purported factor of , a certificate for..

Given an arithmetic progression of terms , for , 2, ..., the series contains an infinite number of primes if and are relatively prime, i.e., . This result had been conjectured by Gauss (Derbyshire 2004, p. 96), but was first proved by Dirichlet (1837).Dirichlet proved this theorem using Dirichlet L-series, but the proof is challenging enough that, in their classic text on number theory, the usually explicit Hardy and Wright (1979) report "this theorem is too difficult for insertion in this book."

Riemann defined the function by(1)(2)(3)(Hardy 1999, p. 30; Borwein et al. 2000; Havil 2003, pp. 189-191 and 196-197; Derbyshire 2004, p. 299), sometimes denoted , (Edwards 2001, pp. 22 and 33; Derbyshire 2004, p. 298), or (Havil 2003, p. 189). Note that this is not an infinite series since the terms become zero starting at , and where is the floor function and is the base-2 logarithm. For , 2, ..., the first few values are 0, 1, 2, 5/2, 7/2, 7/2, 9/2, 29/6, 16/3, 16/3, ... (OEIS A096624 and A096625). As can be seen, when is a prime, jumps by 1; when it is the square of a prime, it jumps by 1/2; when it is a cube of a prime, it jumps by 1/3; and so on (Derbyshire 2004, pp. 300-301), as illustrated above.Amazingly, the prime counting function is related to by the Möbius transform(4)where is the Möbius function (Riesel 1994, p. 49; Havil 2003, p. 196; Derbyshire 2004, p. 302). More amazingly..

The Legendre symbol is a number theoretic function which is defined to be equal to depending on whether is a quadratic residue modulo . The definition is sometimes generalized to have value 0 if ,(1)If is an odd prime, then the Jacobi symbol reduces to the Legendre symbol. The Legendre symbol is implemented in the Wolfram Language via the Jacobi symbol, JacobiSymbol[a, p].The Legendre symbol obeys the identity(2)Particular identities include(3)(4)(5)(6)(Nagell 1951, p. 144), as well as the general(7)when and are both odd primes.In general,(8)if is an odd prime.

The least common multiple of two numbers and , variously denoted (this work; Zwillinger 1996, p. 91; Råde and Westergren 2004, p. 54), (Gellert et al. 1989, p. 25; Graham et al. 1990, p. 103; Bressoud and Wagon 2000, p. 7; D'Angelo and West 2000, p. 135; Yan 2002, p. 31; Bronshtein et al. 2007, pp. 324-325; Wolfram Language), l.c.m. (Andrews 1994, p. 22; Guy 2004, pp. 312-313), or , is the smallest positive number for which there exist positive integers and such that(1)The least common multiple of more than two numbers is similarly defined.The least common multiple of , , ... is implemented in the Wolfram Language as LCM[a, b, ...].The least common multiple of two numbers and can be obtained by finding the prime factorization of each(2)(3)where the s are all prime factors of and , and if does not occur in one factorization, then the corresponding exponent is taken as 0. The least..

The extended greatest common divisor of two integers and can be defined as the greatest common divisor of and which also satisfies the constraint for and given integers. It is used in solving linear Diophantine equations, and is implemented in the Wolfram Language as ExtendedGCD[m, n].

The Mangoldt function is the function defined by(1)sometimes also called the lambda function. has the explicit representation(2)where denotes the least common multiple. The first few values of for , 2, ..., plotted above, are 1, 2, 3, 2, 5, 1, 7, 2, ... (OEIS A014963).The Mangoldt function is implemented in the WolframLanguage as MangoldtLambda[n].It satisfies the divisor sums(3)(4)(5)(6)where is the Möbius function (Hardy and Wright 1979, p. 254).The Mangoldt function is related to the Riemann zeta function by(7)where (Hardy 1999, p. 28; Krantz 1999, p. 161; Edwards 2001, p. 50).The summatory Mangoldt function, illustratedabove, is defined by(8)where is the Mangoldt function, and is also known as the second Chebyshev function (Edwards 2001, p. 51). is given by the so-called explicit formula(9)for and not a prime or prime power (Edwards 2001, pp. 49, 51, and 53), and the sum is over all nontrivial..

The function(1)where is the number of not necessarily distinct prime factors of , with . The values of for , 2, ... are 1, , , 1, , 1, , , 1, 1, , , ... (OEIS A008836). The values of such that are 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, ... (OEIS A026424), while then values such that are 1, 4, 6, 9, 10, 14, 15, 16, 21, 22, 24, ... (OEIS A028260).The Liouville function is implemented in the WolframLanguage as LiouvilleLambda[n].The Liouville function is connected with the Riemannzeta function by the equation(2)(Lehman 1960). It has the Lambert series(3)(4)where is a Jacobi theta function.Consider the summatory function(5)the values of which for , 2, ... are 1, 0, , 0, , 0, , , , 0, , , , , , 0, , , , , ... (OEIS A002819).Lehman (1960) gives the formulas(6)and(7)where , , and are variables ranging over the positive integers, is the Möbius function, is Mertens function, and , , and are positive real numbers with .The conjecture that satisfies for is called the..

There are two definitions of the Carmichael function. One is the reduced totient function (also called the least universal exponent function), defined as the smallest integer such that for all relatively prime to . The multiplicative order of (mod ) is at most (Ribenboim 1989). The first few values of this function, implemented as CarmichaelLambda[n], are 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, ... (OEIS A002322).It is given by the formula(1)where are primaries.It can be defined recursively as(2)Some special values include(3)and(4)where is a primorial (S. M. Ruiz, pers. comm., Jul. 5, 2009).The second Carmichael's function is given by the least common multiple (LCM) of all the factors of the totient function , except that if , then is a factor instead of . The values of for the first few are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 2, 12, ... (OEIS A011773).This function has the special value(5)for an odd prime and ...

The Möbius function is a number theoretic function defined by(1)so indicates that is squarefree (Havil 2003, p. 208). The first few values of are therefore 1, , , 0, , 1, , 0, 0, 1, , 0, ... (OEIS A008683). Similarly, the first few values of for , 2, ... are 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, ... (OEIS A008966).The function was introduced by Möbius (1832), and the notation was first used by Mertens (1874). However, Gauss considered the Möbius function more than 30 years before Möbius, writing "The sum of all primitive roots [of a prime number ] is either (when is divisible by a square), or (mod ) (when is the product of unequal prime numbers; if the number of these is even the sign is positive but if the number is odd, the sign is negative)" (Gauss 1801, Pegg 2003).The Möbius function is implemented in the WolframLanguage as MoebiusMu[n].The summatory function of the Möbius function(2)is called the Mertens function.The..

The Mertens function is the summary function(1)where is the Möbius function (Mertens 1897; Havil 2003, p. 208). The first few values are 1, 0, , , , , , , , , , , ... (OEIS A002321). is also given by the determinant of the Redheffer matrix.Values of for , 1, 2, ... are given by 1, , 1, 2, , , 212, 1037, 1928, , ... (OEIS A084237; Deléglise and Rivat 1996).The following table summarizes the first few values of at which for various OEIS such that 13, 19, 20, 30, 33, 43, 44, 45, 47, 48, 49, 50, ...5, 7, 8, 9, 11, 12, 14, 17, 18, 21, 23, 24, 25, 29, ...3, 4, 6, 10, 15, 16, 22, 26, 27, 28, 35, 36, 38, ...0A0284422, 39, 40, 58, 65, 93, 101, 145, 149, 150, ...1A1186841, 94, 97, 98, 99, 100, 146, 147, 148, 161, ...295, 96, 217, 229, 335, 336, 339, 340, 345, 347, 348, ...3218, 223, 224, 225, 227, 228, 341, 342, 343, 344, 346, ...An analytic formula for is not known, although Titchmarsh (1960) showed that if the Riemann hypothesis holds and if there are no multiple Riemann..

The divisor function for an integer is defined as the sum of the th powers of the (positive integer) divisors of ,(1)It is implemented in the Wolfram Language as DivisorSigma[k, n].The notations (Hardy and Wright 1979, p. 239), (Ore 1988, p. 86), and (Burton 1989, p. 128) are sometimes used for , which gives the number of divisors of . Rather surprisingly, the number of factors of the polynomial are also given by . The values of can be found as the inverse Möbius transform of 1, 1, 1, ... (Sloane and Plouffe 1995, p. 22). Heath-Brown (1984) proved that infinitely often. The numbers having the incrementally largest number of divisors are called highly composite numbers. The function satisfies the identities(2)(3)where the are distinct primes and is the prime factorization of a number .The divisor function is odd iff is a square number.The function that gives the sum of the divisors of is commonly written without the..

Let the divisor function be the number of divisors of (including itself). For a prime , . In general,where is the Euler-Mascheroni constant. Dirichlet originally gave (Hardy and Wright 1979, p. 264; Hardy 1999, pp. 67-68), and Hardy and Landau showed in 1916 that (Hardy 1999, p. 81). The following table summarizes incremental progress on the upper limit (updating Hardy 1999, p. 81).approx.citation1/20.50000Dirichlet1/30.33333Voronoi (1903), Sierpiński (1906), van der Corput (1923)37/1120.33036Littlewood and Walfisz (1925)33/1000.33000van der Corput (1922)27/820.32927van der Corput (1928)15/460.3260912/370.32432Chen (1963), Kolesnik (1969)35/1080.32407Kolesnik (1982)139/4290.32401Kolesnik17/530.32075Vinogradov (1935)7/220.31818Iwaniec and Mozzochi (1988)23/730.31507Huxley (1993)131/4160.31490Huxley (2003)..

A number theoretic character, also called a Dirichlet character (because Dirichlet first introduced them in his famous proof that every arithmetic progression with relatively prime initial term and common difference contains infinitely many primes), modulo is a complex function for positive integer such that(1)(2)(3)for all , and(4)if . can only assume values which are roots of unity, where is the totient function.Number theoretic characters are implemented in the Wolfram Language as DirichletCharacter[k, j, n], where is the modulus and is the index.

The integer sequence beginning with a single digit in which the next term is obtained by describing the previous term. Starting with 1, the sequence would be defined by "1, one 1, two 1s, one 2 one 1," etc., and the result is 1, 11, 21, 1211, 111221, .... Similarly, starting the sequence instead with the digit for gives , 1, 111, 311, 13211, 111312211, 31131122211, 1321132132211, ..., as summarized in the following table.OEISsequence1A0051501, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...2A0067512, 12, 1112, 3112, 132112, 1113122112, 311311222112, ...3A0067153, 13, 1113, 3113, 132113, 1113122113, 311311222113, ...The number of digits in the th term of the sequence for are 1, 2, 2, 4, 6, 6, 8, 10, 14, 20, 26, 34, 46, 62, ... (OEIS A005341). Similarly, the numbers of digits for the th term of the sequence for , 3, ..., are 1, 2, 4, 4, 6, 10, 12, 14, 22, 26, ... (OEIS A022471). These sequences are asymptotic to , where(1)(2)(3)The quantity..

What is the maximum number of queens that can be placed on an chessboard such that no two attack one another? The answer is queens for or and queens otherwise, which gives eight queens for the usual board (Madachy 1979; Steinhaus 1999, p. 29). The number of different ways the queens can be placed on an chessboard so that no two queens may attack each other for the first few are 1, 0, 0, 2, 10, 4, 40, 92, ... (OEIS A000170; Madachy 1979; Steinhaus 1999, p. 29). The number of rotationally and reflectively distinct solutions of these are 1, 0, 0, 1, 2, 1, 6, 12, 46, 92, ... (OEIS A002562; Dudeney 1970; p. 96). The 12 distinct solutions for are illustrated above, and the remaining 80 are generated by rotation and reflection (Madachy 1979, Steinhaus 1999).The minimum number of queens needed to occupy or attack all squares of an chessboard (i.e., domination numbers for the queen graphs) are given for , 2, ... by 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 8, 9,..

