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Let a convex polygon be inscribed in a circle and divided into triangles from diagonals from one polygon vertex. The sum of the radii of the circles inscribed in these triangles is the same independent of the polygon vertex chosen (Johnson 1929, p. 193).If a triangle is inscribed in a circle, another circle inside the triangle, a square inside the circle, another circle inside the square, and so on. Then the equation relating the inradius and circumradius of a regular polygon,(1)gives the ratio of the radii of the final to initial circles as(2)Numerically,(3)(OEIS A085365), where is the corresponding constant for polygon circumscribing. This constant is termed the Kepler-Bouwkamp constant by Finch (2003). Kasner and Newman's (1989) assertion that is incorrect, as is the value of 0.8700... given by Prudnikov et al. (1986, p. 757)...

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..

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,..

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..

A Chaitin's constant, also called a Chaitin omega number, introduced by Chaitin (1975), is the halting probability of a universal prefix-free (self-delimiting) Turing machine. Every Chaitin constant is simultaneously computably enumerable (the limit of a computable, increasing, converging sequence of rationals), and algorithmically random (its binary expansion is an algorithmic random sequence), hence uncomputable (Chaitin 1975).A Chaitin's constant can therefore be defined as(1)which gives the probability that for any set of instructions, a particular prefix-free universal Turing machine will halt, where is the size in bits of program .The value of a Chaitin constant is highly machine-dependent. In some cases, it can even be proved that not a single bit can be computed (Solovay 2000).Chaitin constants are perhaps the most obvious specific example of uncomputable numbers. They are also known to be transcendental.Calude et..

A Pierpont prime is a prime number of the form . The first few Pierpont primes are 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, ... (OEIS A005109).A regular polygon of sides can be constructed by ruler, compass and angle-trisector iffwhere , , ..., are distinct Pierpont primes and (Gleason 1998).The numbers of Pierpont primes less than , , ... are 4, 10, 18, 25, 32, 42, 50, 58, ... (OEIS A113420) and the number less than , , , , ... are 4, 10, 25, 58, 125, 250, 505, 1020, 2075, 4227, ... (OEIS A113412; Caldwell).As of Apr. 2010, the largest known Pierpont prime is , which has decimal digits (https://primes.utm.edu/primes/page.php?id=87449).

The problem of finding the mean triangle area of a triangle with vertices picked inside a triangle with unit area was proposed by Watson (1865) and solved by Sylvester. It solution is a special case of the general formula for polygon triangle picking.Since the problem is affine, it can be solved by considering for simplicity an isosceles right triangle with unit leg lengths. Integrating the formula for the area of a triangle over the six coordinates of the vertices (and normalizing to the area of the triangle and region of integration by dividing by the integral of unity over the region) gives(1)(2)where(3)is the triangle area of a triangle with vertices , , and .The integral can be solved using computer algebra by breaking up the integration region using cylindrical algebraic decomposition. This results in 62 regions, 30 of which have distinct integrals, each of which can be directly integrated. Combining the results then gives the result(4)(Pfiefer..

The totient function , also called Euler's totient function, is defined as the number of positive integers that are relatively prime to (i.e., do not contain any factor in common with) , where 1 is counted as being relatively prime to all numbers. Since a number less than or equal to and relatively prime to a given number is called a totative, the totient function can be simply defined as the number of totatives of . For example, there are eight totatives of 24 (1, 5, 7, 11, 13, 17, 19, and 23), so .The totient function is implemented in the WolframLanguage as EulerPhi[n].The number is called the cototient of and gives the number of positive integers that have at least one prime factor in common with . is always even for . By convention, , although the Wolfram Language defines EulerPhi[0] equal to 0 for consistency with its FactorInteger[0] command. The first few values of for , 2, ... are 1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, ... (OEIS A000010). The totient function is..

The (not necessarily regular) tetrahedron of least volume circumscribed around a convex body with volume is not known. If is a parallelepiped, then the smallest-volume tetrahedron containing it has volume 9/2. It is conjectured that this is the worst possible fit for the general problem, but this remains unproved.

Pick three points , , and distributed independently and uniformly in a unit disk (i.e., in the interior of the unit circle). Then the average area of the triangle determined by these points is(1)Using disk point picking, this can be writtenas(2)where(3)A trigonometric substitution can then be used to remove the trigonometric functions and split the integral into(4)where(5)(6)However, the easiest way to evaluate the integral is using Crofton's formula and polar coordinates to yield a mean triangle area(7)for unit-radius disks (OEIS A189511), or(8)for unit-area disks (OEIS A093587; Woolhouse 1867; Solomon 1978; Pfiefer 1989; Zinani 2003). This problem is very closely related to Sylvester's four-point problem, and can be derived as the limit as of the general polygon triangle picking problem.The distribution of areas, illustrated above, is apparently not known exactly.The probability that three random points in a disk form an acute..

The mean triangle area of a triangle picked at random inside a unit cube is , with variance .The distribution of areas, illustrated above, is apparently not known exactly.The probability that a random triangle in a cube is obtuse is approximately .

Sphere tetrahedron picking is the selection of quadruples of of points corresponding to vertices of a tetrahedron with vertices on the surface of a sphere. random tetrahedra can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 4].Pick four points on a sphere. What is the probability that the tetrahedron having these points as polyhedron vertices contains the center of the sphere? In the one-dimensional case, the probability that a second point is on the opposite side of 1/2 is 1/2. In the two-dimensional case, pick two points. In order for the third to form a triangle containing the center, it must lie in the quadrant bisected by a line segment passing through the center of the circle and the bisector of the two points. This happens for one quadrant, so the probability is 1/4. Similarly, for a sphere the probability is one octant, or 1/8.Pick four points at random on the surface of a unit sphereusing(1)(2)(3)with..

Given a simplex of unit content in Euclidean -space, pick points uniformly and independently at random, and denote the expected content of their convex hull by . Exact values are known only for and 2.(1)(2)(Buchta 1984, 1986), giving the first few values 0, 1/3, 1/2, 3/5, 2/3, 5/7, ...(OEIS A026741 and A026741).(3)(4)where is a harmonic number (Buchta 1984, 1986), giving the first few values 0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, ... (OEIS A093762 and A093763).Not much is known about , although(5)(Buchta 1983, 1986) and(6)(Buchta 1986).Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of points chosen at random in a three-dimensional simplex for arbitrary .

Let two points and be picked randomly from a unit -dimensional hypercube. The expected distance between the points , i.e., the mean line segment length, is then(1)This multiple integral has been evaluated analytically only for small values of . The case corresponds to the line line picking between two random points in the interval .The first few values for are given in the following table.OEIS1--0.3333333333...2A0915050.5214054331...3A0730120.6617071822...4A1039830.7776656535...5A1039840.8785309152...6A1039850.9689420830...7A1039861.0515838734...8A1039871.1281653402...The function satisfies(2)(Anderssen et al. 1976), plotted above together with the actual values.M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the -dimensional integral to an integral over a 1-dimensional integrand such that(3)The first few values are(4)(5)(6)(7)In the limit as , these..

Ball triangle picking is the selection of triples of points (corresponding to vertices of a general triangle) randomly placed inside a ball. random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].The distribution of areas of a triangle with vertices picked at random in a unit ball is illustrated above. The mean triangle area is(1)(Buchta and Müller 1984, Finch 2010). random triangles can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 3].The determination of the probability for obtaining an acute triangle by picking three points at random in the unit disk was generalized by Hall (1982) to the -dimensional ball. Buchta (1986) subsequently gave closed form evaluations for Hall's integrals. Let be the probability that three points chosen independently and uniformly from the -ball form an acute triangle, then (2)(3)These can be combined..

