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A plane curve given by the parametric equations(1)(2)The plots above show curves for values of from 0 to 7.The teardrop curve has area(3)

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..

There are two definitions of the supersingular primes: one group-theoretic, and the other number-theoretic.Group-theoretically, let be the modular group Gamma0, and let be the compactification (by adding cusps) of , where is the upper half-plane. Also define to be the Fricke involution defined by the block matrix . For a prime, define . Then is a supersingular prime if the genus of .The number-theoretic definition involves supersingular elliptic curves defined over the algebraic closure of the finite field . They have their j-invariant in .Supersingular curves were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight" of the television crime drama NUMB3RS.There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (OEIS A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group...

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..

A bubble is a minimal-energy surface of the type that is formed by soap film. The simplest bubble is a single sphere, illustrated above (courtesy of J. M. Sullivan). More complicated forms occur when multiple bubbles are joined together. The simplest example is the double bubble, and beautiful configurations can form when three or more bubbles are conjoined (Sullivan).An outstanding problem involving bubbles is the determination of the arrangements of bubbles with the smallest surface area which enclose and separate given volumes in space.

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,..

The Reuleaux tetrahedron, sometimes also called the spherical tetrahedron, is the three-dimensional solid common to four spheres of equal radius placed so that the center of each sphere lies on the surface of the other three. The centers of the spheres are therefore located at the vertices of a regular tetrahedron, and the solid consists of an "inflated" tetrahedron with four curved edges.Note that the name, coined here for the first time, is based on the fact that the geometric shape is the three-dimensional analog of the Reuleaux triangle, not the fact that it has constant width. In fact, the Reuleaux tetrahedron is not a solid of constant width. However, Meißner (1911) showed how to modify the Reuleaux tetrahedron to form a surface of constant width by replacing three of its edge arcs by curved patches formed as the surfaces of rotation of a circular arc. Depending on which three edge arcs are replaced (three that have a common..

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..

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 term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...

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 fact noticed by physicist G. Gamow when he had an office on the second floor and physicist M. Stern had an office on the sixth floor of a seven-story building (Gamow and Stern 1958, Gardner 1986). Gamow noticed that about 5/6 of the time, the first elevator to stop on his floor was going down, whereas about the same fraction of time, the first elevator to stop on the sixth floor was going up. This actually makes perfect sense, since 5 of the 6 floors 1, 3, 4, 5, 6, 7 are above the second, and 5 of the 6 floors 1, 2, 3, 4, 5, 7 are below the sixth. However, the situation takes some unexpected turns if more than one elevator is involved, as discussed by Gardner (1986). Furthermore, even worse, the analysis by Gamow and Stern (1958) turns out to be incorrect (Knuth 1969)!Main character Charles Eppes discusses the elevator paradox in the Season 4 episode "Chinese Box" of the television crime drama NUMB3RS...

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..

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 king graph is a graph with vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a king.The number of edges in the king graph is , so for , 2, ..., the first few values are 0, 6, 20, 42, 72, 110, ... (OEIS A002943).The order graph has chromatic number for and for . For , 3, ..., the edge chromatic numbers are 3, 8, 8, 8, 8, ....King graphs are implemented in the Wolfram Language as GraphData["King", m, n].All king graphs are Hamiltonian and biconnected. The only regular king graph is the -king graph, which is isomorphic to the tetrahedral graph . The -king graphs are planar only for (with the case corresponding to path graphs) and , some embeddings of which are illustrated above.The -king graph is perfect iff (S. Wagon, pers. comm., Feb. 22, 2013).Closed formulas for the numbers of -cycles of with are given by(1)(2)(3)(4)where the formula for appears in Perepechko and Voropaev.The..

The queen graph is a graph with vertices in which each vertex represents a square in an chessboard, and each edge corresponds to a legal move by a queen. The -queen graphs have nice embeddings, illustrated above. In general, the default embedding with vertices corresponding to squares of the chessboard has degenerate superposed edges, the only nontrivial exception being the -queen graph.Queen graphs are implemented in the Wolfram Language as GraphData["Queen", m, n].The following table summarized some special cases of queen graphs.namecomplete graph tetrahedral graph The following table summarizes some named graph complements of queen graphs.-queen graph-knight graph-queen graph-queen graph-knight graphAll queen graphs are Hamiltonian and biconnected. The only planar and only regular queen graph is the -queen graph, which is isomorphic to the tetrahedral graph .The only perfect queen graphs are , , and .A closed formula..

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 quantity twelve (12) is sometimes known as a dozen.It is in turn one twelfth of a gross.Base-12 is known as duodecimal.The Schoolhouse Rock segment "Little Twelvetoes" discusses the usefulness of multiplying by 12: "Well, with twelve digits, I mean fingers, He probably would've invented two more digits When he invented his number system. Then, if he'd saved the zero for the end, He could count and multiply by 12's, Just as easily as you and I do by 10's. Now, if man Had been born with six fingers on each hand, He's probably count: 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el, do. Dek and el being two entirely new signs meaning 10 and 11 - single digits. And his 12, do, would've been written: one - zero. Get it? That'd be swell, to multiply by 12."

1729 is sometimes called the Hardy-Ramanujan number. It is the smallest taxicab number, i.e., the smallest number which can be expressed as the sum of two cubes in two different ways:A more obscure appearance of 1729 is as the average of the greatest member in each pair of (known) Brown numbers (5, 4), (11, 5), and (71, 7):(K. MacMillan, pers. comm., Apr. 29, 2007).This property of 1729 was mentioned by the character Robert the sometimes insane mathematician, played by Anthony Hopkins, in the 2005 film Proof. The number 1729 also appeared with no mention of its special property as the number associated with gambler Nick Fisher (Sam Jaeger) in the betting books of The Boss (Morgan Freeman) in the 2006 film Lucky Number Slevin.1729 was also part of the designation of the spaceship Nimbus BP-1729 appearing in Season 2 of the animated television series Futurama episode DVD 2ACV02 (Greenwald; left figure), as well as the robot character..

The second Mersenne prime , which is itself the exponent of Mersenne prime . It gives rise to the perfect number It is a Gaussian prime, but not an Eisenstein prime, since it factors as , where is a primitive cube root of unity. It is the smallest non-Sophie Germain prime. It is also the smallest non-Fermat prime, and as such is the smallest number of faces of a regular polygon (the heptagon) that is not constructible by straightedge and compass.It occurs as a sacred number in the Bible and in various other traditions. In Babylonian numerology it was considered as the perfect number, the only number between 2 and 10 which is not generated (divisible) by any other number, nor does it generate (divide) any other number.Words referring to number seven may have the prefix hepta-, derived from the Greek -) (heptic), or sept- (septuple), derived from the Latin septem...

According to the novel The Hitchhiker's Guide to the Galaxy (Adams 1997), 42 is the ultimate answer to life, the universe, and everything. Unfortunately, it is left as an exercise to the reader to determine the actual question.On Feb. 18, 2005, the 42nd Mersenne prime was discovered (Weisstein 2005), leading to humorous speculation that the answer to life, the universe, and everything is somehow contained in the 7.8 million decimal digits of that number.It is also amusing that 042 occurs as the digit string ending at the 50 billionth decimal place in each of and , providing another excellent answer to the ultimate question (Berggren et al. 1997, p. 729).

The base 8 notational system for representing real numbers. The digits used are 0, 1, 2, 3, 4, 5, 6, and 7, so that (8 in base 10) is represented as () in base 8. The following table gives the octal equivalents of the first few decimal numbers.11111321252212142226331315232744141624305515172531661620263277172127338101822283491119232935101220243036The song "New Math" by Tom Lehrer (That Was the Year That Was, 1965) explains how to compute in octal. (The answer is .)

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."..

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...

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).

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..

The tesseract is the hypercube in , also called the 8-cell or octachoron. It has the Schläfli symbol , and vertices . The figure above shows a projection of the tesseract in three-space (Gardner 1977). The tesseract is composed of 8 cubes with 3 to an edge, and therefore has 16 vertices, 32 edges, 24 squares, and 8 cubes. It is one of the six regular polychora.The tesseract has 261 distinct nets (Gardner 1966, Turney 1984-85, Tougne 1986, Buekenhout and Parker 1998).In Madeleine L'Engle's novel A Wrinkle in Time, the characters in the story travel through time and space using tesseracts. The book actually uses the idea of a tesseract to represent a fifth dimension rather than a four-dimensional object (and also uses the word "tesser" to refer to movement from one three dimensional space/world to another).In the science fiction novel Factoring Humanity by Robert J. Sawyer, a tesseract is used by humans on Earth to enter the fourth..

The hypercube is a generalization of a 3-cube to dimensions, also called an -cube or measure polytope. It is a regular polytope with mutually perpendicular sides, and is therefore an orthotope. It is denoted and has Schläfli symbol .The following table summarizes the names of -dimensional hypercubes.object1line segment2square3cube4tesseractThe number of -cubes contained in an -cube can be found from the coefficients of , namely , where is a binomial coefficient. The number of nodes in the -hypercube is therefore (OEIS A000079), the number of edges is (OEIS A001787), the number of squares is (OEIS A001788), the number of cubes is (OEIS A001789), etc.The numbers of distinct nets for the -hypercube for , 2, ... are 1, 11, 261, ... (OEIS A091159; Turney 1984-85).The above figure shows a projection of the tesseract in three-space. A tesseract has 16 polytope vertices, 32 polytope edges, 24 squares, and eight cubes.The dual of the tesseract..

Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the mice in the mice problem, and form logarithmic spirals.The square whirl appears on the cover of Freund (1993).

A star polygon , with positive integers, is a figure formed by connecting with straight lines every th point out of regularly spaced points lying on a circumference. The number is called the polygon density of the star polygon. Without loss of generality, take . The star polygons were first systematically studied by Thomas Bradwardine.The circumradius of a star polygon with and unit edge lengths is given by(1)and its characteristic polynomial is a factor of the resultant with respect to of the polynomials(2)(3)where is a Chebyshev polynomial of the first kind (Gerbracht 2008).The usual definition (Coxeter 1969) requires and to be relatively prime. However, the star polygon can also be generalized to the star figure (or "improper" star polygon) when and share a common divisor (Savio and Suryanaroyan 1993). For such a figure, if all points are not connected after the first pass, i.e., if , then start with the first unconnected point..

The Star of Lakshmi is the star figure , that is used in Hinduism to symbolize Ashtalakshmi, the eight forms of wealth. This symbol appears prominently in the Lugash national museum portrayed in the fictional film The Return of the Pink Panther.The interior of a Star of Lakshmi with edges of length is a regular octagon with side lengths(1)The areas of the intersection and union of the two constituent squares are(2)(3)

The pentagram, also called the five-point star, pentacle, pentalpha, or pentangle, is the star polygon .It is a pagan religious symbol that is one of the oldest symbols on Earth and is known to have been used as early as 4000 years B.C. It represents the "sacred feminine" or "divine goddess" (Brown 2003, pp. 35-37). However, in modern American pop culture, it more commonly represents devil worship. In the novel The Da Vinci Code, dying Louvre museum curator Jacque Saunière draws a pentagram on his abdomen with his own blood as a clue to identify his murderer (Brown 2003, p. 35).In the above figure, let the length from one tip to another connected tip be unity, the length from a tip to an adjacent dimple be , the edge lengths of the inner pentagon be , the inradius of the inner pentagon be , the circumradius of the inner pentagon be , the circumradius of the pentagram be , and the additional horizontal and vertical..

A star polygon-like figure for which and are not relatively prime. Examples include the hexagram , star of Lakshmi , and nonagram .

Also called Chvátal's art gallery theorem. If the walls of an art gallery are made up of straight line segments, then the entire gallery can always be supervised by watchmen placed in corners, where is the floor function. This theorem was proved by Chvátal (1975). It was conjectured that an art gallery with walls and holes requires watchmen, which has now been proven by Bjorling-Sachs and Souvaine (1991, 1995) and Hoffman et al. (1991).In the Season 2 episode "Obsession" (2006) of the television crime drama NUMB3RS, Charlie mentions the art gallery theorem while building an architectural model.

The number two (2) is the second positive integer and the first prime number. It is even, and is the only even prime (the primes other than 2 are called the odd primes). The number 2 is also equal to its factorial since . A quantity taken to the power 2 is said to be squared. The number of times a given binary number is divisible by 2 is given by the position of the first , counting from the right. For example, is divisible by 2 twice, and is divisible by 2 zero times.The only known solutions to the congruenceare summarized in the following table (OEIS A050259). M. Alekseyev explored all solutions below on Jan. 27 2007, finding no other solutions in this range.reference4700063497Guy (1994)3468371109448915M. Alekseyev (pers. comm., Nov. 13, 2006)8365386194032363Crump (pers. comm., 2000)10991007971508067Crump (2007)63130707451134435989380140059866138830623361447484274774099906755Montgomery (1999)In general,..

A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure above. A more exotic configuration in which one bubble is torus-shaped and the other is shaped like a dumbbell is illustrated at right (illustrations courtesy of J. M. Sullivan).In the plane, the analog of the double bubble consists of three circular arcs meeting in two points. It has been proved that the configuration of arcs meeting at equal angles) has the minimum perimeter for enclosing two equal areas (Alfaro et al. 1993, Morgan 1995).It had been conjectured that two equal partial spheres sharing a boundary of a flat disk separate two volumes of air using a total surface area that is less than any other boundary. This equal-volume case was proved by Hass et al. (1995), who reduced the problem to a set of integrals which they carried out on an ordinary PC. Frank Morgan,..