Prince Rupert's cube is the largest cube that can be made to pass through a given cube. In other words, the cube having a side length equal to the side length of the largest hole of a square cross section that can be cut through a unit cube without splitting it into two pieces.Prince Rupert's cube cuts a hole of the shape indicated in the above illustration (Wells 1991). Curiously, it is slightly larger than the original cube, with side length (OEIS A093577). Any cube this size or smaller can be made to pass through the original cube.

The turning of an object or coordinate system by an angle about a fixed point. A rotation is an orientation-preserving orthogonal transformation. Euler's rotation theorem states that an arbitrary rotation can be parameterized using three parameters. These parameters are commonly taken as the Euler angles. Rotations can be implemented using rotation matrices.Rotation in the plane can be concisely described in the complex plane using multiplication of complex numbers with unit modulus such that the resulting angle is given by . For example, multiplication by represents a rotation to the right by and by represents rotation to the left by . So starting with and rotating left twice gives , which is the same as rotating right twice, , and . For multiplication by multiples of , the possible positions are then concisely represented by , , , and .The rotation symmetry operation for rotation by is denoted "." For periodic arrangements of points..

The four parameters , , , and describing a finite rotation about an arbitrary axis. The Euler parameters are defined by(1)(2)(3)where is the unit normal vector, and are a quaternion in scalar-vector representation(4)Because Euler's rotation theorem states that an arbitrary rotation may be described by only three parameters, a relationship must exist between these four quantities(5)(6)(Goldstein 1980, p. 153). The rotation angle is then related to the Euler parameters by(7)(8)(9)and(10)The Euler parameters may be given in terms of the Eulerangles by(11)(12)(13)(14)(Goldstein 1980, p. 155).Using the Euler parameters, the rotation formulabecomes(15)and the rotation matrix becomes(16)where the elements of the matrix are(17)Here, Einstein summation has been used, is the Kronecker delta, and is the permutation symbol. Written out explicitly, the matrix elements are(18)(19)(20)(21)(22)(23)(24)(25)(26)..

According to Euler's rotation theorem, any rotation may be described using three angles. If the rotations are written in terms of rotation matrices , , and , then a general rotation can be written as(1)The three angles giving the three rotation matrices are called Euler angles. There are several conventions for Euler angles, depending on the axes about which the rotations are carried out. Write the matrix as(2)The so-called "-convention," illustrated above, is the most common definition. In this convention, the rotation given by Euler angles , where 1. the first rotation is by an angle about the z-axis using , 2. the second rotation is by an angle about the former x-axis (now ) using , and 3. the third rotation is by an angle about the former z-axis (now ) using . Note, however, that several notational conventions for the angles are in common use. Goldstein (1980, pp. 145-148) and Landau and Lifschitz (1976) use , Tuma (1974) says is..

The solid angle subtended by a surface is defined as the surface area of a unit sphere covered by the surface's projection onto the sphere. This can be written as(1)where is a unit vector from the origin, is the differential area of a surface patch, and is the distance from the origin to the patch. Written in spherical coordinates with the colatitude (polar angle) and for the longitude (azimuth), this becomes(2)Solid angle is measured in steradians, and the solid angle corresponding to all of space being subtended is steradians.To see how the solid angle of simple geometric shapes can be computed explicitly, consider the solid angle subtended by one face of a cube of side length centered at the origin. Since the cube is symmetrical and has six sides, one side obviously subtends steradians. To compute this explicitly, rewrite (1) in Cartesian coordinates using(3)(4)and(5)(6)Considering the top face of the cube, which is located at and has sides..

An insphere is a sphere inscribed in a given solid. The radius of the insphere is called the inradius.Platonic solids (whose duals are themselves Platonic solids) and Archimedean duals have inspheres that touch all their faces, but Archimedean solids do not. Note that the insphere is not necessarily tangent at the centroid of the faces of a dual polyhedron, but is rather only tangent at some point lying on the face.The figures above depict the inspheres of the Platonicsolids.

The -hypersphere (often simply called the -sphere) is a generalization of the circle (called by geometers the 2-sphere) and usual sphere (called by geometers the 3-sphere) to dimensions . The -sphere is therefore defined (again, to a geometer; see below) as the set of -tuples of points (, , ..., ) such that(1)where is the radius of the hypersphere.Unfortunately, geometers and topologists adopt incompatible conventions for the meaning of "-sphere," with geometers referring to the number of coordinates in the underlying space ("thus a two-dimensional sphere is a circle," Coxeter 1973, p. 125) and topologists referring to the dimension of the surface itself ("the -dimensional sphere is defined to be the set of all points in satisfying ," Hocking and Young 1988, p. 17; "the -sphere is ," Maunder 1997, p. 21). A geometer would therefore regard the object described by(2)as a 2-sphere,..

Half of a sphere cut by a plane passing through its center. A hemisphere of radius can be given by the usual spherical coordinates(1)(2)(3)where and . All cross sections passing through the z-axis are semicircles.The volume of the hemisphere is(4)(5)(6)The weighted mean of over the hemisphere is(7)The geometric centroid is then given by(8)(Beyer 1987).

The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres.The spheres can be used to show that the intersection of the plane with the cone is an ellipse. Let be a plane intersecting a right circular cone with vertex in the curve . Call the spheres tangent to the cone and the plane and , and the circles on which the spheres are tangent to the cone and . Pick a line along the cone which intersects at , at , and at . Call the points on the plane where the sphere are tangent and . Because intersecting tangents have the same length,(1)(2)Therefore,(3)which is a constant independent of , so is an ellipse with .

A sliver of the surface of a sphere of radius cut out by two planes through the azimuthal axis with dihedral angle . The surface area of the lune iswhich is just the area of the sphere times . The volume of the associated spherical wedge has volume

A sphere circumscribed in a given solid. Its radius is called the circumradius. By analogy with the equation of the circumcircle, the equation for the circumsphere of the tetrahedron with polygon vertices for , ..., 4 is(1)Expanding the determinant,(2)where(3) is the determinant obtained from the matrix(4)by discarding the column (and taking a plus sign) and similarly for (this time taking the minus sign) and (again taking the plus sign)(5)(6)(7)and is given by(8)Completing the square gives(9)which is a sphere of the form(10)with circumcenter(11)(12)(13)and circumradius(14)The figures above depict the circumspheres of the Platonicsolids.

Find the tunnel between two points and on a gravitating sphere which gives the shortest transit time under the force of gravity. Assume the sphere to be nonrotating, of radius , and with uniform density . Then the standard form Euler-Lagrange differential equation in polar coordinates is(1)along with the boundary conditions , , , and . Integrating once gives(2)But this is the equation of a hypocycloid generated by a circle of radius rolling inside the circle of radius , so the tunnel is shaped like an arc of a hypocycloid. The transit time from point to point is(3)where(4)is the surface gravity with the universal gravitational constant.

Place two solid spheres of radius 1/2 inside a hollow sphere of radius 1 so that the two smaller spheres touch each other at the center of the large sphere and are tangent to the large sphere on the extremities of one of its diameters. This arrangement is called the "bowl of integers" (Soddy 1937) since the bend of each of the infinite chain of spheres that can be packed into it such that each successive sphere is tangent to its neighbors is an integer. The first few bends are then , 2, 5, 6, 9, 11, 14, 15, 18, 21, 23, ... (OEIS A046160). The sizes and positions of the first few rings of spheres are given in the table below.100--220--3546059611071481591801021112312270, 1330143315380Spheres can also be packed along the plane tangent to the two spheres of radius 2 (Soddy 1937). The sequence of integers for can be found using the equation of five tangent spheres. Letting givesFor example, , , , , , and so on, giving the sequence , 2, 3, 11, 15, 27, 35, 47,..

A spherical sector is a solid of revolution enclosed by two radii from the center of a sphere. The spherical sector may either be "open" and have a conical hole (left figure; Beyer 1987), or may be a "closed" spherical cone (right figure; Harris and Stocker 1998). The volume of a spherical sector in either case is given bywhere is the vertical distance between where the upper and lower radii intersect the sphere and is the sphere's radius.

A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment. However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for what is here called a spherical cap and "zone" for spherical segment.Let the sphere have radius , then the volume of a spherical cap of height and base radius is given by the equation of a spherical segment(1)with , giving(2)Using the Pythagorean theorem gives(3)which can be solved for as(4)so the radius of the base circle is(5)and plugging this in gives the equivalent formula(6)In terms of the so-called contact angle (the anglebetween the normal to the sphere at the bottom of the cap and the base plane)(7)(8)so(9)The geometric centroid occurs at a distance(10)above the center of the..

A hosohedron is a regular tiling or map on a sphere composed of digons or spherical lunes, all with the same two vertices and the same vertex angles, . Its Schläfli symbol is . Its dual is the dihedron .

A cork plug is a three-dimensional solid that can stopper a square, triangular, or circular hole. There is an infinite family of such shapes.The shape with smallest volume has triangular cross sections.The plug with the largest volume is made using two cuts from the top diameter to the edge, as illustrated above. Such a plug has to obtain a square cross section. For a general such a plug of height and radius , the volume of the plug is

The truncated square pyramid is a special case of a pyramidal frustum for a square pyramid. Let the base and top side lengths of the truncated pyramid be and , and let the height be . Then the volume of the solid isThis formula was known to the Egyptians ca. 1850 BC. The Egyptians cannot have proved it without calculus, however, since Dehn showed in 1900 that no proof of this equation exists which does not rely on the concept of continuity (and therefore some form of integration).

A pyramidal frustum is a frustum made by chopping thetop off a pyramid. It is a special case of a prismatoid.For a right pyramidal frustum, let be the slant height, the height, the bottom base perimeter, the top base perimeter, the bottom area, and the top area. Then the surface area (of the sides) and volume of a pyramidal frustum are given by(1)(2)The geometric centroid of a right pyramidalfrustum occurs at a height(3)above the bottom base (Harris and Stocker 1998).The bases of a right -gonal frustum are regular polygons of side lengths and with circumradii(4)where is the side length, so the diagonal connecting corresponding vertices on top and bottom has length(5)and the edge length is(6)(7)The triangular () and square () right pyramidal frustums therefore have side surface areas(8)(9)The area of a regular -gon is(10)so the volumes of these frustums are(11)(12)..

The superellipsoid is a generalization of the ellipsoid.The version also called the superquadratic ellipsoid is defined by the equation(1)where and are the east-west and north-south exponents, respectively. The superellipsoid can be rendered in POVRay® with the command superellipsoid{ <e, n> }The generalization(2)of the surface considered by Gray (1997) might also be called a superellipsoid. This surface can be given parametrically by(3)(4)(5)for and . Some special cases of this surface are summarized in the following table.ellipsoid, sphere, Hauser's "cube"The volume of the solid with is(6)(7)As , the solid becomes a cube, so(8)as it must. This is a special case of the integral 3.2.2.2(9)in Prudnikov et al. (1986, p. 583). The cases and appear to be the only integers whose corresponding solids have simple moment of inertia tensors, given by(10)(11)..