Andrica's conjecture states that, for the th prime number, the inequalityholds, where the discrete function is plotted above. The high-water marks for occur for , 2, and 4, with , with no larger value among the first primes. Since the Andrica function falls asymptotically as increases, a prime gap of ever increasing size is needed to make the difference large as becomes large. It therefore seems highly likely the conjecture is true, although this has not yet been proven. bears a strong resemblance to the prime difference function, plotted above, the first few values of which are 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, ... (OEIS A001223).A generalization of Andrica's conjecture considers the equationand solves for . The smallest such is (OEIS A038458), known as the Smarandache constant, which occurs for and (Perez)...

Let be the number of (0,1)-matrices with no adjacent 1s (in either columns or rows). For , 2, ..., is given by 2, 7, 63, 1234, ... (OEIS A006506).The hard square entropy constant is defined by(OEIS A085850). It is not known if this constanthas an exact representation.The quantity arises in statistical physics (Baxter et al. 1980, Pearce and Seaton 1988), and is known as the entropy per site of hard squares. A related constant known as the hard hexagon entropy constant can also be defined.

Let be the probability that a random walk on a -D lattice returns to the origin. In 1921, Pólya proved that(1)but(2)for . Watson (1939), McCrea and Whipple (1940), Domb (1954), and Glasser and Zucker (1977) showed that(3)(OEIS A086230), where(4)(5)(6)(7)(8)(9)(OEIS A086231; Borwein and Bailey 2003, Ch. 2, Ex. 20) is the third of Watson's triple integrals modulo a multiplicative constant, is a complete elliptic integral of the first kind, is a Jacobi theta function, and is the gamma function.Closed forms for are not known, but Montroll (1956) showed that for ,(10)where(11)(12)and is a modified Bessel function of the first kind.Numerical values of from Montroll (1956) and Flajolet (Finch 2003) are given in the following table.OEIS3A0862300.3405374A0862320.1932065A0862330.1351786A0862340.1047157A0862350.08584498A0862360.0729126..

A surface of revolution is a surface generated by rotating a two-dimensional curve about an axis. The resulting surface therefore always has azimuthal symmetry. Examples of surfaces of revolution include the apple surface, cone (excluding the base), conical frustum (excluding the ends), cylinder (excluding the ends), Darwin-de Sitter spheroid, Gabriel's horn, hyperboloid, lemon surface, oblate spheroid, paraboloid, prolate spheroid, pseudosphere, sphere, spheroid, and torus (and its generalization, the toroid).The area element of the surface of revolution obtained by rotating the curve from to about the x-axis is(1)(2)so the surface area is(3)(4)(Apostol 1969, p. 286; Kaplan 1992, p. 251; Anton 1999, p. 380). If the curve is instead specified parametrically by , the surface area obtained by rotating the curve about the x-axis for if in this interval is given by(5)Similarly, the area of the surface of revolution..

Given four points chosen at random inside a unit cube, the average volume of the tetrahedron determined by these points is given by(1)where the polyhedron vertices are located at where , ..., 4, and the (signed) volume is given by the determinant(2)The integral is extremely difficult to compute, but the analytic result for the mean tetrahedron volume is(3)(OEIS A093524; Zinani 2003). Note that the result quoted in the reply to Seidov (2000) actually refers to the average volume for tetrahedron tetrahedron picking.

Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle. This problem is not affine, so a simple formula in terms of the area or linear properties of the original triangle apparently does not exist.However, if the original triangle is chosen to be an isosceles right triangle with unit legs, then the average length of a line with endpoints chosen at random inside it is given by(1)(2)(3)(OEIS A093063; M. Trott, pers. comm., Mar. 10, 2004), which is numerically surprisingly close to .Similarly, if the original triangle is chosen to be an equilateral triangle with unit side lengths, then the average length of a line with endpoints chosen at random inside it is given by(4)(5)The integrand can be split up into the four pieces(6)(7)(8)(9)As illustrated above, symmetry immediately gives and , so(10)With some effort, the integrals and can be done analytically to give..

An obtuse triangle is a triangle in which one of the angles is an obtuse angle. (Obviously, only a single angle in a triangle can be obtuse or it wouldn't be a triangle.) A triangle must be either obtuse, acute, or right.From the law of cosines, for a triangle with side lengths , , and ,(1)with the angle opposite side . For an angle to be obtuse, . Therefore, an obtuse triangle satisfies one of , , or .An obtuse triangle can be dissected into no fewer than seven acutetriangles (Wells 1986, p. 71).A famous problem is to find the chance that three points picked randomly in a plane are the polygon vertices of an obtuse triangle (Eisenberg and Sullivan 1996). Unfortunately, the solution of the problem depends on the procedure used to pick the "random" points (Portnoy 1994). In fact, it is impossible to pick random variables which are uniformly distributed in the plane (Eisenberg and Sullivan 1996). Guy (1993) gives a variety of solutions to the..

Consider the distribution of distances between a point picked at random in the interior of a unit cube and on a face of the cube. The probability function, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form(1)The first even raw moments for , 2, 4, ... are 1, 2/3, 11/18, 211/315, 187/225, 11798/10395, ....

Instead of picking two points from the interior of the cube, instead pick two points on different faces of the unit cube. In this case, the average distance between the points is(1)(OEIS A093066; Borwein and Bailey 2003, p. 26;Borwein et al. 2004, pp. 66-67). Interestingly,(2)as apparently first noted by M. Trott (pers. comm., Mar. 21, 2008).The two integrals above can be written in terms of sums as(3)(4)(Borwein et al. 2004, p. 67), where however appears to be classically divergent and perhaps must be interpreted in some regularized sense.Consider a line whose endpoints are picked at random on opposite sides of the unit cube. The probability density function for the length of this line is given by(5)(Mathai 1999; after simplification). The mean length is(6)(7)The first even raw moments for , 2, 4, ... are 1, 4/3, 167/90, 284/105, 931/225, 9868/1485, ....Consider a line whose endpoints are picked at random..

The average distance between two points chosen at random inside a unit cube (the case of hypercube line picking), sometimes known as the Robbins constant, is(1)(2)(3)(OEIS A073012; Robbins 1978, Le Lionnais 1983).The probability function as a function of line length, illustrated above, was found in (nearly) closed form by Mathai et al. (1999). After simplifying, correcting typos, and completing the integrals, gives the closed form(4)The first even raw moments for , 2, ... are 1, 1/2, 11/30, 211/630, 187/525, 3524083/6306300, ... (OEIS A160693 and A160694).Pick points on a cube, and space them as far apart as possible. The best value known for the minimum straight line distance between any two points is given in the following table. 51.118033988749861.0606601482100718190.86602540378463100.74999998333331110.70961617562351120.70710678118660130.70710678118660140.70710678118660150.625..

Square triangle picking is the selection of triples of points (corresponding to endpoints of a triangle) randomly placed inside a square. random triangles can be picked in a unit square in the Wolfram Language using the function RandomPoint[Rectangle[], n, 3].Given three points chosen at random inside a unit square, the average area of the triangle determined by these points is given analytically by the multiple integrals(1)(2)Here, represent the polygon vertices of the triangle for , 2, 3, and the (signed) area of these triangles is given by the determinant(3)(4)The solution was first given by Woolhouse (1867). Since attempting to do the integrals by brute force result in intractable integrands, the best approach using computer algebra is to divide the six-dimensional region of integration into subregions using cylindrical algebraic decomposition such that the sign of does not change, do the integral in each region directly, and then..