An antimagic square is an array of integers from 1 to such that each row, column, and main diagonal produces a different sum such that these sums form a sequence of consecutive integers. It is therefore a special case of a heterosquare. It was defined by Lindon (1962) and appeared in Madachy's collection of puzzles (Madachy 1979, p. 103), originally published in 1966. Antimagic squares of orders 4-9 are illustrated above (Madachy 1979). For the square, the sums are 30, 31, 32, ..., 39; for the square they are 59, 60, 61, ..., 70; and so on.Let an antimagic square of order have entries 0, 1, ..., , , and letbe the magic constant. Then if an antimagic square of order exists, it is either positive with sums , or negative with sums (Madachy 1979).Antimagic squares of orders one, two, and three are impossible. In the case of the square, there is no known method of proof of this fact except by case analysis or enumeration by computer. There are 18 families of..

A heterosquare is an array of the integers from 1 to such that the rows, columns, and diagonals have different sums. (By contrast, in a magic square, they have the same sum.) There are no heterosquares of order two, but heterosquares of every odd order exist. They can be constructed by placing consecutive integers in a spiral pattern (Fults 1974, Madachy 1979).An antimagic square is a special case of a heterosquare for which the sums of rows, columns, and main diagonals form a sequence of consecutive integers.

The binomial coefficient is the number of ways of picking unordered outcomes from possibilities, also known as a combination or combinatorial number. The symbols and are used to denote a binomial coefficient, and are sometimes read as " choose ." therefore gives the number of k-subsets possible out of a set of distinct items. For example, The 2-subsets of are the six pairs , , , , , and , so . The number of lattice paths from the origin to a point ) is the binomial coefficient (Hilton and Pedersen 1991).The value of the binomial coefficient for nonnegative and is given explicitly by(1)where denotes a factorial. Writing the factorial as a gamma function allows the binomial coefficient to be generalized to noninteger arguments (including complex and ) as(2)For nonnegative integer arguments, the gamma function reduces to factorials, leading to(3)which is Pascal's triangle. Using the symmetryformula(4)for integer , and complex , this..

The Costa surface is a complete minimal embedded surface of finite topology (i.e., it has no boundary and does not intersect itself). It has genus 1 with three punctures (Schwalbe and Wagon 1999). Until this surface was discovered by Costa (1984), the only other known complete minimal embeddable surfaces in with no self-intersections were the plane (genus 0), catenoid (genus 0 with two punctures), and helicoid (genus 0 with two punctures), and it was conjectured that these were the only such surfaces.Rather amazingly, the Costa surface belongs to the dihedral group of symmetries.The Costa minimal surface appears on the cover of Osserman (1986; left figure) as well as on the cover of volume 2, number 2 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).It has also been constructed as a snow sculpture (Ferguson et al. 1999, Wagon1999).On Feb. 20, 2008, a large stone sculpture by Helaman Ferguson was..

Scherk's two minimal surfaces were discovered by Scherk in 1834. They were the first new surfaces discovered since Meusnier in 1776. Beautiful images of wood sculptures of Scherk surfaces are illustrated by Séquin.Scherk's first surface is doubly periodic and is defined by the implicit equation(1)(Osserman 1986, Wells 1991, von Seggern 1993). It has been observed to form in layers of block copolymers (Peterson 1988).Scherk's second surface is the surface generated by Enneper-Weierstrassparameterization with(2)(3)It can be written parametrically as(4)(5)(6)(7)(8)(9)for , and . With this parametrization, the coefficients of the first fundamental form are(10)(11)(12)and of the second fundamental form are(13)(14)(15)The Gaussian and mean curvatures are(16)(17)

A catenary of revolution. The catenoid and plane are the only surfaces of revolution which are also minimal surfaces. The catenoid can be given by the parametric equations(1)(2)(3)where .The line element is(4)The first fundamental form has coefficients(5)(6)(7)and the second fundamental form has coefficients(8)(9)(10)The principal curvatures are(11)(12)The mean curvature of the catenoid is(13)and the Gaussian curvature is(14)The helicoid can be continuously deformed into a catenoid with by the transformation(15)(16)(17)where corresponds to a helicoid and to a catenoid.This deformation is illustrated on the cover of issue 2, volume 2 of The MathematicaJournal.

The gyroid, illustrated above, is an infinitely connected periodic minimal surface containing no straight lines (Osserman 1986) that was discovered by Schoen (1970). Große-Brauckmann and Wohlgemuth (1996) proved that the gyroid is embedded.The gyroid is the only known embedded triply periodic minimal surface with triple junctions. In addition, unlike the five triply periodic minimal surfaces studied by Anderson et al. (1990), the gyroid does not have any reflectional symmetries (Große-Brauckmann 1997).The image above shows a metal print of the gyroid created by digital sculptor BathshebaGrossman (https://www.bathsheba.com/).

Minimal surfaces are defined as surfaces with zero mean curvature. A minimal surface parametrized as therefore satisfies Lagrange's equation,(1)(Gray 1997, p. 399).Finding a minimal surface of a boundary with specified constraints is a problem in the calculus of variations and is sometimes known as Plateau's problem. Minimal surfaces may also be characterized as surfaces of minimal surface area for given boundary conditions. A plane is a trivial minimal surface, and the first nontrivial examples (the catenoid and helicoid) were found by Meusnier in 1776 (Meusnier 1785). The problem of finding the minimum bounding surface of a skew quadrilateral was solved by Schwarz in 1890 (Schwarz 1972).Note that while a sphere is a "minimal surface" in the sense that it minimizes the surface area-to-volume ratio, it does not qualify as a minimal surface in the sense used by mathematicians.Euler proved that a minimal surface is planar..

A Markov chain is collection of random variables (where the index runs through 0, 1, ...) having the property that, given the present, the future is conditionally independent of the past.In other words,If a Markov sequence of random variates take the discrete values , ..., , thenand the sequence is called a Markov chain (Papoulis 1984, p. 532).A simple random walk is an example of a Markovchain.The Season 1 episode "Man Hunt" (2005) of the television crime drama NUMB3RS features Markov chains.

The term "cylinder" has a number of related meanings. In its most general usage, the word "cylinder" refers to a solid bounded by a closed generalized cylinder (a.k.a. cylindrical surface) and two parallel planes (Kern and Bland 1948, p. 32; Harris and Stocker 1998, p. 102). A cylinder of this sort having a polygonal base is therefore a prism (Zwillinger 1995, p. 308). Harris and Stocker (1998, p. 103) use the term "general cylinder" to refer to the solid bounded a closed generalized cylinder.Unfortunately, the term "cylinder" is commonly used not only to refer to the solid bounded by a cylindrical surface, but to the cylindrical surface itself (Zwillinger 1995, p. 311). To make matters worse, according to topologists, a cylindrical surface is not even a true surface, but rather a so-called surface with boundary (Henle 1994, pp. 110 and 129).As if this were..

A regular polygon is an -sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular and equilateral). Only certain regular polygons are "constructible" using the classical Greek tools of the compass and straightedge.The terms equilateral triangle and square refer to the regular 3- and 4-polygons, respectively. The words for polygons with sides (e.g., pentagon, hexagon, heptagon, etc.) can refer to either regular or non-regular polygons, although the terms generally refer to regular polygons in the absence of specific wording.A regular -gon is implemented in the Wolfram Language as RegularPolygon[n], or more generally as RegularPolygon[r, n], RegularPolygon[x, y, rspec, n], etc.The sum of perpendiculars from any point to the sides of a regular polygon of sides is times the apothem.Let be the side length, be the inradius, and the circumradius..

RSA numbers are difficult to-factor composite numbers having exactly two prime factors (i.e., so-called semiprimes) that were listed in the Factoring Challenge of RSA Security®--a challenge that is now withdrawn and no longer active.While RSA numbers are much smaller than the largest known primes, their factorization is significant because of the curious property of numbers that proving or disproving a number to be prime ("primality testing") seems to be much easier than actually identifying the factors of a number ("prime factorization"). Thus, while it is trivial to multiply two large numbers and together, it can be extremely difficult to determine the factors if only their product is given. With some ingenuity, this property can be used to create practical and efficient encryption systems for electronic data.RSA Laboratories sponsored the RSA Factoring Challenge to encourage research into computational..

An invariant defined using the angles of a three-dimensional polyhedron. It remains constant under solid dissection and reassembly. Solids with the same volume can have different Dehn invariants.Two polyhedra can be dissected into each other only if they have the same volume and the same Dehn invariant. In 1902, Dehn showed that two interdissectable polyhedra must have equal Dehn invariants, settling the third of Hilbert's problems, and Sydler (1965) showed that two polyhedra with the same Dehn invariants are interdissectable.

A Mrs. Perkins's quilt is a dissection of a square of side into a number of smaller squares. The name "Mrs. Perkins's Quilt" comes from a problem in one of Dudeney's books, where he gives a solution for . Unlike a perfect square dissection, however, the smaller squares need not be all different sizes. In addition, only prime dissections are considered so that patterns which can be dissected into lower-order squares are not permitted.The smallest numbers of squares needed to create relatively prime dissections of an quilt for , 2, ... are 1, 4, 6, 7, 8, 9, 9, 10, 10, 11, 11, 11, 11, 12, ... (OEIS A005670), the first few of which are illustrated above.On October 9-10, L. Gay (pers. comm. to E. Pegg, Jr.) discovered 18-square quilts for side lengths 88, 89, and 90, thus beating all previous records. The following table summarizes the smallest numbers of squares known to be needed for various side lengths , with those for (and possibly..

French curves are plastic (or wooden) templates having an edge composed of several different curves. French curves are used in drafting (or were before computer-aided design) to draw smooth curves of almost any desired curvature in mechanical drawings. Several typical French curves are illustrated above.While an undergraduate at MIT, Feynman (1997, p. 23) used a French curve to illustrate the fallacy of learning without understanding. When he pointed out to his colleagues in a mechanical drawing class the "amazing" fact that the tangent at the lowest (or highest) point on the curve was horizontal, none of his classmates realized that this was trivially true, since the derivative (tangent) at an extremum (lowest or highest point) of any curve is zero (horizontal), as they had already learned in calculus class...

The points of intersection of the adjacent angle trisectors of the angles of any triangle are the polygon vertices of an equilateral triangle known as the first Morley triangle. Taylor and Marr (1914) give two geometric proofs and one trigonometric proof.A line is parallel to a side of the first Morley triangle if and only ifin directed angles modulo (Ehrmann and Gibert 2001).An even more beautiful result is obtained by taking the intersections of the exterior, as well as interior, angle trisectors, as shown above. In addition to the interior equilateral triangle formed by the interior trisectors, four additional equilateral triangles are obtained, three of which have sides which are extensions of a central triangle (Wells 1991).A generalization of Morley's theorem was discovered by Morley in 1900 but first published by Taylor and Marr (1914). Each angle of a triangle has six trisectors, since each interior angle trisector has two associated..

A hyperboloid is a quadratic surface which may be one- or two-sheeted. The one-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the perpendicular bisector to the line between the foci, while the two-sheeted hyperboloid is a surface of revolution obtained by rotating a hyperbola about the line joining the foci (Hilbert and Cohn-Vossen 1991, p. 11).

The partitioning of a plane with points into convex polygons such that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other. A Voronoi diagram is sometimes also known as a Dirichlet tessellation. The cells are called Dirichlet regions, Thiessen polytopes, or Voronoi polygons.Voronoi diagrams were considered as early at 1644 by René Descartes and were used by Dirichlet (1850) in the investigation of positive quadratic forms. They were also studied by Voronoi (1907), who extended the investigation of Voronoi diagrams to higher dimensions. They find widespread applications in areas such as computer graphics, epidemiology, geophysics, and meteorology. A particularly notable use of a Voronoi diagram was the analysis of the 1854 cholera epidemic in London, in which physician John Snow determined a strong correlation of deaths with proximity to a particular..

The hexagram is the star polygon , also known as the star of David or Solomon's seal, illustrated at left above.It appears as one of the clues in the novel TheDa Vinci Code (Brown 2003, p. 455).For a hexagram with circumradius (red circle), the inradius (green circle) is(1)and the circle passing through the intersections of the triangles has radius(2)The interior of a hexagram is a regular hexagon with side lengths equal to 1/3 that of the original hexagram. Given a hexagram with line segments of length , the areas of the intersection and union of the two constituent triangles are(3)(4)There is a "nonregular" hexagram that can be obtained by spacing the integers 1 to 6 evenly around a circle and connecting . The resulting figure is called a "unicursal hexagram" and was evidently discovered in the 19th century. It is not regular because there are some edges going from to (mod 6) and some edges going from to (mod 6). However,..

A Greek cross, also called a square cross, is a cross inthe shape of a plus sign. It is a non-regular dodecagon.A square cross appears on the flag of Switzerland, and also on the key to the Swiss Bank deposit box in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 146 and 171-172).Greek crosses can tile the plane, as noted by the protagonist Christopher in The Curious Incident of the Dog in the Night-Time (Haddon 2003, pp. 203-204).

A demiregular tessellation, also called a polymorph tessellation, is a type of tessellation whose definition is somewhat problematical. Some authors define them as orderly compositions of the three regular and eight semiregular tessellations (which is not precise enough to draw any conclusions from), while others defined them as a tessellation having more than one transitivity class of vertices (which leads to an infinite number of possible tilings).The number of demiregular tessellations is commonly given as 14 (Critchlow 1970, pp. 62-67; Ghyka 1977, pp. 78-80; Williams 1979, p. 43; Steinhaus 1999, pp. 79 and 81-82). However, not all sources apparently give the same 14. Caution is therefore needed in attempting to determine what is meant by "demiregular tessellation."A more precise term of demiregular tessellations is 2-uniform tessellations (Grünbaum and Shephard 1986, p. 65)...