A spheroid is an ellipsoid having two axes of equal length, making it a surface of revolution. By convention, the two distinct axis lengths are denoted and , and the spheroid is oriented so that its axis of rotational symmetric is along the -axis, giving it the parametric representation(1)(2)(3)with , and .The Cartesian equation of the spheroid is(4)If , the spheroid is called oblate (left figure). If , the spheroid is prolate (right figure). If , the spheroid degenerates to a sphere.In the above parametrization, the coefficients of the firstfundamental form are(5)(6)(7)and of the second fundamental form are(8)(9)(10)The Gaussian curvature is given by(11)the implicit Gaussian curvature by(12)and the mean curvature by(13)The surface area of a spheroid can be variously writtenas(14)(15)(16)(17)where(18)(19)and is a hypergeometric function.The volume of a spheroid can be computed from the formula for a general ellipsoid with ,(20)(21)(Beyer..

The general ellipsoid, also called a triaxial ellipsoid, is a quadratic surface which is given in Cartesian coordinates by(1)where the semi-axes are of lengths , , and . In spherical coordinates, this becomes(2)If the lengths of two axes of an ellipsoid are the same, the figure is called a spheroid (depending on whether or , an oblate spheroid or prolate spheroid, respectively), and if all three are the same, it is a sphere. Tietze (1965, p. 28) calls the general ellipsoid a "triaxial ellipsoid."There are two families of parallel circular cross sections in every ellipsoid. However, the two coincide for spheroids (Hilbert and Cohn-Vossen 1999, pp. 17-19). If the two sets of circles are fastened together by suitably chosen slits so that they are free to rotate without sliding, the model is movable. Furthermore, the disks can always be moved into the shape of a sphere (Hilbert and Cohn-Vossen 1999, p. 18).In 1882, Staude..

The cylindrical hoof is a special case of the cylindrical wedge given by a wedge passing through a diameter of the base (so that ). Let the height of the wedge be and the radius of the cylinder from which it is cut be . Then plugging the points , , and into the 3-point equation for a plane gives the equation for the plane as(1)Combining with the equation of the circle that describes the curved part remaining of the cylinder (and writing ) then gives the parametric equations of the "tongue" of the wedge as(2)(3)(4)for . To examine the form of the tongue, it needs to be rotated into a convenient plane. This can be accomplished by first rotating the plane of the curve by about the x-axis using the rotation matrix and then by the angle(5)above the z-axis. The transformed plane now rests in the -plane and has parametric equations(6)(7)and is shown below. The length of the tongue (measured down its middle) is obtained by plugging into the above equation for..

A ruled surface is called a generalized cylinder if it can be parameterized by , where is a fixed point. A generalized cylinder is a regular surface wherever . The above surface is a generalized cylinder over a cardioid. A generalized cylinder is a flat surface, and is sometimes called a "cylindrical surface" (Kern and Bland 1948, p. 32) or "cylinder surface" (Harris and Stocker 1998, p. 102).A generalized cylinder need not be closed (Kern and Bland 1948, p. 32).Kern and Bland (1948, p. 32) define a cylinder as a solid bounded by a generalized cylinder and two parallel planes. However, when used without qualification, the term "cylinder" generally refers to the particular case of a right circular cylinder.

The solid common to two (or three) right circular cylinders of equal radii intersecting at right angles is called the Steinmetz solid. Two cylinders intersecting at right angles are called a bicylinder or mouhefanggai (Chinese for "two square umbrellas"), and three intersecting cylinders a tricylinder. Half of a bicylinder is called a vault.For two cylinders of radius oriented long the - and -axes gives the equations(1)(2)which can be solved for and gives the parametric equations of the edges of the solid,(3)(4)The surface area can be found as , where(5)(6)Taking the range of integration as a quarter or one face and then multiplying by 16 gives(7)The volume common to two cylinders was known to Archimedes (Heath 1953, Gardner 1962) and the Chinese mathematician Tsu Ch'ung-Chih (Kiang 1972), and does not require calculus to derive. Using calculus provides a simple derivation, however. Noting that the solid has a square cross section..

A spherical ring is a sphere with a cylindrical hole cut so that the centers of the cylinder and sphere coincide, also called a napkin ring. Let the sphere have radius and the cylinder radius .From the right diagram, the surface area of the sphericalring is equal to twice that of a cylinder of half-height(1)and radius plus twice that of the zone of radius and height , giving(2)(3)Note that as illustrated above, the hole cut out consists of a cylindrical portion plus two spherical caps. The volume of the entire cylinder is(4)and the volume of the upper segment is(5)The volume removed upon drilling of a cylindricalhole is then(6)(7)where the expressions(8)(9)obtained from trigonometry have been used to re-express the result.The volume of the spherical ring itself is then givenby(10)(11)(12)By the final equation, the remaining volume of any center-drilled sphere can be calculated given only the length of the hole. In particular, if the sphere..

A wedge is cut from a cylinder by slicing with a plane that intersects the base of the cylinder. The volume of a cylindrical wedge can be found by noting that the plane cutting the cylinder passes through the three points illustrated above (with ), so the three-point form of the plane gives the equation(1)(2)Solving for gives(3)Here, the value of is given by(4)(5)The volume is therefore given as an integralover rectangular areas along the x-axis,(6)Using the identities(7)(8)(9)(10)gives the equivalent alternate forms(11)(12)(Harris and Stocker 1998, p. 104). This simplifies in the case of to(13)The lateral surface areacan be found from(14)where is simply with , so(15)(16)(17)(18)(Harris and Stocker 1998, p. 104).A special case of the cylindrical wedge, also called a cylindrical hoof, is a wedge passing through a diameter of the base (so that )...

A cylindrical segment, sometimes also called a truncated cylinder, is the solid cutfrom a circular cylinder by two (or more) planes.If there are two cutting planes, one perpendicular to the axis of the cylinder and the other titled with respect to it, the resulting solid is known as a cylindrical wedge.If the plane is titled with respect to a circular cross section but does not cut the bottom base, the resulting cylindrical segment has one circular cap and one elliptical cap (see above figure). Consider a cylinder of radius and minimum and maximum heights and . Set up a coordinate system with lower cap in the -plane, origin at the center of the lower cap, and the -axis passing through the center of the lower cap parallel to the projection of the semimajor axis of the upper cap. Then the height of the solid at distance is given byThe volume of the cylindrical section can be obtained instantly by noting that two such sections can be fitted together to form a cylinder..

A sphericon is the solid formed from a bicone with opening angle of (and therefore with ) obtained by slicing the solid with a plane containing the rotational axes resulting in a square cross section, then rotating the two pieces by and reconnecting them. It was constructed by Israeli game and toy inventor David Hirsch who patented the shape in Israel in 1984. It was given the name "sphericon" by Colin Roberts, who independently discovered the solid in the 1960s while attempting to carve a Möbius strip without a hole in the middle out of a block of wood.The solid is not as widely known as it should be.The above net shows another way the sphericon can be constructed. In this figure radians . A sphericon has a single continuous face and rolls by wobbling along that face, resulting in straight-line motion. In addition, one sphericon can roll around another.The sphericon with radius has surface area and volume(1)(2)The centroid is at the..

The Penrose tiles are a pair of shapes that tile the plane only aperiodically (when the markings are constrained to match at borders). These two tiles, illustrated above, are called the "kite" and "dart," respectively. In strict Penrose tiling, the tiles must be placed in such a way that the colored markings agree; in particular, the two tiles may not be combined into a rhombus (Hurd).Two additional types of Penrose tiles known as the rhombs (of which there are two varieties: fat and skinny) and the pentacles (or which there are six type) are sometimes also defined that have slightly more complicated matching conditions (McClure 2002).In 1997, Penrose sued the Kimberly Clark Corporation over their quilted toilet paper, which allegedly resembles a Penrose aperiodic tiling (Mirsky 1997). The suit was apparently settled out of court.To see how the plane may be tiled aperiodically using the kite and dart, divide the kite into..

The salinon is the figure illustrated above formed from four connected semicircles. The word salinon is Greek for "salt cellar," which the figure resembles. If the radius of the large enclosing circle is and the radius of the small central circle is , then the radii of the two small side circles are .In his Book of Lemmas, Archimedes proved that the salinon has an area equal to the circle having the line segment joining the top and bottom points as its diameter (Wells 1991), namely

A lune is a plane figure bounded by two circular arcs of unequal radii, i.e., a crescent. (By contrast, a plane figure bounded by two circular arcs of equal radius is known as a lens.) For circles of radius and whose centers are separated by a distance , the area of the lune is given by(1)where(2)is the area of the triangle with side lengths , , and . The second of these can be obtained directly by subtracting the areas of the two half-lenses whose difference producing the colored region above.In each of the figures above, the area of the lune is equal to the area of the indicated triangle. Hippocrates of Chios squared the above left lune (Dunham 1990, pp. 19-20; Wells 1991, pp. 143-144), as well as two others, in the fifth century BC. Two more squarable lunes were found by T. Clausen in the 19th century (Shenitzer and Steprans 1994; Dunham 1990 attributes these discoveries to Euler in 1771). In the 20th century, N. G. Tschebatorew..

The perpendicular distance from an arc's midpoint to the chord across it, equal to the radius minus the apothem ,(1)For a regular polygon of side length ,(2)(3)(4)(5)(6)where is the circumradius, the inradius, is the side length, and is the number of sides.

A (general, asymmetric) lens is a lamina formed by the intersection of two offset disks of unequal radii such that the intersection is not empty, one disk does not completely enclose the other, and the centers of curvatures are on opposite sides of the lens. If the centers of curvature are on the same side, a lune results.The area of a general asymmetric lens obtained from circles of radii and and offset can be found from the formula for circle-circle intersection, namely(1)(2)Similarly, the height of such a lens is(3)(4)A symmetric lens is lens formed by the intersection of two equal disk. The area of a symmetric lens obtained from circles with radii and offset is given by(5)and the height by(6)A special type of symmetric lens is the vesica piscis (Latin for "fish bladder"), corresponding to a disk offset which is equal to the disk radii.A lens-shaped region also arises in the study of Bessel functions, is very important in the theory of..

A portion of a disk whose upper boundary is a (circular) arc and whose lower boundary is a chord making a central angle radians (), illustrated above as the shaded region. The entire wedge-shaped area is known as a circular sector.Circular segments are implemented in the Wolfram Language as DiskSegment[x, y, r, q1, q2]. Elliptical segments are similarly implemented as DiskSegment[x, y, r1, r2, q1, q2].Let be the radius of the circle, the chord length, the arc length, the height of the arced portion, and the height of the triangular portion. Then the radius is(1)the arc length is(2)the height is(3)(4)(5)and the length of the chord is(6)(7)(8)(9)From elementary trigonometry, the angle obeys the relationships(10)(11)(12)(13)The area of the (shaded) segment is then simply given by the area of the circular sector (the entire wedge-shaped portion) minus the area of the bottom triangular portion,(14)Plugging in gives(15)(16)(17)(18)where..

Half a circle. The area of a semicircle of radius is given by(1)(2)(3)The weighted mean of is(4)(5)The semicircle is the cross section of a hemispherefor any plane through the z-axis.The perimeter of the curved boundary is given by(6)With , this gives(7)The perimeter of the semicircular lamina is then(8)The weighted value of of the semicircular curve is given by(9)(10)(11)so the geometric centroid is(12)The geometric centroid of the semicircularlamina is given by(13)(Kern and Bland 1948, p. 113).