Square line picking is the selection of pairs of points (corresponding to endpoints of a line segment) randomly placed inside a square. random line segments can be picked in a unit square in the Wolfram Language using the function RandomPoint[Rectangle[], n, 2].Picking two points at random from the interior of a unit square, the average distance between them is the case of hypercube line picking, i.e.,(1)(2)(3)(OEIS A091505).The exact probability function is given by(4)(M. Trott, pers. comm., Mar. 11, 2004), and the corresponding distribution function by(5)From this, the mean distance can be computed, as can the variance of lengths,(6)(7)The statistical median is given by the rootof the quartic equation(8)which is approximately .The th raw moment is given for , 4, 6, ... as 1/3, 17/90, 29/210, 187/1575, 239/207, ... (OEIS A103304 and A103305).If, instead of picking two points from the interior of a square, two points are..

The mean triangle area of a triangle picked inside a regular hexagon with unit area is (Woolhouse 1867, Pfiefer 1989). This is a special case of a general polygon triangle picking result due to Alikoski (1939).The distribution of areas, illustrated above, is apparently not known exactly.

Sphere line picking is the selection of pairs of points corresponding to vertices of a line segment with endpoints on the surface of a sphere. random line segments can be picked on a unit sphere in the Wolfram Language using the function RandomPoint[Sphere[], n, 2].Pick two points at random on a unit sphere. The first one can be placed at the north pole, i.e., assigned the coordinate (0, 0, 1), without loss of generality. The second point is then chosen at random using sphere point picking, and so can be assigned coordinates(1)(2)(3)with and . The distance between first and second points is then(4)and solving for gives(5)Now the probability function for distance is then given by(6)(Solomon 1978, p. 163), since and . Here, .Therefore, somewhat surprisingly, large distances are the most common, contrary to most people's intuition. A plot of 15 random lines is shown above. The raw moments are(7)giving the first few as(8)(9)(10)(11)(OEIS..

Finch (2010) gives an overview of known results for random Gaussian triangles.Let the vertices of a triangle in dimensions be normal (normal) variates. The probability that a Gaussian triangle in dimensions is obtuse is(1)(2)(3)(4)(5)where is the gamma function, is the hypergeometric function, and is an incomplete beta function.For even ,(6)(Eisenberg and Sullivan 1996).The first few cases are explicitly(7)(8)(9)(10)(OEIS A102519 and A102520). The even cases are therefore 3/4, 15/32, 159/512, 867/4096, ... (OEIS A102556 and A102557) and the odd cases are , where , 9/8, 27/20, 837/560, ... (OEIS A102558 and A102559).

The Robbins constant is the mean line segment length, i.e., the expected distance between two points chosen at random in cube line picking, namely(1)(2)(3)(OEIS A073012; Robbins 1978, Le Lionnais 1983).

Ball tetrahedron picking is the selection of quadruples of points (corresponding to vertices of a general tetrahedron) randomly placed inside a ball. random tetrahedra can be picked in a unit ball in the Wolfram Language using the function RandomPoint[Ball[], n, 4].The mean tetrahedron volume of a tetrahedronformed by four random points in a unit ball is(OEIS A093591; Hostinsky 1925; Solomon 1978,p. 124; Zinani 2003).

Given an -ball of radius , find the distribution of the lengths of the lines determined by two points chosen at random within the ball. The probability distribution of lengths is given by(1)where(2)and(3)is a regularized beta function, with is an incomplete beta function and is a beta function (Tu and Fischbach 2000).The first few are(4)(5)(6)(7)The mean line segment lengths for and the first few dimensions are given by(8)(9)(10)(11)(OEIS A093530 and A093531 and OEIS A093532 and A093533), corresponding to line line picking, disk line picking, (3-D) ball line picking, and so on.

The mean triangle area of a triangle picked inside a regular -gon of unit area is(1)where (Alikoski 1939; Solomon 1978, p. 109; Croft et al. 1991, p. 54). Prior to Alikoski's work, only the special cases , 4, 6, 8, and had been determined. The first few cases are summarized in the following table, where is the largest root of(2)and is the largest root of(3)problem3triangle triangle picking4square triangle picking5pentagon triangle picking6hexagon triangle picking78910Amazingly, the algebraic degree of is equal to , where is the totient function, giving the first few terms for , 4, ... as 1, 1, 2, 1, 3, 2, 3, 2, 5, 2, 6, 3, 4, 4, 8, ... (OEIS A023022). Therefore, the only values of for which is rational are , 4, and 6.

The probability that a random integer between 1 and will have its greatest prime factor approaches a limiting value as , where for and is defined through the integral equation(1)for (Dickman 1930, Knuth 1998), which is almost (but not quite) a homogeneous Volterra integral equation of the second kind. The function can be given analytically for by(2)(3)(4)(Knuth 1998).Amazingly, the average value of such that is(5)(6)(7)(8)(9)which is precisely the Golomb-Dickman constant , which is defined in a completely different way!The Dickman function can be solved numerically by converting it to a delay differential equation. This can be done by noting that will become upon multiplicative inversion, so define to obtain(10)Now change variables under the integral sign by defining(11)(12)so(13)Plugging back in gives(14)To get rid of the s, define , so(15)But by the first fundamental theoremof calculus,(16)so differentiating both sides of equation..

Liouville's constant, sometimes also called Liouville's number, is the real number defined by(OEIS A012245). Liouville's constant is a decimal fraction with a 1 in each decimal place corresponding to a factorial , and zeros everywhere else. Liouville (1844) constructed an infinite class of transcendental numbers using continued fractions, but the above number was the first decimal constant to be proven transcendental (Liouville 1850). However, Cantor subsequently proved that "almost all" real numbers are in fact transcendental.A recurrence plot of the binary digits is illustratedabove.Liouville's constant nearly satisfieswhich has solution 0.1100009999... (OEIS A093409), but plugging into this equation gives instead of 0.Liouville's constant has continued fraction [0, 9, 11, 99, 1, 10, 9, 999999999999, 1, 8, 10, 1, 99, 11, 9, 999999999999999999999999999999999999999999999999999999999999999999999999,..

Planck's's radiation function is the function(1)which is normalized so that(2)However, the function is sometimes also defined without the numerical normalization factor of (e.g., Abramowitz and Stegun 1972, p. 999).The first and second raw moments are(3)(4)where is Apéry's constant, but higher order raw moments do not exist since the corresponding integrals do not converge.It has a maximum at (OEIS A133838), where(5)and inflection points at (OEIS A133839) and (OEIS A133840), where(6)

Mertens' second theorem states that the asymptotic form of the harmonic series for the sum of reciprocal primes is given bywhere is a prime, is a constant known as the Mertens constant, and is a Landau symbol.

Legendre's constant is the number 1.08366 in Legendre's guess at the primenumber theoremwith . Legendre first published a guess the formin his Essai sur la Théorie des Nombres (Edwards 2001, p. 3; Havil 2003, p. 177), but in the third edition (renamed Théorie des nombres), modified it to the form above (Derbyshire 2004, pp. 55 and 369).This expression is correct to leading term only, since it is actually true that this limit approaches 1 (Rosser and Schoenfeld 1962, Panaitopol 1999).

The hexanacci constant is the limiting ratio of adjacent hexanaccinumbers. It is the algebraic number(1)(2)(OEIS A118427), where denotes a polynomial root.

The heptanacci constant is the limiting ratio of adjacent heptanaccinumbers. It is the algebraic number(1)(2)(OEIS A118428), where denotes a polynomial root.

The tribonacci constant is ratio to which adjacent tribonaccinumbers tend, and is given by(1)(2)(3)(OEIS A058265).The tribonacci constant satisfies the identities(4)(5)(6)(7)(P. Moses, pers. comm., Feb. 21, 2005).The tribonacci constant is extremely prominent in the properties of the snubcube.