Regular tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex are called semiregular tessellations, or sometimes Archimedean tessellations. In the plane, there are eight such tessellations, illustrated above (Ghyka 1977, pp. 76-78; Williams 1979, pp. 37-41; Steinhaus 1999, pp. 78-82; Wells 1991, pp. 226-227). Williams (1979, pp. 37-41) also illustrates the dual tessellations of the semiregular tessellations. The dual tessellation of the tessellation of squares and equilateral triangles is called the Cairo tessellation (Williams 1979, p. 38; Wells 1991, p. 23).

Consider a two-dimensional tessellation with regular -gons at each polygon vertex. In the plane,(1)(2)so(3)(Ball and Coxeter 1987), and the only factorizations are(4)(5)(6)Therefore, there are only three regular tessellations (composed of the hexagon, square, and triangle), illustrated above (Ghyka 1977, p. 76; Williams 1979, p. 36; Wells 1991, p. 213).There do not exist any regular star polygon tessellations in the plane. Regular tessellations of the sphere by spherical triangles are called triangular symmetry groups.

An aperiodic tiling is a non-periodic tiling in which arbitrarily large periodic patches do not occur. A set of tiles is said to be aperiodic if they can form only non-periodic tilings. The most widely known examples of aperiodic tilings are those formed by Penrose tiles.The Federation Square buildings in Melbourne, Australia feature an aperiodic pinwheel tiling attributed to Charles Radin. The tiling is illustrated above in a pair of photographs by P. Bourke.

A plane tiling is said to be isohedral if the symmetry group of the tiling acts transitively on the tiles, and -isohedral if the tiles fall into n orbits under the action of the symmetry group of the tiling. A -anisohedral tiling is a tiling which permits no -isohedral tiling with .The numbers of anisohedral polyominoes with , 9, 10, ... are 1, 9, 44, 108, 222, ... (OEIS A075206), the first few of which are illustrated above (Myers).

The P versus NP problem is the determination of whether all NP-problems are actually P-problems. If P and NP are not equivalent, then the solution of NP-problems requires (in the worst case) an exhaustive search, while if they are, then asymptotically faster algorithms may exist.The answer is not currently known, but determination of the status of this question would have dramatic consequences for the potential speed with which many difficult and important problems could be solved.In the Season 1 episode "Uncertainty Principle" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes uses the game minesweeper as an analogy for the P vs. NP problem.

Percolation theory deals with fluid flow (or any other similar process) in random media.If the medium is a set of regular lattice points, then there are two main types of percolation: A site percolation considers the lattice vertices as the relevant entities; a bond percolation considers the lattice edges as the relevant entities. These two models are examples of discrete percolation theory, an umbrella term used to describe any percolation model which takes place on a regular point lattice or any other discrete set, and while they're most certainly the most-studied of the discrete models, others such as AB percolation and mixed percolation do exist and are reasonably well-studied in their own right.Contrarily, one may also talk about continuum percolation models, i.e.,models which attempt to define analogous tools and results with respect to continuous, uncountable domains. In particular, continuum percolation theory involves notions..

A permutation problem invented by Cayley. Let the numbers 1, 2, ..., be written on a set of cards, and shuffle this deck of cards. Now, start counting using the top card. If the card chosen does not equal the count, move it to the bottom of the deck and continue counting forward. If the card chosen does equal the count, discard the chosen card and begin counting again at 1. The game is won if all cards are discarded, and lost if the count reaches .The number of ways the cards can be arranged such that at least one card is in the proper place for , 2, ... are 1, 1, 4, 15, 76, 455, ... (OEIS A002467).

A generalization to a quartic three-dimensional surface is the quartic surface of revolution(1)illustrated above. With , this surface is termed the "zeck" surface by Hauser. It has volume(2)geometric centroid(3)(4)(5)and inertia tensor(6)for constant density and mass .

Consider the plane quartic curve defined bywhere homogeneous coordinates have been used here so that can be considered a parameter (the plot above shows the curve for a number of values of between and 2), over a field of characteristic 3. Hartshorne (1977, p. 305) terms this "a funny curve" since it is nonsingular, every point is an inflection point, and the dual curve is isomorphic to but the natural map is purely inseparable.The surface in complex projective coordinates (Levy 1999, p. ix; left figure), and with the ideal surface determined by the equation(Thurston 1999, p. 3; right figure) is more properly known as the Klein quarticor Klein curve. It has constant zero Gaussian curvature.Klein (1879; translation reprinted in 1999) discovered that this surface has a number of remarkable properties, including an incredible 336-fold symmetry when mirror reflections are allowed (Levy 1999, p. ix; Thurston..

A dragon curve is a recursive nonintersecting curve whose name derives from its resemblance to a certain mythical creature.The curve can be constructed by representing a left turn by 1 and a right turn by 0. The first-order curve is then denoted 1. For higher order curves, append a 1 to the end, then append the string of preceding digits with its middle digit complemented. For example, the second-order curve is generated as follows: , and the third as .Continuing gives 110110011100100... (OEIS A014577), which is sometimes known as the regular paperfolding sequence and written with s instead of 0s (Allouche and Shallit 2003, p. 155). A recurrence plot of the limiting value of this sequence is illustrated above.Representing the sequence of binary digits 1, 110, 1101100, 110110011100100, ... in octal gives 1, 6, 154, 66344, ...(OEIS A003460; Gardner 1978, p. 216).This procedure is equivalent to drawing a right angle and subsequently..

A power is an exponent to which a given quantity is raised. The expression is therefore known as " to the th power." A number of powers of are plotted above (cf. Derbyshire 2004, pp. 68 and 73).The power may be an integer, real number, or complex number. However, the power of a real number to a non-integer power is not necessarily itself a real number. For example, is real only for .A number other than 0 taken to the power 0 is defined to be 1, which followsfrom the limit(1)This fact is illustrated by the convergence of curves at in the plot above, which shows for , 0.4, ..., 2.0. It can also be seen more intuitively by noting that repeatedly taking the square root of a number gives smaller and smaller numbers that approach one from above, while doing the same with a number between 0 and 1 gives larger and larger numbers that approach one from below. For square roots, the total power taken is , which approaches 0 as is large, giving in the limit that..

A cube can be divided into subcubes for only , 8, 15, 20, 22, 27, 29, 34, 36, 38, 39, 41, 43, 45, 46, and (OEIS A014544; Hadwiger 1946; Scott 1947; Gardner 1992, p. 297). This sequence provides the solution to the so-called Hadwiger problem, which asks for the largest number of subcubes (not necessarily different) into which a cube cannot be divided by plane cuts, and has the answer 47 (Gardner 1992, pp. 297-298).If and are in the sequence, so is , since -dissecting one cube in an -dissection gives an -dissection. The numbers 1, 8, 20, 38, 49, 51, 54 are in the sequence because of dissections corresponding to the equations(1)(2)(3)(4)(5)(6)(7)Combining these facts gives the remaining terms in the sequence, and all numbers , and it has been shown that no other numbers occur.It is not possible to cut a cube into subcubes that are all different sizes (Gardner 1961, p. 208; Gardner 1992, p. 298).The seven pieces used to construct..

Gardner showed how to dissect a square into eight and nine acute scalene triangles.W. Gosper discovered a dissection of a unit square into 10 acute isosceles triangles, illustrated above (pers. comm. to Ed Pegg, Jr., Oct 25, 2002). The coordinates can be found from solving the four simultaneous equations (1)(2)(3)(4)for the four unknowns and picking the solutions for which . The solutions are roots of 12th order polynomials with numerical values given approximately by(5)(6)(7)(8)Pegg has constructed a dissection of a square into 22 acute isosceles triangles.Guy (1989) asks if it is possible to triangulate a square with integer side lengths such that the resulting triangles have integer side lengths (Trott 2004, p. 104).

The number of different triangles which have integer side lengths and perimeter is(1)(2)(3)where is the partition function giving the number of ways of writing as a sum of exactly terms, is the nearest integer function, and is the floor function (Andrews 1979, Jordan et al. 1979, Honsberger 1985). A slightly complicated closed form is given by(4)The values of for , 2, ... are 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, ... (OEIS A005044), which is also Alcuin's sequence padded with two initial 0s.The generating function for is given by(5)(6)(7) also satisfies(8)It is not known if a triangle with integer sides, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have shown that there are infinitely many triangles with rational sides (Heronian triangles) with two rational..

A Nash equilibrium of a strategic game is a profile of strategies , where ( is the strategy set of player ), such that for each player , , , where and .Another way to state the Nash equilibrium condition is that solves for each . In words, in a Nash equilibrium, no player has an incentive to deviate from the strategy chosen, since no player can choose a better strategy given the choices of the other players.The Season 1 episode "Dirty Bomb" (2005) of the television crime drama NUMB3RS mentions Nash equilibrium.

The Monty Hall problem is named for its similarity to the Let's Make a Deal television game show hosted by Monty Hall. The problem is stated as follows. Assume that a room is equipped with three doors. Behind two are goats, and behind the third is a shiny new car. You are asked to pick a door, and will win whatever is behind it. Let's say you pick door 1. Before the door is opened, however, someone who knows what's behind the doors (Monty Hall) opens one of the other two doors, revealing a goat, and asks you if you wish to change your selection to the third door (i.e., the door which neither you picked nor he opened). The Monty Hall problem is deciding whether you do.The correct answer is that you do want to switch. If you do not switch, you have the expected 1/3 chance of winning the car, since no matter whether you initially picked the correct door, Monty will show you a door with a goat. But after Monty has eliminated one of the doors for you, you obviously do not improve..

The fundamental theorem of game theory which states that every finite, zero-sum, two-person game has optimal mixed strategies. It was proved by John von Neumann in 1928.Formally, let and be mixed strategies for players A and B. Let be the payoff matrix. Thenwhere is called the value of the game and and are called the solutions. It also turns out that if there is more than one optimal mixed strategy, there are infinitely many.In the Season 4 opening episode "Trust Metric" (2007) of the television crime drama NUMB3RS, math genius Charlie Eppes mentions that he used the minimax theorem in an attempt to derive an equation describing friendship.

For vectors and in , the cross product in is defined by(1)(2)where is a right-handed, i.e., positively oriented, orthonormal basis. This can be written in a shorthand notation that takes the form of a determinant(3)where , , and are unit vectors. Here, is always perpendicular to both and , with the orientation determined by the right-hand rule.Special cases involving the unit vectors in three-dimensionalCartesian coordinates are given by(4)(5)(6)The cross product satisfies the general identity(7)Note that is not a usual polar vector, but has slightly different transformation properties and is therefore a so-called pseudovector (Arfken 1985, pp. 22-23). Jeffreys and Jeffreys (1988) use the notation to denote the cross product.The cross product is implemented in the Wolfram Language as Cross[a, b].A mathematical joke asks, "What do you get when you cross a mountain-climber with a mosquito?" The answer is, "Nothing:..

A Hadamard matrix is a type of square (-1,1)-matrix invented by Sylvester (1867) under the name of anallagmatic pavement, 26 years before Hadamard (1893) considered them. In a Hadamard matrix, placing any two columns or rows side by side gives half the adjacent cells the same sign and half the other sign. When viewed as pavements, cells with 1s are colored black and those with s are colored white. Therefore, the Hadamard matrix must have white squares (s) and black squares (1s).A Hadamard matrix of order is a solution to Hadamard's maximum determinant problem, i.e., has the maximum possible determinant (in absolute value) of any complex matrix with elements (Brenner and Cummings 1972), namely . An equivalent definition of the Hadamard matrices is given by(1)where is the identity matrix.A Hadamard matrix of order corresponds to a Hadamard design (, , ), and a Hadamard matrix gives a graph on vertices known as a Hadamard graphA complete set of Walsh..

The hypotenuse of a right triangle is the triangle's longest side, i.e., the side opposite the right angle. The word derives from the Greek hypo- ("under") and teinein ("to stretch").The length of the hypotenuse of a right trianglecan be found using the Pythagorean theorem.Among his many other talents, Major General Stanley in Gilbert and Sullivan's operetta The Pirates of Penzance impresses the pirates with his knowledge of the hypotenuse in "The Major General's Song" as follows: "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of..

An -gonal trapezohedron, also called an antidipyramid, antibipyramid, or deltohedron (not to be confused with a deltahedron), is a dual polyhedra of an -antiprism. Unfortunately, the name for these solids is not particularly well chosen since their faces are not trapezoids but rather kites. The trapezohedra are isohedra.The 3-trapezohedron (trigonal trapezohedron) is a rhombohedron having all six faces congruent. A special case is the cube (oriented along a space diagonal), corresponding to the dual of the equilateral 3-antiprism (i.e., the octahedron).A 4-trapezohedron (tetragonal trapezohedron) appears in the upper left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).The trapezohedra generated by taking the duals of the equilateral antiprisms have side length , half-heights (half the peak-to-peak distance) , surface areas , and volumes..

"Escher's solid" is the solid illustrated on the right pedestal in M. C. Escher's Waterfall woodcut (Bool et al. 1982, p. 323). It is obtained by augmenting a rhombic dodecahedron until incident edges become parallel, corresponding to augmentation height of for a rhombic dodecahedron with unit edge lengths.It is the first rhombic dodecahedron stellation and is a space-filling polyhedron. Its convex hull is a cuboctahedron.It is implemented in the Wolfram Languageas PolyhedronData["EschersSolid"].It has edge lengths(1)(2)surface area and volume(3)(4)and moment of inertia tensor(5)The skeleton of Escher's solid is the graph of the disdyakis dodecahedron.Escher's solid can also be viewed as a polyhedron compound of three dipyramids (nonregular octahedra) with edges of length 2 and ...