A circular sector is a wedge obtained by taking a portion of a disk with central angle radians (), illustrated above as the shaded region. A sector with central angle of radians would correspond to a filled semicircle. Let be the radius of the circle, the chord length, the arc length, the sagitta (height of the arced portion), and the apothem (height of the triangular portion). Then(1)(2)(3)(4)(5)(6)(7)(8)(9)The angle obeys the relationships(10)(11)(12)(13)The area of the sector is(14)(15)(Beyer 1987). The area can also be found by direct integration as(16)It follows that the weighted mean of the is(17)(18)so the geometric centroid of the circular sectoris(19)(20)(21)(Gearhart and Schulz 1990). Checking shows that this obeys the proper limits for a semicircle () and for an isosceles triangle ()...

There are a number of meanings for the word "arc" in mathematics. In general, an arc is any smooth curve joining two points. The length of an arc is known as its arc length.In a graph, a graph arc isan ordered pair of adjacent vertices.In particular, an arc is any portion (other than the entire curve) of the circumference of a circle. An arc corresponding to the central angle is denoted . Similarly, the size of the central angle subtended by this arc (i.e., the measure of the arc) is sometimes (e.g., Rhoad et al. 1984, p. 421) but not always (e.g., Jurgensen 1963) denoted .The center of an arc is the center of the circle of whichthe arc is a part.An arc whose endpoints lie on a diameter of a circleis called a semicircle.For a circle of radius , the arc length subtended by a central angle is proportional to , and if is measured in radians, then the constant of proportionality is 1, i.e.,(1)The length of the chord connecting the arc's endpointsis(2)As..

The lines containing the three points of the intersection of the three pairs of oppositesides of a (not necessarily regular) hexagon.There are 6! (i.e., 6 factorial) possible ways of taking all polygon vertices in any order, but among these are six equivalent cyclic permutations and two possible orderings, so the total number of different hexagons (not all simple) isThere are therefore a total of 60 Pascal lines created by connecting polygonvertices in any order.The 60 Pascal lines form a very complicated pattern which can be visualized most easily in the degenerate case of a regular hexagon inscribed in a circle, as illustrated above for magnifications ranging over five powers of 2. Only 45 lines are visible in this figure since each of the three thick lines (located at angles to each other) represents a degenerate group of four Pascal lines, and six of the Pascal lines are lines at infinity (Wells 1991). The pattern for a general ellipse and hexagon..

If , , and are three points on one line, , , and are three points on another line, and meets at , meets at , and meets at , then the three points , , and are collinear. Pappus's hexagon theorem is self-dual.The incidence graph of the configuration corresponding to the theorem is the Pappus graph.

The regular tessellation consisting of regular hexagons (i.e., a hexagonal grid).In general, the term honeycomb is used to refer to a tessellation in dimensions for . The only regular honeycomb in three dimensions is , which consists of eight cubes meeting at each polyhedron vertex. The only quasiregular honeycomb (with regular cells and semiregular vertex figures) has each polyhedron vertex surrounded by eight tetrahedra and six octahedra and is denoted .Ball and Coxeter (1987) use the term "sponge" for a solid that can be parameterized by integers , , and that satisfy the equationThe possible sponges are , , , , and .There are many semiregular honeycombs, such as , in which each polyhedron vertex consists of two octahedra and four cuboctahedra .

Given a triangle and the excentral triangle , define the -vertex of the hexyl triangle as the point in which the perpendicular to through the excenter meets the perpendicular to through the excenter , and similarly define and . Then is known as the hexyl triangle of , and forms a hexagon with parallel sides (Kimberling 1998 pp. 79 and 172).The hexyl triangle has trilinear vertex matrix(1)where , , and (Kimberling 1998, p. 172).It has side lengths(2)(3)(4)and area(5)(6)(7)where is the area of the reference triangle, is the circumradius, and is the inradius. It therefore has the same side lengths and area as the excentral triangle.The Cevians triangles with Cevian points corresponding to Kimberling centers with , 20, 21, 27, 63, and 84 are perspective to the hexyl triangle. That anticevian triangles and antipedal triangles corresponding to Kimberling centers for , 9, 19, 40, 57, 63, 84, 610, 1712, and 2184 are also perspective to the..

A paradox mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two concentric circles of different diameters (a wheel within a wheel). There is a 1:1 correspondence of points on the large circle with points on the small circle, so the wheel should travel the same distance regardless of whether it is rolled from left to right on the top straight line or on the bottom one. this seems to imply that the two circumferences of different sized circles are equal, which is impossible.The fallacy lies in the assumption that a one-to-one correspondence of points means that two curves must have the same length. In fact, the cardinalities of points in a line segment of any length (or even an infinite line, a plane, a three-dimensional space, or an infinite dimensional Euclidean space) are all the same: (the cardinality of the continuum), so the points of any of these can be put in a one-to-one..

After a half rotation of the coin on the left around the central coin (of the same radius), the coin undergoes a complete rotation. In other words, a coin makes two complete rotations when rolled around the boundary of an identical coin. This fact is readily apparent in the generation of the cardioid as one disk rolling on another.

A function, or the symbol representing a function, which corresponds to English conjunctions such as "and," "or," "not," etc. that takes one or more truth values as input and returns a single truth value as output. The terms "logical connective" and "propositional connective" (Mendelson 1997, p. 13) are also used. The following table summarizes some common connectives and their notations.connectivesymbolAND, , , , , equivalent, , implies, , NAND, , nonequivalent, , NOR, , NOT, , , OR, , , XNOR XNOR XOR,

A schematic diagram used in logic theory to depict collectionsof sets and represent their relationships.The Venn diagrams on two and three sets are illustrated above. The order-two diagram (left) consists of two intersecting circles, producing a total of four regions, , , , and (the empty set, represented by none of the regions occupied). Here, denotes the intersection of sets and .The order-three diagram (right) consists of three symmetrically placed mutually intersecting circles comprising a total of eight regions. The regions labeled , , and consist of members which are only in one set and no others, the three regions labelled , , and consist of members which are in two sets but not the third, the region consists of members which are simultaneously in all three, and no regions occupied represents .In general, an order- Venn diagram is a collection of simple closed curves in the plane such that 1. The curves partition the plane into connected..

Propositional calculus is the formal basis of logic dealing with the notion and usage of words such as "NOT," "OR," "AND," and "implies." Many systems of propositional calculus have been devised which attempt to achieve consistency, completeness, and independence of axioms. The term "sentential calculus" is sometimes used as a synonym for propositional calculus.Axioms (or their schemata) and rules of inference define a proof theory, and various equivalent proof theories of propositional calculus can be devised. The following list of axiom schemata of propositional calculus is from Kleene (2002). (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)In each schema, , , can be replaced by any sentential formula. The following rule called Modus Ponens is the sole rule of inference:(11)This rule states that if each of and is either an axiom or a theorem formally deduced from axioms by application of..

A statement is in disjunctive normal form if it is a disjunction (sequence of ORs) consisting of one or more disjuncts, each of which is a conjunction (AND) of one or more literals (i.e., statement letters and negations of statement letters; Mendelson 1997, p. 30). Disjunctive normal form is not unique.The Wolfram Language command LogicalExpand[expr] gives disjunctive normal form (with some contractions, i.e., LogicalExpand attempts to shorten output with heuristic simplification).Examples of disjunctive normal forms include (1)(2)(3)(4)(5)where denotes OR, denotes AND, and denotes NOT (Mendelson 1997, p. 30). Some authors also exclude statements containing both statement letters and their negations, which would exclude the third example above.Every statement in logic consisting of a combination of multiple , , and s can be written in disjunctive normal form...

A statement which is rigorously known to be correct. A statement which is not true is called false, although certain statements can be proved to be rigorously undecidable within the confines of a given set of assumptions and definitions. Regular two-valued logic allows statements to be only true or false, but fuzzy logic treats "truth" as a continuum which can have any value between 0 and 1. The symbol is sometimes used to denote "true," although "T" is more commonly used in truth tables.

A tautology is a logical statement in which the conclusion is equivalent to the premise. More colloquially, it is formula in propositional calculus which is always true (Simpson 1992, p. 2015; D'Angelo and West 2000, p. 33; Bronshtein and Semendyayev 2004, p. 288).If is a tautology, it is written . A sentence whose truth table contains only 'T' is called a tautology. The following sentences are examples of tautologies: (1)(2)(3)(Mendelson 1997, p. 26), where denotes AND, denotes "is equivalent to," denotes NOT, denotes OR, and denotes implies.

A formula of first-order logic is said to be in Skolemized form (sometimes also called Skolem standard form or universal form) if it is of the formwhere is a quantifier-free formula in conjunctive normal form known as the matrix of the formula in question. Since is a conjunction of clauses each of which is a disjunction of literals, is often viewed as a set of the clauses. The process of placing a formula in Skolemized form is known as Skolemization.

In combinatorial logic minimization, a device known as a Karnaugh map is frequently used. It is similar to a truth table, but the various variables are represented along two axes and are arranged in such a way that only one input bit changes in going from one square to an adjacent square. It is also known as a Veitch diagram, K-map, or KV-map.The Karnaugh map may be used to quickly eliminate redundant operations in a Boolean function. The easiest to read Karnaugh maps are those drawn for a function in the form of a complete product or "sum of products," where the latter name also implies the use of and for the AND and OR operators. In a typical truth table for such a function, the inputs are enumerated using 0 for false and 1 for true, and ordered as a counting sequence when read as positive binary integers. A truth table for a function of four variables is illustrated below.00000000100010100111010000101001101011111000110010101001011011001110111110011110For..

In December 1920, M. Schönfinkel presented in a report to the Mathematical Society in Göttingen a new type of formal logic based on the concept of a generalized function whose argument is also a function (Schönfinkel 1924). This mathematical discipline was subsequently termed combinatory logic by Curry and "-conversion" or "-calculus" by Church. Combinators can be used in the study of algebra, topology, and category theory, and have found application in the study of programs in algorithmic languages.

The Königsberg bridge problem asks if the seven bridges of the city of Königsberg (left figure; Kraitchik 1942), formerly in Germany but now known as Kaliningrad and part of Russia, over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began. This is equivalent to asking if the multigraph on four nodes and seven edges (right figure) has an Eulerian cycle. This problem was answered in the negative by Euler (1736), and represented the beginning of graph theory.On a practical note, J. Kåhre observes that bridges and no longer exist and that and are now a single bridge passing above with a stairway in the middle leading down to . Even so, there is still no Eulerian cycle on the nodes , , , and using the modern Königsberg bridges, although there is an Eulerian path (right figure). An example Eulerian path is illustrated in the right..

The knight graph is a graph on vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a knight (which may only make moves which simultaneously shift one square along one axis and two along the other).The number of edges in the knight graph is (8 times the triangular numbers), so for , 2, ..., the first few values are 0, 0, 8, 24, 48, 80, 120, ... (OEIS A033996).Knight graphs are bipartite and therefore areperfect.The following table summarizes some named graph complements of knight graphs.-knight graph-queen graph-knight graph-queen graphThe knight graph is implemented in the Wolfram Language as KnightTourGraph[m, n], and precomputed properties are available in using GraphData["Knight", m, n].Closed formulas for the numbers of -graph cycles of the knight graph are given by for odd and(1)(E. Weisstein, Nov. 16, 2014).A knight's path is a sequence of moves by a..