The tetranacci constant is ratio to which adjacent tetranaccinumbers tend, and is given by(1)(2)(OEIS A086088), where denotes a polynomial root.The tetranacci constant satisfies the identity(3)

The pentanacci constant is the limiting ratio of adjacent pentanaccinumbers. It is the algebraic number(1)(2)(OEIS A103814), where denotes a polynomial root.

Let and be the perimeters of the circumscribed and inscribed -gon and and the perimeters of the circumscribed and inscribed -gon. Then(1)(2)The first follows from the fact that side lengths of the polygons on a circle of radius are(3)(4)so(5)(6)But(7)(8)Using the identity(9)then gives(10)The second follows from(11)Using the identity(12)gives(13)(14)(15)(16)Successive application gives the Archimedes algorithm, which can be used to provide successive approximations to pi ().

Successive application of Archimedes' recurrence formula gives the Archimedes algorithm, which can be used to provide successive approximations to (pi). The algorithm is also called the Borchardt-Pfaff algorithm. Archimedes obtained the first rigorous approximation of by circumscribing and inscribing -gons on a circle. From Archimedes' recurrence formula, the circumferences and of the circumscribed and inscribed polygons are(1)(2)where(3)For a hexagon, and(4)(5)where . The first iteration of Archimedes' recurrence formula then gives(6)(7)(8)Additional iterations do not have simple closed forms, but the numerical approximations for , 1, 2, 3, 4 (corresponding to 6-, 12-, 24-, 48-, and 96-gons) are(9)(10)(11)(12)(13)By taking (a 96-gon) and using strict inequalities to convert irrational bounds to rational bounds at each step, Archimedes obtained the slightly looser result(14)..

The Kakeya needle problems asks for the plane figure of least area in which a line segment of width 1 can be freely rotated (where translation of the segment is also allowed). Surprisingly, there is no minimum area (Besicovitch 1928). Another iterative construction which tends to as small an area as desired is called a Perron tree (Falconer 1990, Wells 1991).When the figure is restricted to be convex, the smallest region is an equilateral triangle of unit height. Wells (1991) states that Kakeya discovered this, while Falconer (1990) attributes it to Pál.If convexity is replaced by the weaker assumption of simply-connectedness, then the area can still be arbitrarily small, but if the set is required to be star-shaped, then is a known lower bound (Cunningham 1965).The smallest simple convex domain in which one can put a segment of length 1 which will coincide with itself when rotated by has area(OEIS A093823; Le Lionnais 1983). ..

Select three points at random on the circumference of a unit circle and find the distribution of areas of the resulting triangles determined by these three points.The first point can be assigned coordinates without loss of generality. Call the central angles from the first point to the second and third and . The range of can be restricted to because of symmetry, but can range from . Then(1)so(2)(3)Therefore,(4)(5)(6)(7)But(8)(9)(10)(11)Write (10) as(12)then(13)and(14)From (12),(15)(16)(17)(18)(19)so(20)Also,(21)(22)(23)(24)so(25)Combining (◇) and (◇) gives the meantriangle area as(26)(OEIS A093582).The first few moments are(27)(28)(29)(30)(31)(32)(OEIS A093583 and A093584and OEIS A093585 and A093586).The variance is therefore given by(33)The probability that the interior of the triangle determined by the three points picked at random on the circumference of a circle contains the origin is 1/4...

Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as , and the second by , with (by symmetry, the range can be limited to instead of ). The distance between the two points is then(1)The average distance is then given by(2)The probability density function is obtained from(3)The raw moments are then(4)(5)(6)giving the first few as(7)(8)(9)(10)(11)(OEIS A000984 and OEIS A093581 and A001803), where the numerators of the odd terms are 4 times OEIS A061549.The central moments are(12)(13)(14)giving the skewness and kurtosisexcess as(15)(16)Bertrand's problem asks for the probability that a chord drawn at random on a circle of radius has length .

Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.In the Season 4 episode "Masterpiece" (2008) of the CBS-TV crime drama "Criminal Minds," the agents of the FBI Behavioral Analysis Unit are confronted by a serial killer who uses the Fibonacci number sequence to determine the number of victims for each of his killing episodes. In this episode, character Dr. Reid also notices that locations of the killings lie on the graph of a golden spiral, and going to the center of the spiral allows Reid to determine the location of the killer's base of operations.

The mean tetrahedron volume of a tetrahedron with vertices chosen at random inside another tetrahedron of unit volume is given by(1)(2)(OEIS A093525; Buchta and Reitzner 1992; Mannion1994; Schneider 1997, p. 170; Buchta and Reitzner 2001; Zinani 2003).This provides a disproof of the conjecture that the solution to this problem is a rational number (1/57 had been suggested by Croft et al. 1991, p. 54), and renders obsolete Solomon's statement that "Explicit values for random points in non-spherical regions such as tetrahedrons, parallelepipeds, etc., have apparently not yet been successfully calculated" (Solomon 1978, p. 124).Furthermore, Buchta and Reitzner (2001) give an explicit formula for the expected volume of the convex hull of points chosen at random in a three-dimensional simplex for arbitrary ...

The jinc function is defined as(1)where is a Bessel function of the first kind, and satisfies . The derivative of the jinc function is given by(2)The function is sometimes normalized by multiplying by a factor of 2 so that (Siegman 1986, p. 729).The first real inflection point of the function occurs when(3)namely 2.29991033... (OEIS A133920).The unique real fixed point occurs at 0.48541702373... (OEIS A133921).

The apodization function(1)Its full width at half maximum is .Its instrument function is(2)(3)where is a Bessel function of the first kind. This function has a maximum of . To investigate the instrument function, define the dimensionless parameter and rewrite the instrument function as(4)Finding the full width at half maximumthen amounts to solving(5)which gives , so for , the full width at half maximum is(6)The maximum negative sidelobe of times the peak, and maximum positive sidelobe of 0.356044 times the peak.

An apodization function(1)having instrument function(2)The peak of is . The full width at half maximum of can found by setting to obtain(3)and solving for , yielding(4)Therefore, with ,(5)The extrema are given by taking the derivative of , substituting , and setting equal to 0(6)Solving this numerically gives sidelobes at 0.715148 (), 1.22951 (0.256749), 1.73544 (), ....

An apodization function, also called the Hann function, frequently used to reduce leakage in discrete Fourier transforms. The illustrations above show the Hanning function, its instrument function, and a blowup of the instrument function sidelobes. It is named after the Austrian meteorologist Julius von Hann (Blackman and Tukey 1959, pp. 98-99). The Hanning function is given by(1)(2)Its full width at half maximum is .It has instrument function(3)(4)To investigate the instrument function, define the dimensionless parameter and rewrite the instrument function as(5)The half-maximum can then be seen to occur at(6)so for , the full width at half maximum is(7)To find the extrema, take the derivative(8)and equate to zero. The first two roots are and 10.7061..., corresponding to the first sidelobe minimum () and maximum (), respectively...

An apodization function chosen to minimize the height of the highest sidelobe (Hamming and Tukey 1949, Blackman and Tukey 1959). The Hamming function is given by(1)and its full width at half maximum is .The corresponding instrument function is(2)This apodization function is close to the one produced by the requirement that the instrument function goes to 0 at . The FWHM is , the peak is 1.08, and the peak negative and positive sidelobes (in units of the peak) are and 0.00734934, respectively.From the apodization function, a general symmetric apodization function can be written as a Fourier series(3)where the coefficients satisfy(4)The corresponding instrument function is(5)To obtain an apodization function with zero at , use(6)so(7)(8)(9)(10)(11)

Min Max Min Max Re Im In one dimension, the Gaussian function is the probabilitydensity function of the normal distribution,(1)sometimes also called the frequency curve. The full width at half maximum (FWHM) for a Gaussian is found by finding the half-maximum points . The constant scaling factor can be ignored, so we must solve(2)But occurs at , so(3)Solving,(4)(5)(6)(7)The full width at half maximum is thereforegiven by(8)In two dimensions, the circular Gaussian function is the distribution function for uncorrelated variates and having a bivariate normal distribution and equal standard deviation ,(9)The corresponding elliptical Gaussian function corresponding to is given by(10)The Gaussian function can also be used as an apodizationfunction(11)shown above with the corresponding instrumentfunction. The instrument function is(12)which has maximum(13)As , equation (12) reduces to(14)The hypergeometric function is also..