The pentagonal dipyramid is one of the convex deltahedra, and Johnson solid . It is also the dual polyhedron of the pentagonal prism and is an isohedron.It is implemented in the Wolfram Language as PolyhedronData["Dipyramid", 5].A pentagonal dipyramid appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).For a pentagonal dipyramid having a base with unit edge lengths, the circumradiusof the base pentagon is(1)In order for the top and bottom edges to also be of unit length, the polyhedron must be of height(2)The ratio of is therefore given by(3)where is the golden ratio.The surface area and volume of a unit pentagonal dipyramid are(4)(5)

The elongated square dipyramid with unit edge lengths is Johnson Solid .An elongated square dipyramid (having a central ribbon composed of rectangles instead of squares) appears in the top center as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).

Dürer's solid, also known as the truncated triangular trapezohedron, is the 8-faced solid depicted in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989), the same engraving in which Dürer's magic square appears, which depicts a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Although Dürer does not specify how his solid is constructed, Schreiber (1999) has noted that it appears to consist of a distorted cube which is first stretched to give rhombic faces with angles of , and then truncated on top and bottom to yield bounding triangular faces whose vertices lie on the circumsphere of the azimuthal cube vertices.It is implemented in the Wolfram Languageas PolyhedronData["DuererSolid"].The skeleton of Dürer's solid is the Dürer graph (i.e., generalized Petersen graph ).Starting..

The stella octangula is a polyhedron compound composed of a tetrahedron and its dual (a second tetrahedron rotated with respect to the first). The stella octangula is also (incorrectly) called the stellated tetrahedron, and is the only stellation of the octahedron. A wireframe version of the stella octangula is sometimes known as the merkaba and imbued with mystic properties.The name "stella octangula" is due to Kepler (1611), but the solid was known earlier to many others, including Pacioli (1509), who called it the "octaedron elevatum," and Jamnitzer (1568); see Cromwell (1997, pp. 124 and 152).It is implemented in the Wolfram Languageas PolyhedronData["StellaOctangula"].A stella octangula can be inscribed in a cube, deltoidal icositetrahedron, pentagonal icositetrahedron, rhombic dodecahedron, small triakis octahedron, and tetrakis hexahedron, (E. Weisstein, Dec. 24-25,..

The uniform polyhedron whose dual is the great dirhombicosidodecacron. This polyhedron is exceptional because it cannot be derived from Schwarz triangles and because it is the only uniform polyhedron with more than six polygons surrounding each polyhedron vertex (four squares alternating with two triangles and two pentagrams). This unique polyhedron has features in common with both snub forms and hemipolyhedra, and its octagrammic faces pass through the origin.It has pseudo-Wythoff symbol . Its faces are , and its circumradius for unit edge length isThe great dirhombicosidodecahedron appears on the cover of issue 4, volume 3 of TheMathematica Journal.

The curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. The roulettes described by the foci of conics when rolled upon a line are sections of minimal surfaces (i.e., they yield minimal surfaces when revolved about the line) known as unduloids.A particularly interesting case of a roulette is a regular -gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above. This motion is smooth in the sense that the geometric centroid follows a straight line, although in the case of the rolling equilateral triangle, a physical model would be impossible to construct because the vertices of the triangles would get "stuck" in the ruts (Wagon 2000). For the rolling square, the shape of the road is the catenary truncated at (Wagon 2000). For a regular -gon, the Cartesian equation of the corresponding catenary iswhereThe roulette consisting of a square..

The great rhombic triacontahedron, also called the great stellated triacontahedron, is a zonohedron which is the dual of the great icosidodecahedron and Wenninger model . It is one of the rhombic triacontahedron stellations.It appears together with an isometric projection of the 5-hypercube on the cover (and p. 103) of Coxeter's well-known book on polytopes (Coxeter 1973).The great rhombic triacontahedron can be constructed by expanding the size of the faces of a rhombic triacontahedron by a factor of , where is the golden ratio (Kabai 2002, p. 183) and keeping the pieces illustrated in the above stellation diagram.

The uniform polyhedron whose dual polyhedron is the small dodecicosacron. It has Wythoff symbol . Its faces are . Its circumradius for unit edge lengths is

The Desargues graph is the cubic symmetric graph on 20 vertices and 30 edges illustrated above in several embeddings. It is isomorphic to the generalized Petersen graph and to the bipartite Kneser graph . It is the incidence graph of the Desargues configuration. It can be represented in LCF notation as (Frucht 1976). It can also be constructed as the graph expansion of with steps 1 and 3, where is a path graph. It is distance-transitive and distance-regular graph and has intersection array .The Desargues graph is one of three cubic graphs on 20 nodes with smallest possible graph crossing number of 6 (the others being two unnamed graphs denoted CNG 6B and CNG 6C by Pegg and Exoo 2009), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).The Desargues is an integral graph with graph spectrum . It is cospectral with another nonisomorphic graph (Haemers and Spence 1995, van Dam and Haemers 2003).It is also a unit-distance..

The curve produced by fixed point on the circumference of a small circle of radius rolling around the inside of a large circle of radius . A hypocycloid is therefore a hypotrochoid with .To derive the equations of the hypocycloid, call the angle by which a point on the small circle rotates about its center , and the angle from the center of the large circle to that of the small circle . Then(1)so(2)Call . If , then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are(3)(4)(5)(6)If instead so the first point is at maximum radius (on the circle), then the equations of the hypocycloid are(7)(8)The curvature, arc length, and tangential angle of a hypocycloid are given by(9)(10)(11)An -cusped hypocycloid has . For an integer and with , the equations of the hypocycloid therefore become(12)(13)and the arc length and area are therefore(14)(15)A 2-cusped hypocycloid is a line segment (Steinhaus 1999, p. 145;..

The path traced out by a point on the edge of a circle of radius rolling on the outside of a circle of radius . An epicycloid is therefore an epitrochoid with . Epicycloids are given by the parametric equations(1)(2)A polar equation can be derived by computing(3)(4)so(5)But(6)so(7)(8)Note that is the parameter here, not the polar angle. The polar angle from the center is(9)To get cusps in the epicycloid, , because then rotations of bring the point on the edge back to its starting position.(10)(11)(12)(13)so(14)(15)An epicycloid with one cusp is called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid.Epicycloids can also be constructed by beginning with the diameter of a circle and offsetting one end by a series of steps of equal arc length along the circumference while at the same time offsetting the other end along the circumference by steps times as large. After traveling around the circle once,..

The Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedTetrahedron"].The dual of the truncated tetrahedron is the triakis tetrahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances from the center of the solid to the centroids of the triangular and hexagonal faces are given by(4)(5)The surface area and volumeare(6)(7)

The snub cube, also called the cubus simus (Kepler 1619, Weissbach and Martini 2002) or snub cuboctahedron, is an Archimedean solid having 38 faces (32 triangular and 6 square), 60 edges, and 24 vertices. It is a chiral solid, and hence has two enantiomorphous forms known as laevo (left-handed) and dextro (right-handed).It is Archimedean solid , uniform polyhedron , and Wenninger model . It has Schläfli symbol and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["SnubCube"].Surprisingly, the tribonacci constant is intimately related to the metric properties of the snub cube.It can be constructed by snubification of a unit cube with outward offset(1)(2)and twist angle(3)(4)(5)(6)Here, the notation indicates a polynomial root and is the tribonacci constant.An attractive dual of the two enantiomers superposed on one another is illustrated above.Its dual polyhedron is the pentagonalicositetrahedron.The..

The pentagonal icositetrahedron is the 24-faced dual polyhedron of the snub cube and Wenninger dual . The mineral cuprite () forms in pentagonal icositetrahedral crystals (Steinhaus 1999, pp. 207 and 209).Because it is the dual of the chiral snub cube, the pentagonal icositetrahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right). An attractive dual of the two enantiomers superposed on one another is illustrated above.A cube, octahedron, and stella octangula can all be inscribed on the vertices of the pentagonal icositetrahedron (E. Weisstein, Dec. 25, 2009).Surprisingly, the tribonacci constant is intimately related to the metric properties of the pentagonal icositetrahedron cube.Its irregular pentagonal faces have vertex angles of(1)(2)(3)(four times) and(4)(5)(6)(once), where is a polynomial root and is the tribonacci constant.The dual formed from a snub cube with..

The truncated octahedron is the 14-faced Archimedean solid , with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol . It was called the "mecon" by Buckminster Fuller (Rawles 1997).The dual polyhedron of the truncated octahedron is the tetrakis hexahedron. The truncated octahedron has the octahedral group of symmetries. The form of the fluorite () resembles the truncated octahedron (Steinhaus 1999, pp. 207-208).It is implemented in the Wolfram Languageas PolyhedronData["TruncatedOctahedron"].The solid of edge length can be formed from an octahedron of edge length via truncation by removing six square pyramids, each with edge slant height , base on a side, and height . The height and base area of the square pyramid are then(1)(2)(3)and its volume is(4)(5)The volume of the truncated octahedron is then given bythe volume of the octahedron(6)(7)minus..

In general, a triakis octahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of regular octahedron. Such a solid is also known as a trisoctahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting icositetrahedron is not regular, its faces are all identical. The small triakis octahedron, called simply the triakis octahedron by Holden (1971, p. 55), is the 24-faced dual polyhedron of the truncated cube and is Wenninger dual . The addition of the word "small" is necessary to distinguish it from the great triakis octahedron, which is the dual of the stellated truncated hexahedron. The small triakis octahedron It can be constructed by augmentation of a unit edge-length octahedron by a pyramid with height .A small triakis octahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's..

The pentagonal hexecontahedron is the 60-faced dual polyhedron of the snub dodecahedron (Holden 1971, p. 55). It is Wenninger dual .A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the pentagonal hexecontahedron (E. Weisstein, Dec. 25-27, 2009).Its irregular pentagonal faces have vertex angles of(1)(2)(four times) and(3)(4)(once), where is a polynomial root.Because it is the dual of the chiral snub dodecahedron, the pentagonal hexecontahedron also comes in two enantiomorphous forms, known as laevo (left) and dextro (right). An attractive dual of the two enantiomers superposed on one another is illustrated above.Starting with a snub dodecahedron with unit edge lengths, the edges lengths of the pentagonal hexecontahedron are given by the roots of (5)(6)which have approximate values and .The surface area and volume are both given by the roots of 12th-order..

The small rhombicuboctahedron is the 26-faced Archimedean solid consisting of faces . Although this solid is sometimes also called the truncated icosidodecahedron, this name is inappropriate since true truncation would yield rectangular instead of square faces. It is uniform polyhedron and Wenninger model . It has Schläfli symbol r and Wythoff symbol .The solid may also be called an expanded (or cantellated) cube or octahedron sinceit may be constructed from either of these solids by the process of expansion.A small rhombicuboctahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).It is implemented in the Wolfram Languageas PolyhedronData["SmallRhombicuboctahedron"].Its dual polyhedron is the deltoidal icositetrahedron, also called the trapezoidal icositetrahedron. The inradius of the..

The 32-faced Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedDodecahedron"].The dual polyhedron is the triakisicosahedron.To construct the truncated dodecahedron by truncation, note that we want the inradius of the truncated pentagon to correspond with that of the original pentagon, , of unit side length . This means that the side lengths of the decagonal faces in the truncated dodecahedron satisfy(1)giving(2)The length of the corner which is chopped off is therefore given by(3)The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(4)(5)(6)The distances from the center of the solid to the centroids of the triangular and decagonal faces are given by(7)(8)The surface area and volumeare(9)(10)..

The 62-faced Archimedean solid with faces . It is uniform polyhedron and Wenninger model . It has Schläfli symbol r and Wythoff symbol . The small dodecicosidodecahedron and small rhombidodecahedron are faceted versions.It is implemented in the Wolfram Languageas PolyhedronData["SmallRhombicosidodecahedron"].Its dual polyhedron is the deltoidal hexecontahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)It has surface area(4)and volume(5)

The 26-faced Archimedean solid consisting of faces . It is sometimes (improperly) called the truncated cuboctahedron (Ball and Coxeter 1987, p. 143), and is also more properly called the rhombitruncated cuboctahedron. It is uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .The great rhombicuboctahedron is an equilateral zonohedron and the Minkowski sum of three cubes. It can be combined with cubes and truncated octahedra into a regular space-filling pattern.The small cubicuboctahedron is a facetedversion of the great rhombicuboctahedron.Its dual is the disdyakis dodecahedron, also called the hexakis octahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)(4)(5)(6)Additional quantities are(7)(8)(9)(10)(11)The distances between the solid center and centroids of the square and octagonal faces are(12)(13)The surface..

The 14-faced Archimedean solid with faces . It is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedCube"].The dual polyhedron of the truncated cube is the small triakis octahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances from the center of the solid to the centroids of the triangular and octagonal faces are(4)(5)The surface area and volumeare(6)(7)

The rhombic triacontahedron is a zonohedron which is the dual polyhedron of the icosidodecahedron (Holden 1971, p. 55). It is Wenninger dual . It is composed of 30 golden rhombi joined at 32 vertices. It is a zonohedron and one of the five golden isozonohedra.The intersecting edges of the dodecahedron-icosahedron compound form the diagonals of 30 rhombi which comprise the triacontahedron. The cube 5-compound has the 30 facial planes of the rhombic triacontahedron and its interior is a rhombic triacontahedron (Wenninger 1983, p. 36; Ball and Coxeter 1987).More specifically, a tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the rhombic triacontahedron (E. Weisstein, Dec. 25-27, 2009).The rhombic triacontahedron is implemented in the WolframLanguage as PolyhedronData["RhombicTriacontahedron"].The short diagonals of the faces..

In general, a triakis tetrahedron is a non-regular dodecahedron that can be constructed as a positive augmentation of a regular tetrahedron. Such a solid is also known as a tristetrahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting dodecahedron is not regular, its faces are all identical. "The" triakis tetrahedron is the dual polyhedron of the truncated tetrahedron (Holden 1971, p. 55) and Wenninger dual . It can be constructed by augmentation of a unit edge-length tetrahedron by a pyramid with height .Five tetrahedra of unit edge length (corresponding to a central tetrahedron and its regular augmentation) and one tetrahedron of edge length 5/3 can be inscribed in the vertices of the unit triakis tetrahedron, forming the configurations illustrated above.The triakis tetrahedron formed by taking the dual of a truncated tetrahedron with unit edge..