The conjecture that every cubic polyhedral graph is Hamiltonian. It was proposed by Tait in 1880 and refuted by Tutte (1946) with the counterexample on 46 vertices (Lederberg 1965) now known as Tutte's graph. Had the conjecture been true, it would have implied the four-color theorem.The following table summarizes some named counterexamples, illustrated above. The smallest examples known has 38 vertices (Lederberg 1965), and was apparently also discovered by D. Barnette and J. Bosák around the same time.graphreference38Barnette-Bośak-Lederberg graphLederberg (1965), Thomassen (1981), Grünbaum (2003, Fig. 17.1.5)42Faulkner-Younger graph 42Faulkner and Younger (1974)42Grinberg graph 42Faulkner and Younger (1974)44Faulkner-Younger graph 44Faulkner and Younger (1974)44Grinberg graph 44Sachs (1968), Berge (1973), Read and Wilson (1998, p. 274)46Grinberg graph 46Bondy..

The game of billiards is played on a rectangular table (known as a billiard table) upon which balls are placed. One ball (the "cue ball") is then struck with the end of a "cue" stick, causing it to bounce into other balls and reflect off the sides of the table. Real billiards can involve spinning the ball so that it does not travel in a straight line, but the mathematical study of billiards generally consists of reflections in which the reflection and incidence angles are the same. However, strange table shapes such as circles and ellipses are often considered. The popular 1959 animated short film Donald in Mathmagic Land features a tutorial by Donald Duck on how to win at billiards using the diamonds normally inscribed around the edge of a real billiard table.Many interesting problems can arise in the detailed study of billiards trajectories. For example, any smooth plane convex set has at least two double normals, so there are..

While many computations admit shortcuts that allow them to be performed more rapidly, others cannot be sped up. Computations that cannot be sped up by means of any shortcut are called computationally irreducible. The principle of computational irreducibility says that the only way to determine the answer to a computationally irreducible question is to perform, or simulate, the computation. Some irreducible computations can be sped up by performing them on faster hardware, as the principle refers only to computation time.According to Wolfram (2002, p. 741), if the behavior of a system is obviously simple--and is say either repetitive or nested--then it will always be computationally reducible. But it follows from the principle of computational equivalence that in practically all other cases it will be computationally irreducible." Here, "practically all" refers to cases that arise naturally or from a simple..

A pairing function is a function that reversibly maps onto , where denotes nonnegative integers. Pairing functions arise naturally in the demonstration that the cardinalities of the rationals and the nonnegative integers are the same, i.e., , where is known as aleph-0, originally due to Georg Cantor. Pairing functions also arise in coding problems, where a vector of integer values is to be folded onto a single integer value reversibly.515202633414101419253236913182423581217112471112345Let(1)then Hopcroft and Ullman (1979, p. 169) define the pairing function(2)(3)illustrated in the table above, where . The inverse may computed from(4)(5)(6)(7)where is the floor function.51522303949604101623314050361117243241237121825331148131926002591420012345The Hopcroft-Ullman function can be reparameterized so that and are in rather than . This function is known as the Cantor function and is given by(8)illustrated in..

A multiway system is a kind of substitution system in which multiple states are permitted at any stage. This accommodates rule systems in which there is more than one possible way to perform an update.A simple example is a string substitution system. For instance, take the rules and the initial condition . There are two choices for how to proceed. Applying the first rule yields the evolution , while applying the second rule yields the evolution . So at the first step, there is a single state (), at the second step there are two states , and at the third step there is a single state .A path through a multiway system arising from a choice of which substitutions to make is called an evolution. Typically, a multiway system will have a large number of possible evolutions. For example, consider strings of s and s with the rule . Then most strings will have more than one occurrence of the substring , and each occurrence leads down another path in the multiway system...

A causal network is an acyclic digraph arising from an evolution of a substitution system, and representing its history. The illustration above shows a causal network corresponding to the rules (applied in a left-to-right scan) and initial condition (Wolfram 2002, p. 498, fig. a).The figure above shows the procedure for diagrammatically creating a causal network from a mobile automaton (Wolfram 2002, pp. 488-489).In an evolution of a multiway system, each substitution event is a vertex in a causal network. Two events which are related by causal dependence, meaning one occurs just before the other, have an edge between the corresponding vertices in the causal network. More precisely, the edge is a directed edge leading from the past event to the future event.Some causal networks are independent of the choice of evolution, and these are calledcausally invariant...

Turing machines are defined by sets of rules that operate on four parameters: (state, tape cell color, operation, state). Let the states and tape cell colors be numbered and represented by quadruples of ordinal numbers. Then there exist algorithmic procedures that sequentially list all consistent sets of Turing machine rules. A set of rules is called consistent if any two quadruples differ in the first or second element out of the four. Any such procedure gives both an algorithm for going from any integer to its corresponding Turing machine and an algorithm for getting the index of any consistent set of Turing machine rules.Assume that one such procedure is selected. If Turing machine is defined by the set of quadruples whose index is , then is called the Gödel number of . The result of application of Turing machine with Godel number to is usually denoted .Given the equivalence of computability and recursiveness, it is common to use Gödel..

The term "recursive function" is often used informally to describe any function that is defined with recursion. There are several formal counterparts to this informal definition, many of which only differ in trivial respects.Kleene (1952) defines a "partial recursive function" of nonnegative integers to be any function that is defined by a noncontradictory system of equations whose left and right sides are composed from (1) function symbols (for example, , , , etc.), (2) variables for nonnegative integers (for example, , , , etc.), (3) the constant 0, and (4) the successor function .For example,(1)(2)(3)(4)defines to be the function that computes the product of and .Note that the equations might not uniquely determine the value of for every possible input, and in that sense the definition is "partial." If the system of equations determines the value of f for every input, then the definition is said to be "total."..

As first shown by Meyer and Ritchie (1967), do-loops (which have a fixed iteration limit) are a special case of while-loops. A function that can be implemented using only do-loops is called primitive recursive. (In contrast, a computable function can be coded using a combination of for- and while-loops, or while-loops only.) Examples of primitive recursive functions include power, greatest common divisor, and (the function giving the th prime).The Ackermann function is the simplest example of a well-defined total function that is computable but not primitive recursive, providing a counterexample to the belief in the early 1900s that every computable function was also primitive recursive (Dötzel 1991; Wolfram 2002, p. 907).

If is an ordinary point of the ordinary differential equation, expand in a Taylor series about . Commonly, the expansion point can be taken as , resulting in the Maclaurin series(1)Plug back into the ODE and group the coefficients by power. Now, obtain a recurrence relation for the th term, and write the series expansion in terms of the s. Expansions for the first few derivatives are(2)(3)(4)(5)(6)If is a regular singular point of the ordinary differential equation,(7)solutions may be found by the Frobenius method or by expansion in a Laurent series. In the Frobenius method, assume a solution of the form(8)so that(9)(10)(11)(12)Now, plug back into the ODE and group the coefficients by power to obtain a recursion formula for the th term, and then write the series expansion in terms of the s. Equating the term to 0 will produce the so-called indicial equation, which will give the allowed values of in the series expansion.As an example, consider the..

For a second-order ordinarydifferential equation,(1)Assume that linearly independent solutions and are known to the homogeneous equation(2)and seek and such that(3)(4)Now, impose the additional condition that(5)so that(6)(7)Plug , , and back into the original equation to obtain(8)which simplifies to(9)Combing equations (◇) and (9) and simultaneously solving for and then gives(10)(11)where(12)is the Wronskian, which is a function of only, so these can be integrated directly to obtain(13)(14)which can be plugged in to give the particular solution(15)Generalizing to an th degree ODE, let , ..., be the solutions to the homogeneous ODE and let , ..., be chosen such that(16)and the particular solution is then(17)

The van der Pol equation is an ordinary differential equation that can be derived from the Rayleigh differential equation by differentiating and setting . It is an equation describing self-sustaining oscillations in which energy is fed into small oscillations and removed from large oscillations. This equation arises in the study of circuits containing vacuum tubes and is given byIf , the equation reduces to the equation of simple harmonic motion

Let be a Euclidean space, be the dot product, and denote the reflection in the hyperplane bywhereThen a subset of the Euclidean space is called a root system in if: 1. is finite, spans , and does not contain 0, 2. If , the reflection leaves invariant, and 3. If , then . The Lie algebra roots of a semisimple Lie algebra are a root system, in a real subspace of the dual vector space to the Cartan subalgebra. In this case, the reflections generate the Weyl group, which is the symmetry group of the root system.

The formal mathematical study of the methods, structure, and validity of mathematical deduction and proof.In Hilbert's day, formal logic sought to devise a complete, consistent formulation of mathematics such that propositions could be formally stated and proved using a small number of symbols with well-defined meanings. The difficulty of formal logic was demonstrated in the monumental Principia Mathematica (1925) of Whitehead and Russell's, in which hundreds of pages of symbols were required before the statement could be deduced.The foundations of this program were obliterated in the mid 1930s when Gödel unexpectedly proved a result now known as Gödel's second incompleteness theorem. This theorem not only showed Hilbert's goal to be impossible, but also proved to be only the first in a series of deep and counterintuitive statements about rigor and provability in mathematics.A very simple form of logic is the study of "truth..

Consider a Boolean algebra of subsets generated by a set , which is the set of subsets of that can be obtained by means of a finite number of the set operations union, intersection, and complementation. Then each of the elements of is called a Boolean function generated by (Comtet 1974, p. 185). Each Boolean function has a unique representation (up to order) as a union of complete products. It follows that there are inequivalent Boolean functions for a set with cardinality (Comtet 1974, p. 187).In 1938, Shannon proved that a two-valued Boolean algebra (whose members are most commonly denoted 0 and 1, or false and true) can describe the operation of two-valued electrical switching circuits. The following table gives the truth table for the possible Boolean functions of two binary variables.00000000000100001111100011001111010101010011111111010000111110001100111101010101The names and symbols for these functions are given..

Maximum likelihood, also called the maximum likelihood method, is the procedure of finding the value of one or more parameters for a given statistic which makes the known likelihood distribution a maximum. The maximum likelihood estimate for a parameter is denoted .For a Bernoulli distribution,(1)so maximum likelihood occurs for . If is not known ahead of time, the likelihood function is(2)(3)(4)where or 1, and , ..., .(5)(6)Rearranging gives(7)so(8)For a normal distribution,(9)(10)so(11)and(12)giving(13)Similarly,(14)gives(15)Note that in this case, the maximum likelihood standard deviation is the sample standard deviation, which is a biased estimator for the population standard deviation.For a weighted normal distribution,(16)(17)(18)gives(19)The variance of the mean isthen(20)But(21)so(22)(23)(24)For a Poisson distribution,(25)(26)(27)(28)..

An estimator is a rule that tells how to calculate an estimate based on the measurements contained in a sample. For example, the sample mean is an estimator for the population mean .The mean square error of an estimator is defined by(1)Let be the estimator bias, then(2)(3)(4)where is the estimator variance.

A statistic which assigns a single number to several individual statistics in order to quantify trends. The best-known index in the United States is the consumer price index, which gives a sort of "average" value for inflation based on price changes for a group of selected products. The Dow Jones and NASDAQ indexes for the New York and American Stock Exchanges, respectively, are also index numbers.Let be the price per unit in period , be the quantity produced in period , and be the value of the units. Let be the estimated relative importance of a product. There are several types of indices defined, among them those listed in the following table. indexabbr.formulaBowley indexFisher indexgeometric mean indexharmonic mean indexLaspeyres' indexmarshall-Edgeworth indexmitchell indexPaasche's indexWalsh index..

For a given bounded function over a partition of a given interval, the upper sum is the sum of box areas using the supremum of the function in each subinterval .