The apodization functionIts full width at half maximum is .Its instrument function iswhich has a maximum of and full width at half maximum .

The apodization functionIts full width at half maximum is .Its instrument function iswhere is a Bessel function of the first kind. This has a maximum of , and full width at half maximum of .

An apodization function given by(1)which has full width at half maximum of . This function is defined so that the coefficients are approximations in the general expansion(2)to(3)(4)(5)which produce zeros of at and .The corresponding instrument function is(6)where is the sinc function. It is full width at half maximum is .

The apodization function(1)which is a generalization of the one-argument triangle function. Its full width at half maximum is .It has instrument function(2)where is the sinc function. The peak of is , and the full width at half maximum is given by setting and numerically solving(3)for , yielding(4)Therefore, with ,(5)The function is always positive, so there are no negative sidelobes. The extrema are given by differentiating with respect to , defining , and setting equal to 0,(6)Solving this numerically gives minima of 0 at , 2, 3, ..., and sidelobes of 0.047190, 0.01648, 0.00834029, ... at , 2.45892, 3.47089, ....

The cosine function is one of the basic functions encountered in trigonometry (the others being the cosecant, cotangent, secant, sine, and tangent). Let be an angle measured counterclockwise from the x-axis along the arc of the unit circle. Then is the horizontal coordinate of the arc endpoint.The common schoolbook definition of the cosine of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the lengths of the side of the triangle adjacent to the angle and the hypotenuse, i.e.,(1)A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).As a result of its definition, the cosine function is periodic with period . By the Pythagorean theorem, also obeys the identity(2) Min Max Re Im The definition of the cosine function can be extended to..

An almost integer is a number that is very close to an integer.Surprising examples are given by(1)which equals to within 5 digits and(2)which equals to within 16 digits (M. Trott, pers. comm., Dec. 7, 2004). The first of these comes from the half-angle formula identity(3)where 22 is the numerator of the convergent 22/7 to , so . It therefore follows that any pi approximation gives a near-identity of the form .Another surprising example involving both e andpi is(4)which can also be written as(5)(6)Here, is Gelfond's constant. Applying cosine a few more times gives(7)This curious near-identity was apparently noticed almost simultaneously around 1988 by N. J. A. Sloane, J. H. Conway, and S. Plouffe, but no satisfying explanation as to "why" is true has yet been discovered.Another nested cosine almost integer is given by(8)(P. Rolli, pers. comm., Feb. 19, 2004).An..

Pairs of primes of the form (, ) are called cousin primes. The first few are (3, 7), (7, 11), (13, 17), (19, 23), (37, 41), (43, 47), (67, 71), ... (OEIS A023200 and A046132).A large pair of cousin (proven) primes start with(1)where is a primorial. These primes have 10154 digits and were found by T. Alm, M. Fleuren, and J. K. Andersen (Andersen 2005).As of Jan. 2006, the largest known pair of cousin (probable) primes are(2)which have 11311 digits and were found by D. Johnson in May 2004.According to the first Hardy-Littlewood conjecture, the cousin primes have the same asymptotic density as the twin primes,(3)(4)where (OEIS A114907) is the twin primes constant.An analogy to Brun's constant, the constant(5)(omitting the initial term ) can be defined. Using cousin primes up to , the value of is estimated as(6)..

The number obtained by adding the reciprocals of the odd twinprimes,(1)By Brun's theorem, the series converges to a definite number, which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1989, p. 201). By contrast, the series of all prime reciprocals diverges to infinity, as follows from the Mertens second theorem by letting (which provides a stronger characterization of the divergence than Euler's proof that , obtained more than a century before Mertens' proof).Shanks and Wrench (1974) used all the twin primes among the first 2 million numbers. Brent (1976) calculated all twin primes up to 100 billion and obtained (Ribenboim 1989, p. 146)(2)assuming the truth of the first Hardy-Littlewood conjecture. Using twin primes up to , Nicely (1996) obtained(3)(Cipra 1995, 1996), in the process discovering a bug in Intel's® PentiumTM microprocessor. Using twin primes up to , Nicely..

For a real number , let be the number of terms in the convergent to a regular continued fraction that are required to represent decimal places of . Then for almost all ,(1)(2)(OEIS A086819; Lochs 1964). This number issometimes known as Lochs' constant.Therefore, the regular continued fraction is only slightly more efficient at representing real numbers than is the decimal expansion. The set of for which this statement does not hold is of measure 0.

The nth root of the denominator of the th convergent of a number tends to a constant(1)(2)(3)(OEIS A086702) for all but a set of of measure zero (Lévy 1936, Lehmer 1939), where(4)(5)Some care is needed in terminology and notation related to this constant. Most authors call "Lévy's constant" (e.g., Le Lionnais 1983, p. 51; Sloane) and some (S. Plouffe) call the "Khinchin-Lévy constant." Other authors refer to (e.g., Finch 2003, p. 60) or (e.g., Wu 2008) without specifically naming the expression in question.Taking the multiplicative inverse of gives another related constant,(6)(7)(OEIS A089729).Corless (1992) showed that(8)with an analogous formula for Khinchin's constant.The Lévy Constant is related to Lochs' constant by(9)or(10)The plot above shows for the first 500 terms in the continued fractions of , , the Euler-Mascheroni constant , and the Copeland-Erdős..

A number of closed-form constants can be obtained for generalized continued fractions having particularly simple partial numerators and denominators.The Ramanujan continued fractions provide a fascinating class of continued fraction constants. The Trott constants are unexpected constants whose partial numerators and denominators correspond to their decimal digits (though to achieve this, it is necessary to allow some partial numerators to equal 0).The first in a series of other famous continued fraction constants is the infiniteregular continued fraction(1)(2)The first few convergents of the constant are 0, 1, 2/3, 7/10, 30/43, 157/225, 972/1393, 6961/9976, ... (OEIS A001053 and A001040).Both numerator and denominator satisfy the recurrence relation(3)where has the initial conditions , and has the initial conditions , . These can be solved exactly to yield(4)(5)(6)(7)where is a modified Bessel function of the first kind..

The end of the last gap in the Lagrange spectrum,given by(OEIS A118472).Real numbers greater than are members of the Markov spectrum.

A Pisot number is a positive algebraic integer greater than 1 all of whose conjugate elements have absolute value less than 1. A real quadratic algebraic integer greater than 1 and of degree 2 or 3 is a Pisot number if its norm is equal to . The golden ratio (denoted when considered as a Pisot number) is an example of a Pisot number since it has degree two and norm .The smallest Pisot number is given by the positive root (OEIS A060006) of(1)known as the plastic constant. This number was identified as the smallest known by Salem (1944), and proved to be the smallest possible by Siegel (1944).Pisot constants give rise to almost integers. For example, the larger the power to which is taken, the closer , where is the floor function, is to either 0 or 1 (Trott 2004). For example, the spectacular example is within of an integer (Trott 2004, pp. 8-9).The powers of for which this quantity is closer to 0 are 1, 3, 4, 5, 6, 7, 8, 11, 12, 14, 17, ... (OEIS A051016), while..