The (first) rhombic dodecahedron is the dual polyhedron of the cuboctahedron (Holden 1971, p. 55) and Wenninger dual . Its sometimes also called the rhomboidal dodecahedron (Cotton 1990), and the "first" may be included when needed to distinguish it from the Bilinski dodecahedron (Bilinski 1960, Chilton and Coxeter 1963).A rhombic dodecahedron appears in the upper right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).The rhombic dodecahedron is implemented in the WolframLanguage as PolyhedronData["RhombicDodecahedron"].The 14 vertices of the rhombic dodecahedron are joined by 12 rhombiof the dimensions shown in the figure below, where(1)(2)(3)(4)(5)The rhombic dodecahedron can be built up by a placing six cubes on the faces of a seventh, in the configuration of a metal "jack" (left figure). Joining..

The disdyakis triacontahedron is the dual polyhedron of the Archimedean great rhombicosidodecahedron . It is also known as the hexakis icosahedron (Holden 1971, p. 55). It is Wenninger dual .A tetrahedron 10-compound, octahedron 5-compound, cube 5-compound, icosahedron, dodecahedron, and icosidodecahedron can be inscribed in the vertices of a disdyakis triacontahedron (E. Weisstein, Dec. 26-27, 2009).Starting with an Archimedean great rhombicosidodecahedron of unit edge lengths, the edge lengths of the corresponding disdyakis triacontahedron are(1)(2)(3)The corresponding midradius is(4)The surface area and volume are(5)(6)

The 60-faced dual polyhedron of the truncated dodecahedron (Holden 1971, p. 55) and Wenninger dual . Wenninger (1989, p. 46) calls the small triambic icosahedron the triakis octahedron.A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed on the vertices of the triakis icosahedron (E. Weisstein, Dec. 25-27, 2009).Taking the dual of a truncated dodecahedronwith unit edge lengths gives a triakis icosahedron with edge lengths(1)(2)The surface area and volumeare(3)(4)

The disdyakis dodecahedron is the dual polyhedron of the Archimedean great rhombicuboctahedron and Wenninger dual . It is also called the hexakis octahedron (Unkelbach 1940; Holden 1971, p. 55).If the original great rhombicuboctahedronhas unit side lengths, then the resulting dual has edge lengths(1)(2)(3)The inradius is(4)Scaling the disdyakis dodecahedron so that gives a solid with surface area and volume(5)(6)

In general, a tetrakis hexahedron is a non-regular icositetrahedron that can be constructed as a positive augmentation of a cube. Such a solid is also known as a tetrahexahedron, especially to mineralogists (Correns 1949, p. 41; Berry and Mason 1959, p. 127). While the resulting icositetrahedron is not regular, its faces are all identical. "The" tetrakis hexahedron is the 24-faced dual polyhedron of the truncated octahedron (Holden 1971, p. 55) and Wenninger dual . It can be constructed by augmentation of a unit cube by a pyramid with height 1/4.A cube, octahedron, and stella octangula can all be inscribed in the vertices of the tetrakis hexahedron (E. Weisstein, Dec. 25, 2009).The edge lengths for the tetrakis hexahedron constructed as the dual of the truncatedoctahedron with unit edge lengths are(1)(2)Normalizing so that gives a tetrakis hexahedron with surface area and volume(3)(4)..

The deltoidal icositetrahedron is the 24-faced dual polyhedron of the small rhombicuboctahedron and Wenninger dual . It is also called the trapezoidal icositetrahedron (Holden 1971, p. 55).A deltoidal icositetrahedron appears in the middle right as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43).A stella octangula, attractive octahedron 4-compound (whose dual is an attractive cube 4-compound), and cube can all be inscribed in a deltoidal icositetrahedron (E. Weisstein, Dec. 24, 2009). Superposing all these solids gives the beautiful compound illustrated above.For a small rhombicuboctahedron withunit edge length, the deltoidal icositetrahedron has edge lengths(1)(2)and inradius(3)Normalizing so the smallest edge has unit edge length gives a deltoidal icositetrahedron with surface area and volume(4)(5)..

The square antiprism is the antiprism with square bases whose dual is the tetragonal trapezohedron. The square antiprism has 10 faces.The square antiprism with unit edge lengths has surfacearea and volume(1)(2)

The deltoidal hexecontahedron is the 60-faced dual polyhedron of the small rhombicosidodecahedron . It is sometimes also called the trapezoidal hexecontahedron (Holden 1971, p. 55), strombic hexecontahedron, or dyakis hexecontahedron (Unkelbach 1940). It is Wenninger dual .A tetrahedron 10-compound, octahedron 5-compound, cube 5-compound, icosahedron, dodecahedron, and icosidodecahedron can all be inscribed in the vertices of the deltoidal hexecontahedron (E. W. Weisstein, Dec. 24-27, 2009). The resulting compound of all these inscriptable solids is also illustrated above.Starting from a small rhombicosidodecahedron of unit edge length, the edge lengths of the corresponding deltoidal hexecontahedron are(1)(2)The corresponding midradius is(3)The surface area and volume are(4)(5)..

A cuboctahedron, also called the heptaparallelohedron or dymaxion (the latter according to Buckminster Fuller; Rawles 1997), is Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .A cuboctahedron appears in the lower left as one of the polyhedral "stars" in M. C. Escher's 1948 wood engraving "Stars" (Forty 2003, Plate 43), as well is in the mezzotint "Crystal" (Bool et al. 1982, p. 293).It is implemented in the Wolfram Languageas PolyhedronData["Cuboctahedron"].It is shown above in a number of symmetric projections.The dual polyhedron is the rhombic dodecahedron. The cuboctahedron has the octahedral group of symmetries. According to Heron, Archimedes ascribed the cuboctahedron to Plato (Heath 1981; Coxeter 1973, p. 30). The polyhedron..

The snub dodecahedron is an Archimedean solid consisting of 92 faces (80 triangular, 12 pentagonal), 150 edges, and 60 vertices. It is sometimes called the dodecahedron simum (Kepler 1619, Weissbach and Martini 2002) or snub icosidodecahedron. It is a chiral solid, and therefore exists in two enantiomorphous forms, commonly called laevo (left-handed) and dextro (right-handed).It is Archimedean solid , uniform polyhedron and Wenninger model . It has Schläfli symbol s and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["SnubDodecahedron"].An attractive dual of the two enantiomers superposed on one another is illustrated above.The dual polyhedron of the snub dodecahedron isthe pentagonal hexecontahedron.It can be constructed by snubification of a dodecahedron of unit edge length with outward offset(1)and twist angle(2)Here, the notation indicates a polynomial root.The inradius..

The pentakis dodecahedron is the 60-faced dual polyhedron of the truncated icosahedron (Holden 1971, p. 55). It is Wenninger dual . It can be constructed by augmentation of a unit edge-length dodecahedron by a pyramid with height .A tetrahedron 10-compound, cube 5-compound, icosahedron, and dodecahedron can be inscribed in the vertices of the pentakis dodecahedron (E. Weisstein, Dec. 25-27, 2009).Taking the dual of a truncated icosahedronwith unit edge lengths gives a pentakis dodecahedron with edge lengths(1)(2)Normalizing so that , the surface area and volume are(3)(4)

A matrix with elements(1)for , 2, ..., . Hilbert matrices are implemented in the Wolfram Language by HilbertMatrix[m, n]. The figure above shows a plot of the Hilbert matrix with elements colored according to their values.Hilbert matrices whose entries are specified as machine-precision numbers are difficult to invert using numerical techniques.The determinants for the first few values of for , 2, ... are given by one divided by 1, 12, 2160, 6048000, 266716800000, ... (OEIS A005249). The terms of sequence have the closed form(2)(3)(4)where is the Glaisher-Kinkelin constant and is the Barnes G-function. The numerical values are given in the following table.det()1123456The elements of the matrix inverse of the Hilbert matrix are given analytically by(5)(Choi 1983, Richardson 1999).

The truncated icosahedron is the 32-faced Archimedean solid corresponding to the facial arrangement . It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Rhodes 1996, p. 195). The truncated icosahedron has 60 vertices, and is also the structure of pure carbon known as buckyballs (a.k.a. fullerenes).The truncated icosahedron is uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedIcosahedron"].Several symmetrical projections of the truncated icosahedron are illustrated above.The dual polyhedron of the truncated icosahedron is the pentakis dodecahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances..

Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for added security against counterfeiting. For currency, the precise techniques used by the governments of Russia, the United States, Brazil, the European Union, Madagascar, Egypt, and all other countries are likely quite different. The figures above show the same guilloche pattern plotted in polar and Cartesian coordinates generated by a series of nested additions and multiplications of sinusoids of various periods.Guilloché machines (alternately called geometric lathes, rose machines, engine-turners, and cycloidal engines) were first used for a watch casing dated 1624, and consist of myriad gears and settings that can produce many different patterns. Many goldsmiths, including Fabergè, employed guilloché machines.The..

A superellipse is a curve with Cartesian equation(1)first discussed in 1818 by Lamé. A superellipse may be described parametrically by(2)(3)The restriction to is sometimes made.Superellipses with are also known as Lamé curves or Lamé ovals, and the case with is sometimes known as the squircle. By analogy, the superellipse with and might be termed the rectellipse.A range of superellipses are shown above, with special cases , 1, and 2 illustrated right above. The following table summarizes a few special cases. Piet Hein used with a number of different ratios for various of his projects. For example, he used for Sergels Torg (Sergel's Square) in Stockholm, Sweden (Vestergaard), and for his table.curve(squashed) astroid1(squashed) diamond2ellipsePiet Hein's "superellipse"4rectellipseIf is a rational, then a superellipse is algebraic. However, for irrational , it is transcendental. For even integers..

A hypotrochoid generated by a fixed point on a circle rolling inside a fixed circle. The curves above correspond to values of , 0.2, ..., 1.0.Additional attractive designs such as the one above can also be made by superposing individual spirographs.The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features spirographs when discussing Guilloché patterns.

Ede (1958) enumerates 13 basic series of stellations of the rhombic triacontahedron, the total number of which is extremely large. Pawley (1973) gave a set of restrictions upon which a complete enumeration of stellations can be achieved (Wenninger 1983, p. 36). Messer (1995) describes 227 stellations (including the original solid in the count as usual), some of which are illustrated above.The Great Stella stellation software reproduces Messer's 227 fully supported stellations. Using Miller's rules gives 358833098 stellations, 84959 of them reflexible and 358748139 of them chiral.The convex hull of the dodecadodecahedron is an icosidodecahedron and the dual of the icosidodecahedron is the rhombic triacontahedron, so the dual of the dodecadodecahedron (the medial rhombic triacontahedron) is one of the rhombic triacontahedron stellations (Wenninger 1983, p. 41). Others include the great rhombic triacontahedron,..

One of the Kepler-Poinsot solids whose dual is the great stellated dodecahedron. It is also uniform polyhedron , Wenninger model , and has Schläfli symbol and Wythoff symbol .The great icosahedron can be constructed from an icosahedron with unit edge lengths by taking the 20 sets of vertices that are mutually spaced by a distance , the golden ratio. The solid therefore consists of 20 equilateral triangles. The symmetry of their arrangement is such that the resulting solid contains 12 pentagrams.The great icosahedron can most easily be constructed by building a "squashed" dodecahedron (top right figure) from the corresponding net (top left). Then, using the net shown in the bottom left figure, build 12 pentagrammic pyramids (bottom middle figure) and affix them into the dimples (bottom right). This method of construction is given in Cundy and Rollett (1989, pp. 98-99). If the edge lengths of the dodecahedron are unity,..

"Spikey" is the logo of Wolfram Research, makers of Mathematica and the Wolfram Language. In its original (Version 1) form, it is an augmented icosahedron with an augmentation height of , not to be confused with the great stellated dodecahedron, which is a distinct solid. This gives it 60 equilateral triangular faces, making it a deltahedron. More elaborate forms of Spikey have been used for subsequent versions of Mathematica. In particular, Spikeys for Version 2 and up are based on a hyperbolic dodecahedron with embellishments rather than an augmented icosahedron (Trott 2007, Weisstein 2009).The "classic" (Version 1) Spikey illustrated above is implemented in theWolfram Language as PolyhedronData["MathematicaPolyhedron"].The skeleton of the classic Spikey is the graph of thetriakis icosahedron.A glyph corresponding to the classic Spikey, illustrated above, is available as the character \[MathematicaIcon]..

The small stellated dodecahedron is the Kepler-Poinsot solids whose dual polyhedron is the great dodecahedron. It is also uniform polyhedron , Wenninger model , and is the first stellation of the dodecahedron (Wenninger 1989). The small stellated dodecahedron has Schläfli symbol and Wythoff symbol . It is concave, and is composed of 12 pentagrammic faces ().The small stellated dodecahedron appeared ca. 1430 as a mosaic by Paolo Uccello on the floor of San Marco cathedral, Venice (Muraro 1955). It was rediscovered by Kepler (who used th term "urchin") in his work Harmonice Mundi in 1619, and again by Poinsot in 1809.The skeleton of the small stellated dodecahedron is isomorphic to the icosahedralgraph.Schläfli (1901, p. 134) did not recognize the small stellated dodecahedron as a regular solid because it violates the polyhedral formula, instead satisfying(1)where is the number of vertices, the number of edges,..