An integral is a mathematical object that can be interpreted as an area or a generalization of area. Integrals, together with derivatives, are the fundamental objects of calculus. Other words for integral include antiderivative and primitive. The Riemann integral is the simplest integral definition and the only one usually encountered in physics and elementary calculus. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."The Riemann integral of the function over from to is written(1)Note that if , the integral is written simply(2)as opposed to .Every definition of an integral is based on a particular measure. For instance, the Riemann integral is based on Jordan measure, and the Lebesgue integral is based..

An improper integral is a definite integral that has either or both limits infinite or an integrand that approaches infinity at one or more points in the range of integration. Improper integrals cannot be computed using a normal Riemann integral.For example, the integral(1)is an improper integral. Some such integrals can sometimes be computed by replacing infinite limits with finite values(2)and then taking the limit as ,(3)(4)(5)Improper integrals of the form(6)with one infinite limit and the other nonzero may also be expressed as finite integrals over transformed functions. If decreases at least as fast as , then let(7)(8)(9)(10)and(11)(12)If diverges as for , let(13)(14)(15)(16)(17)and(18)If diverges as for , let(19)(20)(21)and(22)(23)If the integral diverges exponentially, then let(24)(25)(26)and(27)..

The Riemann integral is the definite integral normally encountered in calculus texts and used by physicists and engineers. Other types of integrals exist (e.g., the Lebesgue integral), but are unlikely to be encountered outside the confines of advanced mathematics texts. In fact, according to Jeffreys and Jeffreys (1988, p. 29), "it appears that cases where these methods [i.e., generalizations of the Riemann integral] are applicable and Riemann's [definition of the integral] is not are too rare in physics to repay the extra difficulty."The Riemann integral is based on the Jordan measure,and defined by taking a limit of a Riemann sum,(1)(2)(3)where and , , and are arbitrary points in the intervals , , and , respectively. The value is called the mesh size of a partition of the interval into subintervals .As an example of the application of the Riemann integral definition, find the area under the curve from 0 to . Divide into segments,..

Let be a path given parametrically by . Let denote arc length from the initial point. Then(1)(2)where .

A double integral is a two-fold multiple integral.Examples of definite double integrals evaluating to simple constants include(1)(2)(3)(4)where is Catalan's constant (Borwein et al. 2004, pp. 48-49), and(5)where is the Euler-Mascheroni constant (Sondow 2003, 2005; Borwein et al. 2004, pp. 48-49).

A prime link is a link that cannot be represented as a knot sum of other links. Doll and Hoste (1991) list polynomials for oriented links of nine or fewer crossings, and Rolfsen (1976) gives a table of links with small numbers of components and crossings.The following table summarizes the number of distinct prime -components links having specified crossing numbers. ThecomponentsOEISprime -component links with 1, 2, ... crossings1A0028630, 0, 1, 1, 2, 3, 7, 21, 49, 165, 552, ...2A0489520, 1, 0, 1, 1, 3, 8, 16, 61, 185, 638, ...3A0489530, 0, 0, 0, 0, 3, 1, 10, 21, 74, 329, ...4A0870710, 0, 0, 0, 0, 0, 0, 3, 1, 15, 39, ...50, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, ...totalA0867710, 1, 1, 2, 3, 9, 16, 50, 132, 442, 1559, ...The following table lists some named links. The notation and ordering follows that of Rolfsen (1976), where denotes the th -component link with crossing number .link numbernameunlinkHopf linkWhitehead linkBorromean ringsA listing of the first few simple..

There are several different definition of link.In knot theory, a link is one or more disjointly embedded circles in three-space. More informally, a link is an assembly of knots with mutual entanglements. Kuperberg (1994) has shown that a nontrivial knot or link in has four collinear points (Eppstein). Like knots, links can be decomposed into basic units known as prime links.The term "link" is also used primarily by physicists to refer to a graphedge.

A -torus knot is obtained by looping a string through the hole of a torus times with revolutions before joining its ends, where and are relatively prime. A -torus knot is equivalent to a -torus knot. All torus knots are prime (Hoste et al. 1998, Burde and Zieschang 2002). Torus knots are all chiral, invertible, and have symmetry group (Schreier 1924, Hoste et al. 1998).Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are torus knots using the function KnotData[knot, "Torus"].The link crossing number of a -torus knot is(1)(Williams 1988, Murasugi and Przytycki 1989, Murasugi 1991, Hoste et al. 1998). The unknotting number of a -torus knot is(2)(Adams 1991).The numbers of torus knots with crossings are 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, ... (OEIS A051764). Torus knots with fewer than 11 crossings are summarized in the following table (Adams et al. 1991) and the first few are illustrated above.knotnametrefoil..

In mathematics, a knot is defined as a closed, non-self-intersecting curve that is embedded in three dimensions and cannot be untangled to produce a simple loop (i.e., the unknot). While in common usage, knots can be tied in string and rope such that one or more strands are left open on either side of the knot, the mathematical theory of knots terms an object of this type a "braid" rather than a knot. To a mathematician, an object is a knot only if its free ends are attached in some way so that the resulting structure consists of a single looped strand.A knot can be generalized to a link, which is simply a knottedcollection of one or more closed strands.The study of knots and their properties is known as knot theory. Knot theory was given its first impetus when Lord Kelvin proposed a theory that atoms were vortex loops, with different chemical elements consisting of different knotted configurations (Thompson 1867). P. G. Tait then..

The distribution function , also called the cumulative distribution function (CDF) or cumulative frequency function, describes the probability that a variate takes on a value less than or equal to a number . The distribution function is sometimes also denoted (Evans et al. 2000, p. 6).The distribution function is therefore related to a continuous probability density function by(1)(2)so (when it exists) is simply the derivative of the distribution function(3)Similarly, the distribution function is related to a discrete probability by(4)(5)There exist distributions that are neither continuous nor discrete.A joint distribution function can bedefined if outcomes are dependent on two parameters:(6)(7)(8)Similarly, a multivariate distribution function can be defined if outcomes depend on parameters:(9)The probability content of a closed region can be found much more efficiently than by direct integration of the probability..

A statistical distribution whose variables can take on only discrete values. Abramowitz and Stegun (1972, p. 929) give a table of the parameters of most common discrete distributions.A discrete distribution with probability function defined over , 2, ..., has distribution functionand population mean

The Galton board, also known as a quincunx or bean machine, is a device for statistical experiments named after English scientist Sir Francis Galton. It consists of an upright board with evenly spaced nails (or pegs) driven into its upper half, where the nails are arranged in staggered order, and a lower half divided into a number of evenly-spaced rectangular slots. The front of the device is covered with a glass cover to allow viewing of both nails and slots. In the middle of the upper edge, there is a funnel into which balls can be poured, where the diameter of the balls must be much smaller than the distance between the nails. The funnel is located precisely above the central nail of the second row so that each ball, if perfectly centered, would fall vertically and directly onto the uppermost point of this nail's surface (Kozlov and Mitrofanova 2002). The figure above shows a variant of the board in which only the nails that can potentially be hit by a ball..

Let a set of random variates , , ..., have a probability function(1)where are nonnegative integers such that(2)and are constants with and(3)Then the joint distribution of , ..., is a multinomial distribution and is given by the corresponding coefficient of the multinomial series(4)In the words, if , , ..., are mutually exclusive events with , ..., . Then the probability that occurs times, ..., occurs times is given by(5)(Papoulis 1984, p. 75).The mean and variance of are(6)(7)The covariance of and is(8)

A normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range ,(1)It is related to the probability integral(2)by(3)Let so . Then(4)Here, erf is a function sometimes called the error function. The probability that a normal variate assumes a value in the range is therefore given by(5)Neither nor erf can be expressed in terms of finite additions, subtractions, multiplications, and root extractions, and so must be either computed numerically or otherwise approximated.Note that a function different from is sometimes defined as "the" normal distribution function(6)(7)(8)(9)(Feller 1968; Beyer 1987, p. 551), although this function is less widely encountered than the usual . The notation is due to Feller (1971).The value of for which falls within the interval with a given probability is a related quantity called the confidence interval.For small values..

Given a Poisson distribution with a rate of change , the distribution function giving the waiting times until the th Poisson event is(1)(2)for , where is a complete gamma function, and an incomplete gamma function. With explicitly an integer, this distribution is known as the Erlang distribution, and has probability function(3)It is closely related to the gamma distribution, which is obtained by letting (not necessarily an integer) and defining . When , it simplifies to the exponential distribution.Evans et al. (2000, p. 71) write the distribution using the variables and .

A standard normal distribution is a normal distribution with zero mean () and unit variance (), given by the probability density function and distribution function(1)(2)over the domain .It has mean, variance, skewness,and kurtosis excess given by(3)(4)(5)(6)The first quartile of the standard normal distribution occurs when , which is(7)(8)(OEIS A092678; Kenney and Keeping 1962, p. 134), where is the inverse erf function. The absolute value of this is known as the probable error.

Let be the sum of prime factors (with repetition) of a number . For example, , so . Then for , 2, ... is given by 0, 2, 3, 4, 5, 5, 7, 6, 6, 7, 11, 7, 13, 9, 8, ... (OEIS A001414). The sum of prime factors function is also known as the integer logarithm.The high-water marks are 0, 2, 3, 4, 5, 7, 11, 13, 17, ..., which occur at positions 1, 2, 3, 4, 5, 7, 11, 13, 17, ... (OEIS A046022), which, with the exception of the first term, correspond exactly to the actual values of the high-water marks.If is considered to be 0 for a prime, then the sequence of high-water marks is 0, 4, 5, 6, 7, 9, 10, 13, 15, 19, 21, 25, 31, 33, ... (OEIS A088685), which occur at positions 1, 4, 6, 8, 10, 14, 21, 22, 26, 34, 38, 46, 58, ... (OEIS A088686). Rather amazingly, if the first 7 terms are dropped, then the last digit of the high-water marks and the last digit of their positions fall into one of the four patterns , (3, 2), (5, 6), or (9, 4) (A. Jones, pers. comm., October 5, 2003).Now consider iterating..

Fermat's theorem, sometimes called Fermat's two-square theorem or simply "Fermat's theorem," states that a prime number can be represented in an essentially unique manner (up to the order of addends) in the form for integer and iff or (which is a degenerate case with ). The theorem was stated by Fermat, but the first published proof was by Euler.The first few primes which are 1 or 2 (mod 4) are 2, 5, 13, 17, 29, 37, 41, 53, 61, ... (OEIS A002313) (with the only prime congruent to 2 mod 4 being 2). The numbers such that equal these primes are (1, 1), (1, 2), (2, 3), (1, 4), (2, 5), (1, 6), ... (OEIS A002331 and A002330).The theorem can be restated by lettingthen all relatively prime solutions to the problem of representing for any integer are achieved by means of successive applications of the genus theorem and composition theorem...

There are several types of numbers that are commonly termed "lucky numbers."The first is the lucky numbers of Euler. The second is obtained by writing out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The first odd number is 3, so strike out every third number from the list: 1, 3, 7, 9, 13, 15, 19, .... The first odd number greater than 3 in the list is 7, so strike out every seventh number: 1, 3, 7, 9, 13, 15, 21, 25, 31, ....Numbers remaining after this procedure has been carried out completely are called lucky numbers. The first few are 1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, ... (OEIS A000959). Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is , just as the prime number theorem, and the frequency of twin primes and twin lucky numbers are similar. A version of the Goldbach conjecture also seems to hold.It therefore appears that the sieving process accountsfor many properties of the primes...