Wallis's constant is the real solution (OEIS A007493) to the cubic equationIt was solved by Wallis to illustrate Newton's methodfor numerical equation solving.

Min Max Min Max Re Im The hyperbolic cosine integral, often called the "Chi function" for short, is defined by(1)where is the Euler-Mascheroni constant. The function is given by the Wolfram Language command CoshIntegral[z].The Chi function has a unique real root at (OEIS A133746).The derivative of is(2)and the integral is(3)

The double factorial of a positive integer is a generalization of the usual factorial defined by(1)Note that , by definition (Arfken 1985, p. 547).The origin of the notation appears not to not be widely known and is not mentioned in Cajori (1993).For , 1, 2, ..., the first few values are 1, 1, 2, 3, 8, 15, 48, 105, 384, ... (OEIS A006882). The numbers of decimal digits in for , 1, ... are 1, 4, 80, 1285, 17831, 228289, 2782857, 32828532, ... (OEIS A114488).The double factorial is implemented in the WolframLanguage as n!! or Factorial2[n].The double factorial is a special case of the multifactorial.The double factorial can be expressed in terms of the gammafunction by(2)(Arfken 1985, p. 548).The double factorial can also be extended to negative odd integers using the definition(3)(4)for , 1, ... (Arfken 1985, p. 547). Min Max Re Im Similarly, the double factorial can be extended to complex arguments as(5)There are many identities..

The (complete) gamma function is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by(1)a slightly unfortunate notation due to Legendre which is now universally used instead of Gauss's simpler (Gauss 1812; Edwards 2001, p. 8).It is analytic everywhere except at , , , ..., and the residue at is(2)There are no points at which .The gamma function is implemented in the WolframLanguage as Gamma[z].There are a number of notational conventions in common use for indication of a power of a gamma functions. While authors such as Watson (1939) use (i.e., using a trigonometric function-like convention), it is also common to write .The gamma function can be defined as a definite integral for (Euler's integral form)(3)(4)or(5)The complete gamma function can be generalized to the upper incomplete gamma function and lower incomplete gamma function . Min Max Re Im Plots of the real and imaginary..

Let be defined as the power series whose th term has a coefficient equal to the th prime ,(1)(2)The function has a zero at (OEIS A088751). Now let be defined by(3)(4)(5)(OEIS A030018).Then N. Backhouse conjectured that(6)(7)(OEIS A072508). This limit was subsequently shown to exist by P. Flajolet. Note that , which follows from the radius of convergence of the reciprocal power series.The continued fraction of Backhouse's constant is [1, 2, 5, 5, 4, 1, 1, 18, 1, 1, 1, 1, 1, 2, ...] (OEIS A074269), which is also the same as the continued fraction of except for a leading 0 in the latter.

The goat problem (or bull-tethering problem) considers a fenced circular field of radius with a goat (or bull, or other animal) tied to a point on the interior or exterior of the fence by means of a tether of length , and asks for the solution to various problems concerning how much of the field can be grazed.Tieing a goat to a point on the interior of the fence with radius 1 using a chain of length , consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection(1)Taking gives(2)plotted above. Setting (i.e., half of ) leads to the equation(3)which cannot be solved exactly, but which has approximate solution(4)(OEIS A133731).Now instead consider tieing the goat to the exterior of the fence (or equivalently, to the exterior of a silo whose horizontal cross section is a circle) with radius . Assume that , so that the goat is not..

A sultan has granted a commoner a chance to marry one of his daughters. The commoner will be presented with the daughters one at a time and, when each daughter is presented, the commoner will be told the daughter's dowry (which is fixed in advance). Upon being presented with a daughter, the commoner must immediately decide whether to accept or reject her (he is not allowed to return to a previously rejected daughter). However, the sultan will allow the marriage to take place only if the commoner picks the daughter with the overall highest dowry. Then what is the commoner's best strategy, assuming he knows nothing about the distribution of dowries (Mosteller 1987)?Since the commoner knows nothing about the distribution of the dowries, the best strategy is to wait until a certain number of daughters have been presented, then pick the highest dowry thereafter (Havil 2003, p. 134). The exact number to skip is determined by the condition that the..

Approximations to Khinchin's constant include(1)(2)(3)(4)which are correct to 9, 7, 6, and 5 digits, respectively (M. Hudson, pers. comm., Nov. 20, 2004).

Champernowne's constant(1)(OEIS A033307) is the number obtained by concatenating the positive integers and interpreting them as decimal digits to the right of a decimal point. It is normal in base 10 (Champernowne 1933, Bailey and Crandall 2002). Mahler (1961) showed it to also be transcendental. The constant has been computed to digits by E. W. Weisstein (Jul. 3, 2013) using the Wolfram Language.The infinite sequence of digits in Champernowne's constant is sometimes known as Barbier's infinite word (Allouche and Shallit 2003, pp. 114, 299 and 334).The number of digits after concatenation of the first, second, ... primes are givenby 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, ... (OEIS A068670).The Champernowne constant continued fraction contains sporadic very large terms, making the continued fraction difficult to calculate. However, the size of the continued fraction high-water marks display apparent patterns (Sikora..

Let be a positive nonsquare integer. Then Artin conjectured that the set of all primes for which is a primitive root is infinite. Under the assumption of the generalized Riemann hypothesis, Artin's conjecture was solved by Hooley (1967; Finch 2003, p. 105).Let be not an th power for any such the squarefree part of satisfies (mod 4). Let be the set of all primes for which such an is a primitive root. Then Artin also conjectured that the density of relative to the primes is given independently of the choice of by , where(1)(OEIS A005596), and is the th prime.The significance of Artin's constant is more easily seen by describing it as the fraction of primes for which has a maximal period repeating decimal, i.e., is a full reptend prime (Conway and Guy 1996) corresponding to a cyclic number. is connected with the prime zeta function by(2)where is a Lucas number (Ribenboim 1998, Gourdon and Sebah). Wrench (1961) gave 45 digits of , and Gourdon and Sebah..

The base-2 transcendental number(1)(OEIS A014578), where the th bit is 1 if is not divisible by 3 and is the complement of the th bit if is divisible by 3. It is also given by the substitution system(2)(3)Interpreted as a decimal number, the Thue constant equals 0.8590997969...(OEIS A074071).

The -ball, denoted , is the interior of a sphere , and sometimes also called the -disk. (Although physicists often use the term "sphere" to mean the solid ball, mathematicians definitely do not!)The ball of radius centered at point is implemented in the Wolfram Language as Ball[x, y, z, r].The equation for the surface area of the -dimensional unit hypersphere gives the recurrence relation(1)Using then gives the hypercontent of the -ball of radius as(2)(Sommerville 1958, p. 136; Apostol 1974, p. 430; Conway and Sloane 1993). Strangely enough, the content reaches a maximum and then decreases towards 0 as increases. The point of maximal content of a unit -ball satisfies(3)(4)(5)where is the digamma function, is the gamma function, is the Euler-Mascheroni constant, and is a harmonic number. This equation cannot be solved analytically for , but the numerical solution to(6)is (OEIS A074455) (Wells 1986, p. 67)...

The number (the cube root of 2) which is to be constructed in the cube duplication problem. This number is not a Euclidean number although it is an algebraic number of third degree.It has decimal digits 1.25992104989... (OEIS A002580).Its continued fraction is [1, 3, 1, 5, 1, 1,4, 1, 1, 8, 1, 14, 1, ...] (OEIS A002945).