The regular tetrahedron, often simply called "the" tetrahedron, is the Platonic solid with four polyhedron vertices, six polyhedron edges, and four equivalent equilateral triangular faces, . It is also uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and the Wythoff symbol is . It is an isohedron, and a special case of the general tetrahedron and the isosceles tetrahedron.The regular tetrahedron is implemented in the Wolfram Language as Tetrahedron[], and precomputed properties are available as PolyhedronData["Tetrahedron"].The tetrahedron has 7 axes of symmetry: (axes connecting vertices with the centers of the opposite faces) and (the axes connecting the midpoints of opposite sides).There are no other convex polyhedra other than the tetrahedron having four faces.The tetrahedron has two distinct nets (Buekenhout and Parker 1998). Questions of polyhedron coloring..

The regular octahedron, often simply called "the" octahedron, is the Platonic solid with six polyhedron vertices, 12 polyhedron edges, and eight equivalent equilateral triangular faces, denoted . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and Wythoff symbol . The octahedron of unit side length is the antiprism of sides with height . The octahedron is also a square dipyramid with equal edge lengths.The regular octahedron is implemented in the Wolfram Language as Octahedron[], and precomputed properties are available as PolyhedronData["Octahedron"].There are 11 distinct nets for the octahedron, the same as for the cube (Buekenhout and Parker 1998). Questions of polyhedron coloring of the octahedron can be addressed using the Pólya enumeration theorem.The dual polyhedron of an octahedron with unit edge lengths is a cube with edge lengths .The illustration..

The regular icosahedron (often simply called "the" icosahedron) is the regular polyhedron and Platonic solid illustrated above having 12 polyhedron vertices, 30 polyhedron edges, and 20 equivalent equilateral triangle faces, .The regular icosahedron is also uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and Wythoff symbol . Coxeter et al. (1999) have shown that there are 58 icosahedron stellations (giving a total of 59 solids when the icosahedron itself is included).The regular icosahedron is implemented in the Wolfram Language as Icosahedron[], and precomputed properties are available as PolyhedronData["Icosahedron"].Two icosahedra constructed in origami are illustrated above (Gurkewitz and Arnstein 1995, p. 53). This construction uses 30 triangle edge modules, each made from a single sheet of origami paper.Two icosahedra appears as polyhedral "stars"..

The regular dodecahedron, often simply called "the" dodecahedron, is the Platonic solid composed of 20 polyhedron vertices, 30 polyhedron edges, and 12 pentagonal faces, . It is also uniform polyhedron and Wenninger model . It is given by the Schläfli symbol and the Wythoff symbol .The regular dodecahedron is implemented in the Wolfram Language as Dodecahedron[], and precomputed properties are available as PolyhedronData["Dodecahedron"].There are 43380 distinct nets for the regular dodecahedron, the same number as for the icosahedron (Bouzette and Vandamme, Hippenmeyer 1979, Buekenhout and Parker 1998). Questions of polyhedron coloring of the regular dodecahedron can be addressed using the Pólya enumeration theorem.The image above shows an origami regular dodecahedron constructed using six dodecahedron units, each consisting of a single sheet of paper (Kasahara and Takahama 1987, pp. 86-87).A..

In general, an icosidodecahedron is a 32-faced polyhedron. "The" icosidodecahedron is the 32-faced Archimedean solid with faces . It is one of the two convex quasiregular polyhedra. It is also uniform polyhedron and Wenninger model . It has Schläfli symbol and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["Icosidodecahedron"].Several symmetric projections of the icosidodecahedron are illustrated above. The dual polyhedron is the rhombic triacontahedron. The polyhedron vertices of an icosidodecahedron of polyhedron edge length are , , , , , . The 30 polyhedron vertices of an octahedron 5-compound form an icosidodecahedron (Ball and Coxeter 1987). Faceted versions include the small icosihemidodecahedron and small dodecahemidodecahedron.The icosidodecahedron constructed in origami is illustrated above (Kasahara and Takahama 1987, pp. 48-49). This construction..

The 62-faced Archimedean solid with faces . It is also known as the rhombitruncated icosidodecahedron, and is sometimes improperly called the truncated icosidodecahedron (Ball and Coxeter 1987, p. 143), a name which is inappropriate since truncation would yield rectangular instead of square. The great rhombicosidodecahedron is also uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .The great rhombicosidodecahedron is an equilateral zonohedron and is the Minkowski sum of five cubes.Its dual is the disdyakis triacontahedron, also called the hexakis icosahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The great rhombicosidodecahedron has surface area(4)and volume(5)The great rhombicosidodecahedron constructed by E. K. Herrstrom in origami is illustrated above (Kasahara and Takahama 1987, pp. 46-49)...

Augmentation is the dual operation of truncation which replaces the faces of a polyhedron with pyramids of height (where may be positive, zero, or negative) having the face as the base (Cromwell 1997, p. 124 and 195-197). The operation is sometimes also called accretion, akisation (since it transforms a regular polygon to an -akis polyhedron, i.e., quadruples the number of faces), capping, or cumulation.B. Grünbaum used the terms elevatum and invaginatum for positive-height (outward-pointing) and negative-height (inward-pointing), respectively, pyramids used in augmentation.The term "augmented" is also sometimes used in the more general context of affixing one polyhedral cap over the face of a base solid. An example is the Johnson solid called the augmented truncated cube, for which the affixed shape is a square cupola--not a pyramid.Augmentation is implemented under the misnomer Stellate[poly,..

The Skewes number (or first Skewes number) is the number above which must fail (assuming that the Riemann hypothesis is true), where is the prime counting function and is the logarithmic integral.Isaac Asimov featured the Skewes number in his science fact article "Skewered!"(1974).In 1912, Littlewood proved that exists (Hardy 1999, p. 17), and the upper boundwas subsequently found by Skewes (1933). The Skewes number has since been reduced to by Lehman in 1966 (Conway and Guy 1996; Derbyshire 2004, p. 237), by te Riele (1987), and less than (Bays and Hudson 2000; Granville 2002; Borwein and Bailey 2003, p. 65; Havil 2003, p. 200; Derbyshire 2004, p. 237). The results of Bays and Hudson left open the possibility that the inequality could fail around , and thus established a large range of violation around (Derbyshire 2004, p. 237). More recent work by Demichel establishes that the first crossover..

An algorithm for making tables of primes. Sequentially write down the integers from 2 to the highest number you wish to include in the table. Cross out all numbers which are divisible by 2 (every second number). Find the smallest remaining number . It is 3. So cross out all numbers which are divisible by 3 (every third number). Find the smallest remaining number . It is 5. So cross out all numbers which are divisible by 5 (every fifth number).Continue until you have crossed out all numbers divisible by , where is the floor function. The numbers remaining are prime. This procedure is illustrated in the above diagram which sieves up to 50, and therefore crosses out composite numbers up to . If the procedure is then continued up to , then the number of cross-outs gives the number of distinct prime factors of each number.The sieve of Eratosthenes can be used to compute the primecounting function aswhich is essentially an application of the inclusion-exclusionprinciple..

The trefoil knot , also called the threefoil knot or overhand knot, is the unique prime knot with three crossings. It is a (3, 2)-torus knot and has braid word . The trefoil and its mirror image are not equivalent, as first proved by Dehn (1914). In other words, the trefoil knot is not amphichiral. It is, however, invertible, and has Arf invariant 1.Its laevo form is implemented in the WolframLanguage, as illustrated above, as KnotData["Trefoil"].M. C. Escher's woodcut "Knots" (Bool et al. 1982, pp. 128 and 325; Forty 2003, Plate 71) depicts three trefoil knots composed of differing types of strands. A preliminary study (Bool et al. 1982, p. 123) depicts another trefoil.The animation above shows a series of gears arranged along a Möbiusstrip trefoil knot (M. Trott).The bracket polynomial can be computed as follows.(1)(2)Plugging in(3)(4)gives(5)The corresponding Kauffman polynomial..

The Kuen surface is a special case of Enneper'snegative curvature surfaces which can be given parametrically by(1)(2)(3)(4)(5)for , (Reckziegel et al. 1986; Gray et al. 2006, p. 484).The Kuen surface appears on the cover of volume 2, number 1 of La Gaceta de laReal Sociedad Matemática Española (1999).The coefficients of the first fundamental formare(6)(7)(8)the second fundamental form coefficientsare(9)(10)(11)and the surface area element is(12)The Gaussian and meancurvatures are(13)(14)so the Kuen surface has constant negative Gaussian curvature, and the principal curvatures are(15)(16)(Gray 1997, p. 496).

A surface of constant negative curvature obtained by twisting a pseudosphere and given by the parametric equations(1)(2)(3)The above figure corresponds to , , , and .Dini's surface is pictured in the upper right-hand corner of Gray (1997; left figure), as well as on the cover of volume 2, number 3 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).The coefficients of the first fundamental formare(4)(5)(6)the coefficients of the second fundamentalform are(7)(8)(9)and the area element is(10)The Gaussian and meancurvatures are given by(11)(12)

Togliatti surfaces are quintic surfaces having the maximum possible number of ordinary double points (31).A related surface sometimes known as the dervish can be defined by(1)where(2)(3)(4)(5)(6)(7)(8)and(9)(10)(11)

The Kummer surfaces are a family of quartic surfacesgiven by the algebraic equation(1)where(2), , , and are the tetrahedral coordinates(3)(4)(5)(6)and is a parameter which, in the above plots, is set to .The above plots correspond to (7)(double sphere), 2/3, 1(8)(Roman surface), 2, 3(9)(four planes), and 5. The case corresponds to four real points.The following table gives the number of ordinary double points for various ranges of , corresponding to the preceding illustrations.parameterreal nodescomplex nodes412412160160The Kummer surfaces can be represented parametrically by hyperelliptic theta functions. Most of the Kummer surfaces admit 16 ordinary double points, the maximum possible for a quartic surface. A special case of a Kummer surface is the tetrahedroid.Nordstrand gives the implicit equations as(10)or(11)..

A cubic algebraicsurface given by the equation(1)with the added constraint(2)The implicit equation obtained by taking the plane at infinity as is(3)(Hunt 1996), illustrated above.On Clebsch's diagonal surface, all 27 of the complex lines (Solomon's seal lines) present on a general smooth cubic surface are real. In addition, there are 10 points on the surface where 3 of the 27 lines meet. These points are called Eckardt points (Fischer 1986ab, Hunt 1996), and the Clebsch diagonal surface is the unique cubic surface containing 10 such points (Hunt 1996).If one of the variables describing Clebsch's diagonal surface is dropped, leaving the equations(4)(5)the equations degenerate into two intersecting planes given by the equation(6)

An algebraic surface with affine equation(1)where is a Chebyshev polynomial of the first kind and is a polynomial defined by(2)where the matrices have dimensions . These represent surfaces in with only ordinary double points as singularities. The first few surfaces are given by (3)(4)(5)The th order such surface has(6)singular points (Chmutov 1992), giving the sequence 0, 1, 3, 14, 28, 57, 93, 154, 216, 321, 425, 576, 732, 949, 1155, ... (OEIS A057870) for , 2, .... For a number of orders , Chmutov surfaces have more ordinary double points than any other known equations of the same degree.Based on Chmutov's equations, Banchoff (1991) defined the simpler set of surfaces(7)where is even and is again a Chebyshev polynomial of the first kind. For example, the surfaces illustrated above have orders 2, 4, and 6 and are given by the equations (8)(9)(10)..

A surface with tetrahedral symmetry which looks likean inflatable chair from the 1970s. It is given by the implicit equationThe surface illustrated above has , , and .

The dodecic surface defined by(1)where(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) and are both invariants of order 12. It was discovered by A. Sarti in 1999.The version with arbitrary and has exactly 600 ordinary points (Endraß), and taking gives the surface with 560 real ordinary points illustrated above.The Sarti surface is invariant under the bipolyhedralgroup.

Cayley's cubic surface is the unique cubic surface having four ordinary double points (Hunt), the maximum possible for cubic surface (Endraß). The Cayley cubic is invariant under the tetrahedral group and contains exactly nine lines, six of which connect the four nodes pairwise and the other three of which are coplanar (Endraß).If the ordinary double points in projective three-space are taken as (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), then the equation of the surface in projective coordinates is(1)(Hunt). Defining "affine" coordinates with plane at infinity and(2)(3)(4)then gives the equation(5)plotted in the left figure above (Hunt). The slightly different form(6)is given by Endraß (2003) which, when rewritten in tetrahedralcoordinates, becomes(7)plotted in the right figure above.The Hessian of the Cayley cubic is given by(8)in homogeneous coordinates , , , and . Taking the plane at infinity..

A heart-shaped surface given by the sextic equation(Taubin 1993, 1994). The figures above show a ray-traced rendering (left) and the cross section (right) of the surface.A slight variation of the same surface is given by(Nordstrand, Kuska 2004).

The quartic surface obtained by replacing the constant in the equation of the Cassini ovals with , obtaining(1)As can be seen by letting to obtain(2)(3)the intersection of the surface with the plane is a circle of radius .The Gaussian curvature of the surface is givenimplicitly by(4)Let a torus of tube radius be cut by a plane perpendicular to the plane of the torus's centroid. Call the distance of this plane from the center of the torus hole , let , and consider the intersection of this plane with the torus as is varied. The resulting curves are Cassini ovals, and the surface having these curves as cross sections is the Cassini surface(5)which has a scaled on the right side instead of .

Endraß surfaces are a pair of octic surfaces which have 168 ordinary double points. This is the maximum number known to exist for an octic surface, although the rigorous upper bound is 174. The equations of the surfaces arewhere is a parameter. All ordinary double points of are real, while 24 of those in are complex. The surfaces were discovered in a five-dimensional family of octics with 112 nodes, and are invariant under the group .The surfaces illustrated above take . The first of these has 144 real ordinary double points, and the second of which has 144 complex ordinary double points, 128 of which are real.