An integer is -smooth if it has no prime factors . The following table gives the first few -smooth numbers for small . Berndt (1994, p. 52) called the 7-smooth numbers "highly composite numbers."OEIS-smooth numbers2A0000791, 2, 4, 8, 16, 32, 64, 128, 256, 512, ...3A0035861, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, ...5A0510371, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, ...7A0024731, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, ...11A0510381, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, ...The probability that a random positive integer is -smooth is , where is the number of -smooth numbers . This fact is important in application of Kraitchik's extension of Fermat's factorization method because it is related to the number of random numbers which must be examined to find a suitable subset whose product is a square.Since about -smooth numbers must be found (where is the prime counting function), the number of random numbers which must be examined is about . But because it..

A round number is a number that is the product of a considerable number of comparatively small factors (Hardy 1999, p. 48). Round numbers are very rare. As Hardy (1999, p. 48) notes, "Half the numbers are divisible by 2, one-third by 3, one-sixth by both 2 and 3, and so on. Surely, then we may expect most numbers to have a large number of factors. But the facts seem to show the opposite."A positive integer is sometimes said to be round (or "square root-smooth") if it has no prime factors greater than . The first few such numbers are 1, 4, 8, 9, 12, 16, 18, 24, 25, 27, 30, 32, ... (OEIS A048098). Using this definition, an asymptotic formula for the number of round integers less than or equal to a positive real number is given by(Hildebrand).A different meaning of "round" is used when speaking of "roundinga number."..

To define a recurring digital invariant of order , compute the sum of the th powers of the digits of a number . If this number is equal to the original number , then is called a -Narcissistic number. If not, compute the sums of the th powers of the digits of , and so on. If this process eventually leads back to the original number , the smallest number in the sequence is said to be a -recurring digital invariant. For example,(1)(2)(3)so 55 is an order 3 recurring digital invariant. The following table gives recurring digital invariants of orders 2 to 10 (Madachy 1979).orderRDIcycle lengths248355, 136, 160, 9193, 2, 3, 241138, 21787, 25244, 8294, 8299, 9044, 9045, 10933,28, 10, 6, 10, 22, 4, 12, 2, 224584, 58618, 89883617148, 63804, 93531, 239459, 28259530, 2, 4, 10, 3780441, 86874, 253074, 376762,92, 56, 27, 30, 14, 21922428, 982108, five more86822, 7973187, 86168049322219, 2274831, 20700388, eleven more1020818070, five more..

Start with an integer , known as the digitaddition generator. Add the sum of the digitaddition generator's digits to obtain the digitaddition . A number can have more than one digitaddition generator. If a number has no digitaddition generator, it is called a self number. The sum of all numbers in a digitaddition series is given by the last term minus the first plus the sum of the digits of the last.If the digitaddition process is performed on to yield its digitaddition , on to yield , etc., a single-digit number, known as the digital root of , is eventually obtained. The digital roots of the first few integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, ... (OEIS A010888).If the process is generalized so that the th (instead of first) powers of the digits of a number are repeatedly added, a periodic sequence of numbers is eventually obtained for any given starting number . For example, the 2-digitaddition sequence for is given by 2, , , , , , , and so on.If..

A number is said to be pandigital if it contains each of the digits from 0 to 9 (and whose leading digit must be nonzero). However, "zeroless" pandigital quantities contain the digits 1 through 9. Sometimes exclusivity is also required so that each digit is restricted to appear exactly once. For example, 6729/13458 is a (zeroless, restricted) pandigital fraction and 1023456789 is the smallest (zerofull) pandigital number.The first few zerofull restricted pandigital numbers are 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, ... (OEIS A050278). A 10-digit pandigital number is always divisible by 9 sinceThis passes the divisibility test for 9 since .The smallest unrestricted pandigital primes must therefore have 11 digits (no two of which can be 0). The first few unrestricted pandigital primes are therefore 10123457689, 10123465789, 10123465897, 10123485679, ... (OEIS A050288).If zeros are excluded, the..

Numbers which are not perfect and for whichor equivalentlywhere is the divisor function. Deficient numbers are sometimes called defective numbers (Singh 1997). Primes, prime powers, and any divisors of a perfect or deficient number are all deficient. The first few deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, ... (OEIS A005100).

The Kaprekar routine is an algorithm discovered in 1949 by D. R. Kaprekar for 4-digit numbers, but which can be generalized to -digit numbers. To apply the Kaprekar routine to a number , arrange the digits in descending () and ascending () order. Now compute (discarding any initial 0s) and iterate, where is sometimes called the Kaprekar function. The algorithm reaches 0 (a degenerate case), a constant, or a cycle, depending on the number of digits in and the value of . The list of values is sometimes called a Kaprekar sequence, and the result is sometimes called a Kaprekar number (Deutsch and Goldman 2004), though this nomenclature should be deprecated because of confusing with the distinct sort of Kaprekar number.In base-10, the numbers for which are given by 495, 6174, 549945, 631764, ... (OEIS A099009). Similarly, the numbers for which iterating gives a cycle of length are given by 53955, 59994, 61974, 62964, 63954, 71973, ... (OEIS..

The reversal of a positive integer is . The reversal of a positive integer is implemented in the Wolfram Language as IntegerReverse[n].A positive integer that is the same as its own reversal is known as a palindromicnumber.Ball and Coxeter (1987) consider numbers whose reversals are integral multiples of themselves. Palindromic numbers and numbers ending with a zero are trivial examples.The first few nontrivial examples of numbers whose reversals are multiples of themselves are 8712, 9801, 87912, 98901, 879912, 989901, 8799912, 9899901, 87128712, 87999912, 98019801, 98999901, ... (OEIS A031877). The pattern continues for large numbers, with numbers of the form equal to 4 times their reversals and numbers of the form equal to 9 times their reversals. In addition, runs of numbers of either of these forms can be concatenated to yield numbers of the form , equal to 4 times their reversals, and , equal to 9 times their reversals.The reversals..

An abundant number, sometimes also called an excessive number, is a positive integer for which(1)where is the divisor function and is the restricted divisor function. The quantity is sometimes called the abundance.A number which is abundant but for which all its proper divisors are deficient is called a primitive abundant number (Guy 1994, p. 46).The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (OEIS A005101).Every positive integer with is abundant. Any multiple of a perfect number or an abundant number is also abundant. Prime numbers are not abundant. Every number greater than 20161 can be expressed as a sum of two abundant numbers.There are only 21 abundant numbers less than 100, and they are all even.The first odd abundant number is(2)That 945 is abundant can be seen by computing(3)Define the density function(4)(correcting the expression in Finch 2003, p. 126) for a positive real number where gives the cardinal..

The decimal expansion of a number is its representation in base-10 (i.e., in the decimal system). In this system, each "decimal place" consists of a digit 0-9 arranged such that each digit is multiplied by a power of 10, decreasing from left to right, and with a decimal place indicating the s place. For example, the number with decimal expansion 1234.56 is defined as(1)(2)Expressions written in this form (where negative are allowed as exemplified above but usually not considered in elementary education contexts) are said to be in expanded notation.Other examples include the decimal expansion of given by 625, of given by 3.14159..., and of given by 0.1111.... The decimal expansion of a number can be found in the Wolfram Language using the command RealDigits[n], or equivalently, RealDigits[n, 10].The decimal expansion of a number may terminate (in which case the number is called a regular number or finite decimal, e.g., ), eventually..

The base 2 method of counting in which only the digits 0 and 1 are used. In this base, the number 1011 equals . This base is used in computers, since all numbers can be simply represented as a string of electrically pulsed ons and offs. In computer parlance, one binary digit is called a bit, two digits are called a crumb, four digits are called a nibble, and eight digits are called a byte.An integer may be represented in binary in the Wolfram Language using the command BaseForm[n, 2], and the first digits of a real number may be obtained in binary using RealDigits[x, 2, d]. Finally, a list of binary digits can be converted to a decimal rational number or integer using FromDigits[l, 2].The illustration above shows the binary numbers from 0 to 63 represented graphically (Wolfram 2002, p. 117), and the following table gives the binary equivalents of the first few decimal numbers.1111101121101012101211002210110311131101231011141001411102411000510115111125110016110161000026110107111171000127110118100018100102811100910011910011291110110101020101003011110A..

It is possible to perform multiplication of large numbers in (many) fewer operations than the usual brute-force technique of "long multiplication." As discovered by Karatsuba (Karatsuba and Ofman 1962), multiplication of two -digit numbers can be done with a bit complexity of less than using identities of the form(1)Proceeding recursively then gives bit complexity , where (Borwein et al. 1989). The best known bound is steps for (Schönhage and Strassen 1971, Knuth 1998). However, this algorithm is difficult to implement, but a procedure based on the fast Fourier transform is straightforward to implement and gives bit complexity (Brigham 1974, Borodin and Munro 1975, Borwein et al. 1989, Knuth 1998).As a concrete example, consider multiplication of two numbers each just two "digits" long in base ,(2)(3)then their product is(4)(5)(6)Instead of evaluating products of individual digits, now write(7)(8)(9)The..

A multiplication table is an array showing the result of applying a binary operator to elements of a given set . For example, the following table is the multiplication table for ordinary multiplication. 12345678910112345678910224681012141618203369121518212427304481216202428323640551015202530354045506612182430364248546077142128354249566370881624324048566472809918273645546372819010102030405060708090100The results of any binary mathematical operation can be written as a multiplication table. For example, groups have multiplication tables, where the group operation is understood as multiplication. However, different labelings and orderings of a multiplication table may describe the same abstract group. For example, the multiplication table for the cyclic group C4 may be written in three equivalent ways--denoted here by , , and --by permuting the symbols used for the group elements (Cotton 1990, p. 11).The..

Also called "Ethiopian multiplication." To multiply two numbers and , write and in two columns. Under , write , where is the floor function, and under , write . Continue until . Then cross out any entries in the column which are opposite an even number in the column and add the column. The result is the desired product. For example, for Russian multiplication works because it implements binarymultiplication: 1. If , accumulate . 2. Right-shift one bit. 3. If , exit. 4. Left-shift one bit. 5. Loop.

In simple algebra, multiplication is the process of calculating the result when a number is taken times. The result of a multiplication is called the product of and , and each of the numbers and is called a factor of the product . Multiplication is denoted , , , or simply . The symbol is known as the multiplication sign. Normal multiplication is associative, commutative, and distributive.More generally, multiplication can also be defined for other mathematical objects such as groups, matrices, sets, and tensors.Karatsuba and Ofman (1962) discovered that multiplication of two digit numbers can be done with a bit complexity of less than using an algorithm now known as Karatsuba multiplication.Eddy Grant's pop song "Electric Avenue" (Electric Avenue, 2001) includes the commentary: "Who is to blame in one country; Never can get to the one; Dealin' in multiplication; And they still can't feed everyone, oh no."..

Division by zero is the operation of taking the quotient of any number and 0, i.e., . The uniqueness of division breaks down when dividing by zero, since the product is the same for any , so cannot be recovered by inverting the process of multiplication. 0 is the only number with this property and, as a result, division by zero is undefined for real numbers and can produce a fatal condition called a "division by zero error" in computer programs.To the persistent but misguided reader who insists on asking "What happens if I do divide by zero," Derbyshire (2004, p. 36) provides the slightly flippant but firm and concise response, "You can't. It is against the rules." Even in fields other than the real numbers, division by zero is never allowed (Derbyshire 2004, p. 266).There are, however, contexts in which division by zero can be considered as defined. For example, division by zero for in the extended complex..