What is the sofa of greatest area which can be moved around a right-angled hallway of unit width? Hammersley (Croft et al. 1994) showed that(1)(OEIS A086118). Gerver (1992) found a sofa with larger area and provided arguments indicating that it is either optimal or close to it. The boundary of Gerver's sofa is a complicated shape composed of 18 arcs. Its area can be given by defining the constants , , , and by solving(2)(3)(4)(5)This gives(6)(7)(8)(9)Now define(10)where(11)(12)(13)Finally, define the functions(14)(15)(16)The area of the optimal sofa is then given by(17)(18)(Finch 2003).

Bezdek and Kuperberg (1991) have constructed packings of identical ellipsoidsof densities arbitrarily close to(OEIS A093824), greater than the maximum density of (OEIS A093825) that is possible for sphere packing (Sloane 1998), as established by proof of the Kepler conjecture. Furthermore, J. Wills has modified the ellipsoid packing to yield an even higher density of (Bezdek and Kuperberg 1991).Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.

Define the packing density of a packing of spheres to be the fraction of a volume filled by the spheres. In three dimensions, there are three periodic packings for identical spheres: cubic lattice, face-centered cubic lattice, and hexagonal lattice. It was hypothesized by Kepler in 1611 that close packing (cubic or hexagonal, which have equivalent packing densities) is the densest possible, and this assertion is known as the Kepler conjecture. The problem of finding the densest packing of spheres (not necessarily periodic) is therefore known as the Kepler problem, where(OEIS A093825; Steinhaus 1999, p. 202;Wells 1986, p. 29; Wells 1991, p. 237).In 1831, Gauss managed to prove that the face-centered cubic is the densest lattice packing in three dimensions (Conway and Sloane 1993, p. 9), but the general conjecture remained open for many decades.While the Kepler conjecture is intuitively obvious, the proof remained..

A circle packing is an arrangement of circles inside a given boundary such that no two overlap and some (or all) of them are mutually tangent. The generalization to spheres is called a sphere packing. Tessellations of regular polygons correspond to particular circle packings (Williams 1979, pp. 35-41). There is a well-developed theory of circle packing in the context of discrete conformal mapping (Stephenson).The densest packing of circles in the plane is the hexagonal lattice of the bee's honeycomb (right figure; Steinhaus 1999, p. 202), which has a packing density of(1)(OEIS A093766; Wells 1986, p. 30). Gauss proved that the hexagonal lattice is the densest plane lattice packing, and in 1940, L. Fejes Tóth proved that the hexagonal lattice is indeed the densest of all possible plane packings.Surprisingly, the circular disk is not the least economical region for packing the plane. The "worst"..

Find the plane lamina of least area which is capable of covering any plane figure of unit generalized diameter. A unit circle is too small, but a hexagon circumscribed on the unit circle is larger than necessary. Pál (1920) showed that the hexagon can be reduced by cutting off two isosceles triangles on the corners of the hexagon which are tangent to the hexagon's incircle (Wells 1991; left figure above). Sprague subsequently demonstrated that an additional small curvilinear region could be removed (Wells 1991; right figure above). These constructions give upper bounds.The hexagon having inradius (giving a diameter of 1) has side length(1)and the area of this hexagon is(2)(OEIS A010527).In the above figure, the sagitta is given by(3)(4)and the other distances by(5)(6)so the area of one of the equilateral triangles removed in Pál's reduction is(7)(8)(9)(10)so the area left after removing two of these triangles is(11)(12)(13)(OEIS..

The answer to the question "which fits better, a round peg in a square hole, or a square peg in a round hole?" can be interpreted as asking which is larger, the ratio of the area of a circle to its circumscribed square, or the area of the square to its circumscribed circle? In two dimensions, the ratios are and , respectively. Therefore, a round peg fits better into a square hole than a square peg fits into a round hole (Wells 1986, p. 74).However, this result is true only in dimensions , and for , the unit -hypercube fits more closely into the -hypersphere than vice versa (Singmaster 1964; Wells 1986, p. 74). This can be demonstrated by noting that the formulas for the content of the unit -ball, the content of its circumscribed hypercube, and the content of its inscribed hypercube are given by(1)(2)(3)The ratios in question are then(4)(5)(Singmaster 1964). The ratio of these ratios is the transcendental equation(6)illustrated..

The expected number of real zeros of a random polynomial of degree if the coefficients are independent and distributed normally is given by(1)(2)(Kac 1943, Edelman and Kostlan 1995). Another form of the equation is given by(3)(Kostlan 1993, Edelman and Kostlan 1995). The plots above show the integrand (left) and numerical values of (red curve in right plot) for small . The first few values are 1, 1.29702, 1.49276, 1.64049, 1.7596, 1.85955, ....As ,(4)where(5)(6)(OEIS A093601; top curve in right plot above).The initial term was derived by Kac (1943).

Min Max Re Im The hyperbolic cotangent is defined as(1)The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix). It is implemented in the Wolfram Language as Coth[z].The hyperbolic cotangent satisfies the identity(2)where is the hyperbolic cosecant.It has a unique real fixed point where(3)at (OEIS A085984), which is related to the Laplace limit in the solution of Kepler's equation.The derivative is given by(4)where is the hyperbolic cosecant, and the indefinite integral by(5)where is a constant of integration.The Laurent series of is given by(6)(7)(OEIS A002431 and A036278), where is a Bernoulli number and is a Bernoulli polynomial. An asymptotic series about infinity on the real line is given by(8)

The nested radical constant is the constant defined by(1)(2)(OEIS A072449).No closed-form expression is known for this constant (Finch 2003, p. 8; S. Plouffe, pers. comm., Aug. 29, 2008).

The Feigenbaum constant is a universal constant for functions approaching chaos via period doubling. It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function(1)and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter is increased for fixed . The plot above is made by iterating equation (1) with several hundred times for a series of discrete but closely spaced values of , discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining.A similar plot that more directly shows the cycle may be constructed by plotting as a function of . The plot above (Trott, pers. comm.) shows the resulting curves for , 2, and 4.Let be the point at which a period -cycle appears, and denote the converged value by . Assuming geometric convergence, the difference between this value and..

Zero is the integer denoted 0 that, when used as a counting number, means that no objects are present. It is the only integer (and, in fact, the only real number) that is neither negative nor positive. A number which is not zero is said to be nonzero. A root of a function is also sometimes known as "a zero of ."The Schoolhouse Rock segment "My Hero, Zero" extols the virtues of zero with such praises as, "My hero, zero Such a funny little hero But till you came along We counted on our fingers and toes Now you're here to stay And nobody really knows How wonderful you are Why we could never reach a star Without you, zero, my hero How wonderful you are."Zero is commonly taken to have the factorization (e.g., in the Wolfram Language's FactorInteger[n] command). On the other hand, the divisors and divisor function are generally taken to be undefined, since by convention, (i.e., divides 0) for every except zero.Because the number of..

A fixed point is a point that does not change upon application of a map, system of differential equations, etc. In particular, a fixed point of a function is a point such that(1)The fixed point of a function starting from an initial value can be computed in the Wolfram Language using FixedPoint[f, x]. Similarly, to get a list of the values obtained by iterating the function until a fixed point is reached, the command FixedPointList[f, x] can be used.The following table lists the smallest positive fixed points for several simple functions.functionfixed pointOEIScosecant1.1141571408A133866cosine0.7390851332A003957cotangent0.8603335890A069855hyperbolic cosecant0.9320200293A133867hyperbolic cosine----hyperbolic cotangent1.1996786402A085984hyperbolic secant0.7650099545A069814hyperbolic sine0--hyperbolic tangent0--inverse cosecant1.1141571408A133866inverse cosine0.7390851332A003957inverse cotangent0.8603335890A069855inverse..