The Barth sextic is a sextic surface in complex three-dimensional projective space having the maximum possible number of ordinary double points, namely 65. The surface was discovered by W. Barth in 1994, and is given by the implicit equationwhere is the golden ratio.Taking gives the surface in 3-space illustrated above, which retains 50 ordinary double points.Of these, 20 nodes are at the vertices of a regular dodecahedron of side length and circumradius (left figure above), and 30 are at the vertices of a concentric icosidodecahedron and circumradius 1 (right figure).The Barth sextic is invariant under the icosahedralgroup. Under the mapthe surface is the eightfold cover of the Cayley cubic(Endraß 2003).The Barth sextic appeared on the cover of the March 1999 issue of Notices of theAmerican Mathematical Society (Dominici 1999)...

The prime spiral, also known as Ulam's spiral, is a plot in which the positive integers are arranged in a spiral (left figure), with primes indicated in some way along the spiral. In the right plot above, primes are indicated in red and composites are indicated in yellow.The plot above shows a larger part of the spiral in which the primes are shown as dots.Unexpected patterns of diagonal lines are apparent in such a plot, as illustrated in the above grid. This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting. While drawing a grid of lines, he decided to number the intersections according to a spiral pattern, and then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam's slightly more formal prose, it "appears to exhibit a strongly..

The length of the polygonal spiral is found by noting that the ratio of inradius to circumradius of a regular polygon of sides is(1)The total length of the spiral for an -gon with side length is therefore(2)(3)Consider the solid region obtained by filling in subsequent triangles which the spiral encloses. The area of this region, illustrated above for -gons of side length , is(4)The shaded triangular polygonal spiral is a rep-4-tile.

The spherical curve taken by a ship which travels from the south pole to the north pole of a sphere while keeping a fixed (but not right) angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles.It is given by the parametric equations(1)(2)(3)where(4)and is a constant. Plugging in therefore gives(5)(6)(7)It is a special case of a loxodrome.The arc length, curvature,and torsion are all slightly complicated expressions.A series of spherical spirals are illustrated in Escher's woodcuts "Sphere Surface with Fish" (Bool et al. 1982, pp. 96 and 318) and "Sphere Spirals" (Bool et al. 1982, p. 319; Forty 2003, Plate 67).

A helix, sometimes also called a coil, is a curve for which the tangent makes a constant angle with a fixed line. The shortest path between two points on a cylinder (one not directly above the other) is a fractional turn of a helix, as can be seen by cutting the cylinder along one of its sides, flattening it out, and noting that a straight line connecting the points becomes helical upon re-wrapping (Steinhaus 1999, p. 229). It is for this reason that squirrels chasing one another up and around tree trunks follow helical paths.Helices come in enantiomorphous left- (coils counterclockwise as it "goes away") and right-handed forms (coils clockwise). Standard screws, nuts, and bolts are all right-handed, as are both the helices in a double-stranded molecule of DNA (Gardner 1984, pp. 2-3). Large helical structures in animals (such as horns) usually appear in both mirror-image forms, although the teeth of a male narwhal, usually..

The Borromean rings, also called the Borromean links (Livingston 1993, p. 10) are three mutually interlocked rings (left figure), named after the Italian Renaissance family who used them on their coat of arms. The configuration of rings is also known as a "Ballantine," and a brand of beer (right figure; Falstaff Brewing Corporation) has been brewed under this name. In the Borromean rings, no two rings are linked, so if any one of the rings is cut, all three rings fall apart. Any number of rings can be linked in an analogous manner (Steinhaus 1999, Wells 1991).The Borromean rings are a prime link. They have link symbol 06-0302, braid word , and are also the simplest Brunnian link.It turns out that rigid Borromean rings composed of real (finite thickness) tubes cannot be physically constructed using three circular rings of either equal or differing radii. However, they can be made from three congruent elliptical rings...

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.

An infinite sequence of circles such that every four consecutive circles are mutually tangent, and the circles' radii ..., , ..., , , , , , , ..., , , ..., are in geometric progression with ratiowhere is the golden ratio (Gardner 1979ab). Coxeter (1968) generalized the sequence to spheres.

A plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, a tiling is a collection of disjoint open sets, the closures of which cover the plane. Given a single tile, the so-called first corona is the set of all tiles that have a common boundary point with the tile (including the original tile itself).Wang's conjecture (1961) stated that if a set of tiles tiled the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a tessellation. The conjecture was refuted in 1966 when R. Berger showed that an aperiodic set of tiles exists. By 1971, R. Robinson had reduced the number to six and, in 1974, R. Penrose discovered an aperiodic set (when color-matching rules are included) of two tiles: the so-called Penrose tiles. It is not known if there is a single aperiodic tile.A spiral tiling using a single piece is illustrated..

The Boy surface is a nonorientable surface that is one possible parametrization of the surface obtained by sewing a Möbius strip to the edge of a disk. Two other topologically equivalent parametrizations are the cross-cap and Roman surface. The Boy surface is a model of the projective plane without singularities and is a sextic surface.A sculpture of the Boy surface as a special immersion of the real projective plane in Euclidean 3-space was installed in front of the library of the Mathematisches Forschungsinstitut Oberwolfach library building on January 28, 1991 (Mathematisches Forschungsinstitut Oberwolfach; Karcher and Pinkall 1997).The Boy surface can be described using the general method for nonorientable surfaces, but this was not known until the analytic equations were found by Apéry (1986). Based on the fact that it had been proven impossible to describe the surface using quadratic polynomials, Hopf had conjectured..

A set of magic circles is a numbering of the intersections of the circles such that the sum over all intersections is the same constant for all circles. The above sets of three and four magic circles have magic constants 14 and 39 (Madachy 1979). For circles, the constant is , for , 2, ... corresponding to 3, 14, 39, 84, 155, 258, ... (OEIS A027444).Another type of magic circle arranges the number 1, 2, ..., in a number of rings, which each ring containing the same number of elements and corresponding elements being connected with radial lines. One of the numbers (which is subsequently ignored) is placed at the center. In a magic circle arrangement, the rings have equal sums and this sum is also equal to the sum of elements along each diameter (excluding the central number). Three magic circles using the numbers 1 to 33 are illustrated above...

Combinatorics is the branch of mathematics studying the enumeration, combination, and permutation of sets of elements and the mathematical relations that characterize their properties.Mathematicians sometimes use the term "combinatorics" to refer to a larger subset of discrete mathematics that includes graph theory. In that case, what is commonly called combinatorics is then referred to as "enumeration."The Season 1 episode "Noisy Edge" (2005) of the television crime drama NUMB3RS mentions combinatorics.

Combinatorial matrix theory is a rich branch of mathematics that combines combinatorics, graph theory, and linear algebra. It includes the theory of matrices with prescribed combinatorial properties, including permanents and Latin squares. It also comprises combinatorial proof of classical algebraic theorems such as Cayley-Hamilton theorem.As mentioned in Season 4 episodes 407 "Primacy" and 412 "Power" of the television crime drama NUMB3RS, professor Amita Ramanujan's primary teaching interest is combinatorial matrix theory.

The rook is a chess piece that may move any number of spaces either horizontally or vertically per move. The maximum number of nonattacking rooks that may be placed on an chessboard is . This arrangement is achieved by placing the rooks along the diagonal (Madachy 1979). The total number of ways of placing nonattacking rooks on an board is (Madachy 1979, p. 47). In general, the polynomialwhose coefficients give the numbers of ways nonattacking rooks can be placed on an chessboard is called a rook polynomial.The number of rotationally and reflectively inequivalent ways of placing nonattacking rooks on an board are 1, 2, 7, 23, 115, 694, ... (OEIS A000903; Dudeney 1970, p. 96; Madachy 1979, pp. 46-54).The minimum number of rooks needed to occupy or attack all spaces on an chessboard is 8 (Madachy 1979), arranged in the same orientation as above.Consider an chessboard with the restriction that, for every subset of , a rook may not..

The rook numbers of an board are the number of subsets of size such that no two elements have the same first or second coordinate. In other word, it is the number of ways of placing rooks on a board such that none attack each other (one form of the so-called rooks problem). The rook number is therefore the leading coefficient of the corresponding rook polynomial .For an board, each permutation matrix corresponds to an allowed configuration of rooks. However, the permutation matrices give only a subset of the total number of solutions, which on an board is simply the factorial . This can be seen easily by noting that there are ways to place the first rook in the first column, ways to place the second rook in the second column, ways to place the third rook, ..., and a single way to place the th rook in the last (th) column.The rook numbers of a board determine the rook numbers of the complementary board , written as . This is known as the rook reciprocity theorem...

Various handshaking problems are in circulation, the most common one being the following. In a room of people, how many different handshakes are possible?The answer is . To see this, enumerate the people present, and consider one person at a time. The first person may shake hands with other people. The next person may shake hands with other people, not counting the first person again. Continuing like this gives us a total number ofhandshakes, which is exactly the answer given above.Another popular handshake problem starts out similarly with people at a party. Not being able to shake hands with yourself, and not counting multiple handshakes with the same person, the problem is to show that there will always be two people at the party, who have shaken hands the same number of times.The solution to this problem uses Dirichlet's box principle. If there exists a person at the party, who has shaken hands zero times, then every person at the party has shaken..

The continuous Fourier transform is definedas(1)(2)Now consider generalization to the case of a discrete function, by letting , where , with , ..., . Writing this out gives the discrete Fourier transform as(3)The inverse transform is then(4)Discrete Fourier transforms (DFTs) are extremely useful because they reveal periodicities in input data as well as the relative strengths of any periodic components. There are however a few subtleties in the interpretation of discrete Fourier transforms. In general, the discrete Fourier transform of a real sequence of numbers will be a sequence of complex numbers of the same length. In particular, if are real, then and are related by(5)for , 1, ..., , where denotes the complex conjugate. This means that the component is always real for real data.As a result of the above relation, a periodic function will contain transformed peaks in not one, but two places. This happens because the periods of the input data..

The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are quasi-doubly periodic, and are most commonly denoted in modern texts, although the notations and (Borwein and Borwein 1987) are sometimes also used. Whittaker and Watson (1990, p. 487) gives a table summarizing notations used by various earlier writers.The theta functions are given in the Wolfram Language by EllipticTheta[n, z, q], and their derivatives are given by EllipticThetaPrime[n, z, q].The translational partition function for an ideal gas can be derived using elliptic theta functions (Golden 1961, pp. 119 and 133; Melzak 1973, p. 122; Levine 2002, p. 838).The theta functions may be expressed in terms of the nome , denoted , or the half-period ratio , denoted , where and and are related by(1)Let the multivalued function be interpreted to stand..

Jenny's constant is the name given (Munroe 2012) to the positive real constant defined by(1)(2)(OEIS A182369), the first few digits of which are 867-5309, corresponding to the fictitious phone number in the song "867-5309/Jenny" performed by Tommy Tutone in 1982.Other "simple" expressions that might vie for that moniker include(3)(4)(5)(6)(7)(8)(9)(10)where is the hard hexagon entropy constant, the first three of which are "better" than the canonical Jenny expression (E. Weisstein, Jul. 12, 2013).

Triskaidekaphobia is the fear of 13, a number commonly associated with bad luck in Western culture. While fear of the number 13 can be traced back to medieval times, the word triskaidekaphobia itself is of recent vintage, having been first coined by Coriat (1911; Simpson and Weiner 1992). It seems to have first appeared in the general media in a Nov. 8, 1953 New York Times article covering discussions of a United Nations committee.This superstition leads some people to fear or avoid anything involving the number 13. In particular, this leads to interesting practices such as the numbering of floors as 1, 2, ..., 11, 12, 14, 15, ... (OEIS A011760; the "elevator sequence"), omitting the number 13, in many high-rise American hotels, the numbering of streets avoiding 13th Avenue, and so on.Apparently, 13 hasn't always been considered unlucky. In fact, it appears that in ancient times, 13 was either considered in a positive light or..

Nice approximations for the golden ratio are given by(1)(2)the last of which is due to W. van Doorn (pers. comm., Jul. 18, 2006) and which are accurate to and , respectively. An even more amazing approximation uses Catalan's constant and the Feigenbaum constant is given by(3)which is accurate to within (D. Ross, cited in Pegg 2005).A curious (although not particularly useful) approximation due to D. Barron is given by(4)where is Catalan's constant and is the Euler-Mascheroni constant, which is good to two digits.

A curious approximation to the Feigenbaum constant is given by(1)where is Gelfond's constant, which is good to 6 digits to the right of the decimal point.M. Trott (pers. comm., May 6, 2008) noted(2)where is Gauss's constant, which is good to 4 decimal digits, and(3)where is the tetranacci constant, which is good to 3 decimal digits.A strange approximation good to five digits is given by the solution to(4)which is(5)where is the Lambert W-function (G. Deppe, pers. comm., Feb. 27, 2003).(6)gives to 3 digits (S. Plouffe, pers. comm., Apr. 10, 2006).M. Hudson (pers. comm., Nov. 20, 2004) gave(7)(8)(9)which are good to 17, 13, and 9 digits respectively.Stoschek gave the strange approximation(10)which is good to 9 digits.R. Phillips (pers. comm., Sept. 14, 2004-Jan. 25, 2005) gave the approximations(11)(12)(13)(14)(15)(16)where e is the base of the natural logarithm and..

A number having 666 digits (where 666 is the beastnumber) is called an apocalypse number.The Fibonacci number is the smallest Fibonacci apocalypse number (Livio 2002, p. 108).Apocalypse primes are given by for , 1837, 6409, 7329, 8569, 8967, 9663, ... (OEIS A115983). The smallest apocalypse prime containing the digits 666 is (Rupinski).