Taking the ratio of two numbers and , also written . Here, is called the dividend, is called the divisor, and is called a quotient. The symbol "/" is called a solidus (sometimes, the "diagonal"), and the symbol "" is called the obelus. If left unevaluated, is called a fraction, with known as the numerator and known as the denominator.Division in which the fractional (remainder) is discarded is called integer division, and is sometimes denoted using a backslash, .Division is the inverse operation of multiplication,so that ifthen can be recovered asas long as . In general, division by zero is not defined since the ability to "invert" to recover breaks down if (in which case is always 0, independent of ).Cutting or separating an object into two or more parts is also called division...

In general, a remainder is a quantity "left over" after performing a particular algorithm. The term is most commonly used to refer to the number left over when two integers are divided by each other in integer division. For example, , with a remainder of 6. Of course in real division, there is no such thing as a remainder since, for example, .The term remainder is also sometimes applied to the residueof a congruence.

Long division is an algorithm for dividing two numbers, obtaining the quotient one digit at a time. The example above shows how the division of 123456/17 is performed to obtain the result 7262.11....The term "long division" is also used to refer to the method of dividing one polynomial by another, as illustrated above. This example illustrates the resultThe symbol separating the dividend from the divisor seems to have no established name, so can be simply referred to as the long division symbol (or sometimes the division bracket).The chorus of the song "Singular Girl" by Rhett Miller (The Believer, 2006) contains the slightly cryptic line "Talking to you girl is like long division, yeah." Coincidentally, Long Division (1995) is also the name of the second album by the band Low...

The lattice method is an alternative to long multiplication for numbers. In this approach, a lattice is first constructed, sized to fit the numbers being multiplied. If we are multiplying an -digit number by an -digit number, the size of the lattice is . The multiplicand is placed along the top of the lattice so that each digit is the header for one column of cells (the most significant digit is put at the left). The multiplier is placed along the right side of the lattice so that each digit is a (trailing) header for one row of cells (the most significant digit is put at the top). Illustrated above is the lattice configuration for computing .Before the actual multiplication can begin, lines must be drawn for every diagonal path in the lattice from upper right to lower left to bisect each cell. There will be 5 diagonals for our lattice array.Now we calculate a product for each cell by multiplying the digit at the top of the column and the digit at the right of the..

The use of percentages is a way of expressing ratios in terms of whole numbers. A ratio or fraction is converted to a percentage by multiplying by 100 and appending a "percentage sign" %. For example, if an investment grows from to , then is times as much as , i.e., 173.08% of . So it is also true that the investment has grown by . A change of a certain percent is sometimes said to be a change of percentage points.

A mixed fraction is an improper fraction written in the form . In common usage such as cooking recipes, is often written as (e.g., 1 ), much to the chagrin of mathematicians, to whom means , which is quite a different beast from .(The author of this work discovered this fact early in his mathematical career after having points marked off a calculus exam for using the recipe-like notation. Future mathematicians are therefore encouraged to avoid mixed fractions, except perhaps in the kitchen.)

An Egyptian fraction is a sum of positive (usually) distinct unit fractions. The famous Rhind papyrus, dated to around 1650 BC contains a table of representations of as Egyptian fractions for odd between 5 and 101. The reason the Egyptians chose this method for representing fractions is not clear, although André Weil characterized the decision as "a wrong turn" (Hoffman 1998, pp. 153-154). The unique fraction that the Egyptians did not represent using unit fractions was 2/3 (Wells 1986, p. 29).Egyptian fractions are almost always required to exclude repeated terms, since representations such as are trivial. Any rational number has representations as an Egyptian fraction with arbitrarily many terms and with arbitrarily large denominators, although for a given fixed number of terms, there are only finitely many. Fibonacci proved that any fraction can be represented as a sum of distinct unit fractions (Hoffman..

If the period of a repeating decimal for , where is prime and is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s. For example, , and .

An irreducible fraction is a fraction for which , i.e., and are relatively prime. For example, in the complex plane, is reducible, while is not.The figure above shows the irreducible fractions plotted in the complex plane (Pickover 1997; Trott 2004, p. 29).

A common fraction is a fraction in which numerator and denominator are both integers, as opposed to fractions. For example, is a common fraction, whileis not. Common fractions are sometimes also called vulgar fractions (Derbyshire 2004, p. 171).

A reducible fraction is a fraction such that , i.e., can be written in reduced form. A fraction that is not reducible is said to be irreducible.For example, in the complex plane, is reducible, while is not.

The ratio of two numbers and is written , where is the numerator and is the denominator. The ratio of to is equivalent to the quotient . Betting odds written as correspond to . A number which can be expressed as a ratio of integers is called a rational number.

The Farey sequence for any positive integer is the set of irreducible rational numbers with and arranged in increasing order. The first few are(1)(2)(3)(4)(5)(OEIS A006842 and A006843). Except for , each has an odd number of terms and the middle term is always 1/2.Let , , and be three successive terms in a Farey series. Then(6)(7)These two statements are actually equivalent (Hardy and Wright 1979, p. 24). For a method of computing a successive sequence from an existing one of terms, insert the mediant fraction between terms and when (Hardy and Wright 1979, pp. 25-26; Conway and Guy 1996; Apostol 1997). Given with , let be the mediant of and . Then , and these fractions satisfy the unimodular relations(8)(9)(Apostol 1997, p. 99).The number of terms in the Farey sequence for the integer is(10)(11)where is the totient function and is the summatory function of , giving 2, 3, 5, 7, 11, 13, 19, ... (OEIS A005728). The asymptotic limit..

Subtraction is the operation of taking the difference of two numbers and . Here, is called the minuend, is called the subtrahend, and the symbol between the and is called the minus sign. The expression "" is read " minus ."Subtraction is the inverse of addition, so .The subtraction of a number from itself gives 0, while the subtraction of a real number from a smaller real number gives a negative real number. Subtraction of real numbers can be naturally extended to complex numbers.

The combining of two or more quantities using the plus operator. The individual numbers being combined are called addends, and the total is called the sum. The first of several addends, or "the one to which the others are added," is sometimes called the augend. The opposite of addition is subtraction.While the usual form of adding two -digit integers (which consists of summing over the columns right to left and "carrying" a 1 to the next column if the sum exceeds 9) requires operations (plus carries), two -digit integers can be added in about steps by processors using carry-lookahead addition (McGeoch 1993). Here, is the lg function, the logarithm to the base 2.

Solomon's seal knot is the prime (5,2)-torus knot with braid word . It is also known as the cinquefoil knot (a name derived from certain herbs and shrubs of the rose family which have five-lobed leaves and five-petaled flowers) or the double overhand knot. It has Arf invariant 1 and is not amphichiral, although it is invertible.The knot group of Solomon's seal knot is(1)(Livingston 1993, p. 127).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the Solomon's seal knot are(2)(3)(4)(5)(6)(7)Surprisingly, the knot 10-132 shares the same Alexander polynomial and Jones polynomial with the Solomon's seal knot. However, no knots on 10 or fewer crossings share the same BLM/Ho polynomial with it.

The operation of exchanging all points of a mathematical object with their mirror images (i.e., reflections in a mirror). Objects that do not change handedness under reflection are said to be amphichiral; those that do are said to be chiral.Consider the geometry of the left figure in which a point is reflected in a mirror (blue line). Then(1)so the reflection of is given by(2)The term reflection can also refer to the reflection of a ball, ray of light, etc. off a flat surface. As shown in the right diagram above, the reflection of a points off a wall with normal vector satisfies(3)If the plane of reflection is taken as the -plane, the reflection in two- or three-dimensional space consists of making the transformation for each point. Consider an arbitrary point and a plane specified by the equation(4)This plane has normal vector(5)and the signed point-plane distance is(6)The position of the point reflected in the given plane is therefore given by(7)(8)The..

A transformation (a.k.a., map, function) over a domain takes the elements to elements , where the range (a.k.a., image) of is defined asNote that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system, with space and objects embedded in it being viewed as fixed (a so-called alias transformation), or on the space itself relative to a fixed coordinate system (a so-called alibi transformation).Examples of transformations are summarized in the following table.TransformationCharacterizationdilationcenter of dilation, scale decrease factorexpansioncenter of expansion, scale increase factorreflectionmirror line or planerotationcenter of rotation, rotation angleshearinvariant line and shear factorstretch (1-way)invariant line and scale factorstretch (2-way)invariant lines and scale factorstranslationdisplacement..

Let and , thenis a Möbius transformation, where is the complex conjugate of . is a conformal mapping self-map of the unit disk for each , and specifically of the boundary of the unit disk to itself. The same holds for .Any conformal self-map of the unit disk to itself is a composition of a Möbius transformation with a rotation, and any conformal self-map of the unit disk can be written in the formfor some Möbius transformation and some complex number with (Krantz 1999, p. 81).

Min Max Re Im for , where is the floor function. The natural invariant of the map is

A bijective map between twometric spaces that preserves distances, i.e.,where is the map and is the distance function. Isometries are sometimes also called congruence transformations. Two figures that can be transformed into each other by an isometry are said to be congruent (Coxeter and Greitzer 1967, p. 80).An isometry of the plane is a linear transformation which preserves length. Isometries include rotation, translation, reflection, glides, and the identity map. Two geometric figures related by an isometry are said to be geometrically congruent (Coxeter and Greitzer 1967, p. 80).If a plane isometry has more than one fixed point, it must be either the identity transformation or a reflection. Every isometry of period two (two applications of the transformation preserving lengths in the original configuration) is either a reflection or a half-turn rotation. Every isometry in the plane is the product of at most three reflections..

The shear matrix is obtained from the identity matrix by inserting at , e.g.,(1)Bolt and Hobbs (1998) define a shear matrix as a matrix(2)such that(3)(4)

A transformation in which all points along a given line remain fixed while other points are shifted parallel to by a distance proportional to their perpendicular distance from . Shearing a plane figure does not change its area. The shear can also be generalized to three dimensions, in which planes are translated instead of lines.

Expansion is an affine transformation (sometimes called an enlargement or dilation) in which the scale is increased. It is the opposite of a geometric contraction, and is also sometimes called an enlargement. A central dilation corresponds to an expansion plus a translation.Another type of expansion is the process of radially displacing the edges or faces of a polyhedron (while keeping their orientations and sizes constant) while filling in the gaps with new faces (Ball and Coxeter 1987, pp. 139-140). This procedure was devised by Stott (1910), and can be used to construct all 11 amphichiral (out of 13 total) Archimedean solids. The opposite operation of expansion (i.e., inward expansion) is called contraction. Expansion is a special case of snubification in which no twist occurs.The following table summarizes some expansions of some unit edge length Platonic and Archimedean solids, where is the displacement and is the golden ratio.base..

The cross polytope is the regular polytope in dimensions corresponding to the convex hull of the points formed by permuting the coordinates (, 0, 0, ..., 0). A cross-polytope (also called an orthoplex) is denoted and has vertices and Schläfli symbol . The cross polytope is named because its vertices are located equidistant from the origin along the Cartesian axes in Euclidean space, which each such axis perpendicular to all others. A cross polytope is bounded by -simplexes, and is a dipyramid erected (in both directions) into the th dimension, with an -dimensional cross polytope as its base.In one dimension, the cross polytope is the line segment . In two dimensions, the cross polytope is the filled square with vertices , , , . In three dimensions, the cross polytope is the convex hull of the octahedron with vertices , , , , , . In four dimensions, the cross polytope is the 16-cell, depicted in the above figure by projecting onto one of the four mutually..

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