The Dottie number is the name given by Kaplan (2007) to the unique real root of (namely, the unique real fixed point of the cosine function), which is 0.739085... (OEIS A003957). The name "Dottie" is of no fundamental mathematical significance since it refers to a particular French professor who--no doubt like many other calculator uses before and after her--noticed that whenever she typed a number into her calculator and hit the cosine button repeatedly, the result always converged to this value.The number is well-known, having appeared in numerous elementary works on algebra already by the late 1880s (e.g., Bertrand 1865, p. 285; Heis 1886, p. 468; Briot 1881, pp. 341-343), and probably much earlier as well. It is also known simply as the cosine constant, cosine superposition constant, iterated cosine constant, or cosine fixed point constant. Arakelian (1981, pp. 135-136; 1995) has used the Armenian..

Let be the Riemann-Siegel function. The unique value such that(1)where , 1, ... is then known as a Gram point (Edwards 2001, pp. 125-126).An excellent approximation for Gram point can be obtained by using the first few terms in the asymptotic expansion for and inverting to obtain(2)where is the Lambert W-function. This approximation gives as error of for , decreasing to by .The following table gives the first few Gram points.OEIS0A11485717.84559954041A11485823.1702827012227.6701822178331.7179799547435.4671842971538.9992099640642.3635503920745.5930289815848.7107766217951.73384281331054.6752374468The integers closest to these points are 18, 23, 28, 32, 35, 39, 42, 46, 49, 52,55, 58, ... (OEIS A002505).There is a unique point at which , given by the solution to the equation(3)and having numerical value(4)(OEIS A114893).It is usually the case that . Values of for which this does not hold are , 134, 195, 211, 232, 254,..

The tangent function is defined by(1)where is the sine function and is the cosine function. The notation is sometimes also used (Gradshteyn and Ryzhik 2000, p. xxix).The common schoolbook definition of the tangent of an angle in a right triangle (which is equivalent to the definition just given) is as the ratio of the side lengths opposite to the angle and adjacent the angle, i.e.,(2)A convenient mnemonic for remembering the definition of the sine, cosine, and tangent is SOHCAHTOA (sine equals opposite over hypotenuse, cosine equals adjacent over hypotenuse, tangent equals opposite over adjacent).The word "tangent" also has an important related meaning as a line or plane which touches a given curve or solid at a single point. These geometrical objects are then called a tangent line or tangent plane, respectively. Min Max Re Im The definition of the tangent function can be extended to complex arguments using the definition(3)(4)(5)(6)where..

By analogy with the sinc function, define the tancfunction by(1)Since is not a cardinal function, the "analogy" with the sinc function is one of functional structure, not mathematical properties. It is quite possible that a better term than , as introduced here, could be coined, although there appears to be no name previously assigned to this function.The derivative is given by(2)The indefinite integral can apparently not be done in closed form in terms of conventionally defined functions.This function commonly arises in problems in physics, where it is desired to determine values of for which , i.e., . This is a transcendental equation whose first few solutions are given in the following table and illustrated above.OEISroot001A1153654.4934094579090641753...27.7252518369377071642...310.904121659428899827...414.066193912831473480...517.220755271930768739...The positive solutions can be written explicitly..

Replacing the logistic equation(1)with the quadratic recurrence equation(2)where (sometimes also denoted ) is a positive constant sometimes known as the "biotic potential" gives the so-called logistic map. This quadratic map is capable of very complicated behavior. While John von Neumann had suggested using the logistic map as a random number generator in the late 1940s, it was not until work by W. Ricker in 1954 and detailed analytic studies of logistic maps beginning in the 1950s with Paul Stein and Stanislaw Ulam that the complicated properties of this type of map beyond simple oscillatory behavior were widely noted (Wolfram 2002, pp. 918-919).The first few iterations of the logistic map (2) give(3)(4)(5)where is the initial value, plotted above through five iterations (with increasing iteration number indicated by colors; 1 is red, 2 is yellow, 3 is green, 4 is blue, and 5 is violet) for various values of .The..

By analogy with the tanc function, define the tanhcfunction by(1)It has derivative(2)The indefinite integral can apparently not be done in closed form in terms of conventionally defined functions.It has maximum at , and positive inflection point at the solution to(3)which is 0.919937667... (OEIS A133919).It has a unique real fixed point at 0.82242927726... (OEIS A133918).

The functions(1)(2)(3)(4) has an inflection point at(5)which can be solved numerically to give (OEIS A118080).

By analogy with the sinc function, define the sinhcfunction by(1)Since is not a cardinal function, the "analogy" with the sinc function is one of functional structure, not mathematical properties. It is quite possible that a better term than could be coined, although there appears to be no other name previously assigned to this function.The function has derivative(2)and indefinite integral(3)where is the Shi function.The function has real fixed points at 1.31328371835... (OEIS A133916)and 2.63924951389... (OEIS A133917).

The constant that Gelfond's theorem established to be transcendental seems to lack a generally accepted name. As a result, in this work, it will be dubbed Gelfond's constant. Both the Gelfond-Schneider constant and Gelfond's constant were singled out in the 7th of Hilbert's problems as examples of numbers whose transcendence was an open problem (Wells 1986, p. 45).Gelfond's constant has the numerical value(1)(OEIS A039661) and simplecontinued fraction(2)(OEIS A058287).Its digits can be computed efficiently using the iteration(3)with , and then plugging in to(4)(Borwein and Bailey 2003, p. 137).

Let be a compact connected subset of -dimensional Euclidean space. Gross (1964) and Stadje (1981) proved that there is a unique real number such that for all , , ..., , there exists with(1)The magic constant of is defined by(2)where(3)These numbers are also called dispersion numbers and rendezvous values. For any , Gross (1964) and Stadje (1981) proved that(4)If is a subinterval of the line and is a circular disk in the plane, then(5)If is a circle, then(6)(OEIS A060294). An expression for the magic constant of an ellipse in terms of its semimajor and semiminor axes lengths is not known. Nikolas and Yost (1988) showed that for a Reuleaux triangle (7)Denote the maximum value of in -dimensional space by . Thenwhere is the gamma function (Nikolas and Yost 1988).An unrelated quantity characteristic of a given magicsquare is also known as a magic constant...

The conjecture that, for any triangle,(1)where , , and are the vertex angles of the triangle and is the Brocard angle. The Abi-Khuzam inequality states that(2)(Yff 1963, Le Lionnais 1983, Abi-Khuzam and Boghossian 1989), which can be used to prove the conjecture (Abi-Khuzam 1974).The maximum value of occurs when two angles are equal, so taking , and using , the maximum occurs at the maximum of(3)which occurs when(4)Solving numerically gives (OEIS A133844), corresponding to a maximum value of approximately 0.440053 (OEIS A133845).

Extend Hilbert's inequality by letting and(1)so that(2)Levin (1937) and Stečkin (1949) showed that(3)and(4)Mitrinovic et al. (1991) indicate that this constant is the best possible.

Let be the smallest tour length for points in a -D hypercube. Then there exists a smallest constant such that for all optimal tours in the hypercube,(1)and a constant such that for almost all optimal tours in the hypercube,(2)These constants satisfy the inequalities(3)(4)(5)(6)(7)(8)(9)(Fejes Tóth 1940, Verblunsky 1951, Few 1955, Beardwood et al. 1959),where(10) is the gamma function, is an expression involving Struve functions and Bessel functions of the second kind,(11)(OEIS A086306; Karloff 1989), and(12)(OEIS A086307; Goddyn 1990).In the limit ,(13)(14)(15)and(16)where(17)and is the best sphere packing density in -D space (Goddyn 1990, Moran 1984, Kabatyanskii and Levenshtein 1978). Steele and Snyder (1989) proved that the limit exists.Now consider the constant(18)so(19)Nonrigorous numerical estimates give (Johnson et al. 1996) and (Percus and Martin 1996).A certain self-avoiding space-filling function..

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