Convergents of the pi continued fractions are the simplest approximants to . The first few are given by 3, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, ... (OEIS A002485 and A002486), which are good to 0, 2, 4, 6, 9, 9, 9, 10, 11, 11, 12, 13, ... (OEIS A114526) decimal digits, respectively.Two approximations follow from the near-identity function evaluated at and , giving(1)(2)which are good to 2 and 3 digits, respectively.Kochanski's approximation is the rootof(3)given by(4)which is good to 4 digits.Another curious fact is the almost integer(5)which can also be written as(6)(7)Here, is Gelfond's constant. Applying cosine a few more times gives(8)Another approximation involving is given by(9)which is good to 2 decimal digits (A. Povolotsky, pers. comm., Mar. 6, 2008).An apparently interesting near-identity is given by(10)which becomes less surprising when it is noted that 555555 is a repdigit,so the above is..

E. Pegg Jr. (pers. comm., Nov. 8, 2004) found an approximation to Apéry's constant given by(1)which is good to 6 digits.M. Hudson (pers. comm., Nov. 8, 2004) found the approximations(2)(3)(4)(5)(6)(7)where is the Euler-Mascheroni constant and is the golden ratio, which are good to 5, 7, 7, 8, 11, and 12 digits, respectively.A curious approximation to is given by(8)where is the Euler-Mascheroni constant, which is accurate to four digits (P. Galliani, pers. comm., April 19, 2002).Lima (2009) found the approximation(9)where is Catalan's constant, which is correct to 21 digits.

An amazing pandigital approximation to that is correct to 18457734525360901453873570 decimal digits is given by(1)found by R. Sabey in 2004 (Friedman 2004).Castellanos (1988ab) gives several curious approximations to ,(2)(3)(4)(5)(6)(7)which are good to 6, 7, 9, 10, 12, and 15 digits respectively.E. Pegg Jr. (pers. comm., Mar. 2, 2002), found(8)which is good to 7 digits.J. Lafont (pers. comm., MAy 16, 2008) found(9)where is a harmonic number, which is good to 7 digits.N. Davidson (pers. comm., Sept. 7, 2004) found(10)which is good to 6 digits.D. Barron noticed the curious approximation(11)where is Catalan's constant and is the Euler-Mascheroni constant, which however, is only good to 3 digits.

Approximations to Catalan's constant include(1)(2)(3)(4)(5)(6)(M. Hudson, pers. comm., Nov. 19, 2004), where is the golden ratio, which are good to 4, 5, 6, 6, 7, 7, and 9 digits, respectively.Other approximations include(7)(8)(K. Hammond, pers. comm., Dec. 31, 2005), where is the golden ratio, which are good to 5 and 9 digits, respectively.

A Woodall prime is a Woodall numberthat is prime. The first few Woodall primes are 7, 23, 383, 32212254719, 2833419889721787128217599, ... (OEIS A050918), corresponding to , 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, 462, 512, 751, 822, 5312, 7755, 9531, 12379, ... (OEIS A002234). The following table summarizes large known Woodall primes. As of Mar. 2018, all have been checked (PrimeGrid).decimal digitsdate1467763441847Jun. 20072013992606279Aug. 20072367906712818Aug. 200737529481129757Dec. 2007170166025122515Mar. 2018

, sometimes also denoted (Abramowitz and Stegun 1972, p. 825; Comtet 1974, p. 94; Hardy and Wright 1979, p. 273; Conway and Guy 1996, p. 94; Andrews 1998, p. 1), gives the number of ways of writing the integer as a sum of positive integers, where the order of addends is not considered significant. By convention, partitions are usually ordered from largest to smallest (Skiena 1990, p. 51). For example, since 4 can be written(1)(2)(3)(4)(5)it follows that . is sometimes called the number of unrestricted partitions, and is implemented in the Wolfram Language as PartitionsP[n].The values of for , 2, ..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ... (OEIS A000041). The values of for , 1, ... are given by 1, 42, 190569292, 24061467864032622473692149727991, ... (OEIS A070177).The first few prime values of are 2, 3, 5, 7, 11, 101, 17977, 10619863, ... (OEIS A049575), corresponding to indices 2, 3, 4, 5, 6, 13, 36, 77, 132,..

Let a prime number generated by Euler's prime-generating polynomial be known as an Euler prime. Then the first few Euler primes occur for , 2, ..., 39, 42, 43, 45, ... (OEIS A056561), corresponding to the primes 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, 173, ... (OEIS A005846).As of Feb. 2013, the largest known Euler prime is , which has 398204 decimal digits and was found by D. Broadhurst (https://primes.utm.edu/primes/page.php?id=111195).

A prime is called a Wolstenholme prime if the central binomial coefficient(1)or equivalently if(2)where is the th Bernoulli number and the congruence is fractional.A prime is a Wolstenholme prime if and only if(3)where the congruence is again fractional.The only known Wolstenholme primes are 16843 and 2124679 (OEIS A088164). There are no others up to (McIntosh 2004).

A palindromic prime is a number that is simultaneously palindromic and prime. The first few (base-10) palindromic primes are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, ... (OEIS A002385; Beiler 1964, p. 228). The number of palindromic primes less than a given number are illustrated in the plot above. The number of palindromic numbers having , 2, 3, ... digits are 4, 1, 15, 0, 93, 0, 668, 0, 5172, 0, ... (OEIS A016115; De Geest) and the total number of palindromic primes less than 10, , , ... are 4, 5, 20, 20, 113, 113, 781, ... (OEIS A050251). Gupta (2009) has computed the numbers of palindromic primes up to .The following table lists palindromic primes in various small bases. OEISbase- palindromic primes2A11769711, 101, 111, 10001, 11111, 1001001, 1101011, ...3A1176982, 111, 212, 12121, 20102, 22122, ...4A1176992, 3, 11, 101, 131, 323, 10001, 11311, 12121, ...5A1177002, 3, 111, 131, 232, 313, 414, 10301, 12121,..

The Euler numbers, also called the secant numbers or zig numbers, are defined for by(1)(2)where is the hyperbolic secant and sec is the secant. Euler numbers give the number of odd alternating permutations and are related to Genocchi numbers. The base e of the natural logarithm is sometimes known as Euler's number.A different sort of Euler number, the Euler number of a finite complex , is defined by(3)This Euler number is a topological invariant.To confuse matters further, the Euler characteristic is sometimes also called the "Euler number" and numbers produced by the prime-generating polynomial are sometimes called "Euler numbers" (Flannery and Flannery 2000, p. 47). In this work, primes generated by that polynomial are termed Euler primes.Some values of the (secant) Euler numbers are(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(OEIS A000364).The slightly different convention defined by(16)(17)is..

A Wilson prime is a prime satisfyingwhere is the Wilson quotient, or equivalently,The first few Wilson primes are 5, 13, and 563 (OEIS A007540). Crandall et al. (1997) showed there are no others less than (McIntosh 2004), a limit that has subsequently been increased to (Costa et al. 2012).

A Wieferich prime is a prime which is a solution to the congruence equation(1)Note the similarity of this expression to the special case of Fermat'slittle theorem(2)which holds for all odd primes. The first few Wieferich primes are 1093, 3511, ... (OEIS A001220), with none other less than (Lehmer 1981, Crandall 1986, Crandall et al. 1997), a limit since increased to (McIntosh 2004) and subsequently to by PrimeGrid as of November 2015.Interestingly, one less than these numbers have suggestive periodic binaryrepresentations(3)(4)(Johnson 1977).If the first case of Fermat's last theorem is false for exponent , then must be a Wieferich prime (Wieferich 1909). If with and relatively prime, then is a Wieferich prime iff also divides . The conjecture that there are no three consecutive powerful numbers implies that there are infinitely many non-Wieferich primes (Granville 1986; Ribenboim 1996, p. 341; Vardi 1991). In addition, the abc..

In Book IX of The Elements, Euclid gave a method for constructing perfect numbers (Dickson 2005, p. 3), although this method applies only to even perfect numbers. In a 1638 letter to Mersenne, Descartes proposed that every even perfect number is of Euclid's form, and stated that he saw no reason why an odd perfect number could not exist (Dickson 2005, p. 12). Descartes was therefore among the first to consider the existence of odd perfect numbers; prior to Descartes, many authors had implicitly assumed (without proof) that the perfect numbers generated by Euclid's construction comprised all possible perfect numbers (Dickson 2005, pp. 6-12). In 1657, Frenicle repeated Descartes' belief that every even perfect number is of Euclid's form and that there was no reason odd perfect number could not exist. Like Frenicle, Euler also considered odd perfect numbers.To this day, it is not known if any odd perfect numbers exist, although..

A "weird number" is a number that is abundant (i.e., the sum of proper divisors is greater than the number) without being pseudoperfect (i.e., no subset of the proper divisors sums to the number itself). The pseudoperfect part of the definition means that finding weird numbers is a case of the subset sum problem.Since prime numbers are deficient, prime numbers are not weird. Similarly, since multiples of 6 are pseudoperfect, no weird number is a multiple of 6.The smallest weird number is 70, which has proper divisors 1, 2, 5, 7, 10, 14, and 35. These sum to 74, which is greater that the number itself, so 70 is abundant, and no subset of them sums to 70. In contrast, the smallest abundant number is 12, which has proper divisors 1, 2, 3, 4, and 6. These sum to 16, so 12 is abundant, but the subset sum equals 12, so 12 is not weird.The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ...(OEIS A006037).An infinite number of weird..

An extremely fast factorization method developed by Pollard which was used to factor the RSA-130 number. This method is the most powerful known for factoring general numbers, and has complexity(1)reducing the exponent over the continued fraction factorization algorithm and quadratic sieve. There are three values of relevant to different flavors of the method (Pomerance 1996). For the "special" case of the algorithm applied to numbers near a large power,(2)for the "general" case applicable to any odd positive number which is not a power,(3)and for a version using many polynomials (Coppersmith1993),(4)

Euclid's second theorem states that the number of primes is infinite. The proof of this can be accomplished using the numbers(1)(2)known as Euclid numbers, where is the th prime and is the primorial.The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, ... (OEIS A006862; Tietze 1965, p. 19).The indices of the first few prime Euclid numbers are 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, ... (OEIS A014545), so the first few Euclid primes (commonly known as primorial primes) are 3, 7, 31, 211, 2311, 200560490131, ... (OEIS A018239). The largest known Euclid number is , and it is not known if there are an infinite number of prime Euclid numbers (Guy 1994, Ribenboim 1996).The largest factors of for , 2, ... are 3, 7, 31, 211, 2311, 509, 277, 27953, ... (OEIS A002585)...

Elliptic curve primality proving, abbreviated ECPP, is class of algorithms that provide certificates of primality using sophisticated results from the theory of elliptic curves. A detailed description and list of references are given by Atkin and Morain (1990, 1993).Adleman and Huang (1987) designed an independent algorithm using hyperellipticcurves of genus two.ECPP is the fastest known general-purpose primality testing algorithm. ECPP has a running time of . As of 2004, the program PRIMO can certify a 4769-digit prime in approximately 2000 hours of computation (or nearly three months of uninterrupted computation) on a 1 GHz processor using this technique. As of 2009, the largest prime certified using this technique was the 11th Mills' prime (https://primes.utm.edu/primes/page.php?id=77907)which has decimal digits. The proof was performed using a distributed computation that started in September 2005 and ended in June 2006..

A Wagstaff prime is a prime number of the form for a prime number. The first few are given by , 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, and 4031399 (OEIS A000978), with and larger corresponding to probable primes. These values correspond to the primes with indices , 3, 4, 5, 6, 7, 8, 9, 11, 14, 18, 22, 26, ... (OEIS A123176).The Wagstaff primes are featured in the newMersenne prime conjecture.There is no simple primality test analogous to the Lucas-Lehmer test for Wagstaff primes, so all recent primality proofs of Wagstaff primes have used elliptic curve primality proving.A Wagstaff prime can also be interpreted as a repunit prime of base , asif is odd, as it must be for the above number to be prime.Some of the largest known Wagstaff probable primes are summarized in the following..

The numerical value of is given by(OEIS A002392). It was computed to decimal digits by S. Kondo on May 20, 2011 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082, ... (OEIS A228243).-constant primes occur at 1, 2, 40, 242, 842, 1541, 75067, ... decimal digits (OEIS A228240).The starting positions of the first occurrence of , 1, ... in the decimal expansion of (including the initial 2 and counting it as the first digit) are 3, 21, 1, 2, 13, 5, 17, 22, ... (OEIS A229197).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 22, 701, 7486, 88092, 1189434, 13426407, ... (OEIS A229124), which end at digits 7, 38, 351, 8493, 33058, 362945, ... (OEIS A229126).The digit strings 0123456789 first occurs starting at position 3349545080, but 9876543210 does not occur in the first..

The elliptic curve factorization method, abbreviated ECM and sometimes also called the Lenstra elliptic curve method, is a factorization algorithm that computes a large multiple of a point on a random elliptic curve modulo the number to be factored . It tends to be faster than the Pollard rho factorization and Pollard p-1 factorization methods.Zimmermann maintains a table of the largest factors found using the ECM. As of Jan. 2009, the largest prime factor found using the ECM had 67 decimal digits. This factor of was found by B. Dodson on Aug. 24, 2006 (Zimmermann).

A pair of numbers and such thatwhere is the unitary divisor function. Hagis (1971) and García (1987) give 82 such pairs. The first few are (114, 126), (1140, 1260), (18018, 22302), (32130, 40446), ... (OEIS A002952 and A002953; Pedersen).On Jan. 30, 2004, Y. Kohmoto discovered the largest known unitary amicable pair, where each member has 317 digits.Kohmoto calls a unitary amicable pair whose members are squareful a proper unitary amicable pair.

The Earls sequence gives the starting position in the decimal digits of (or in general, any constant), not counting digits to the left of the decimal point, at which a string of copies of the number first occurs. The following table gives generalized Earls sequences for various constants, including .constantOEISsequenceApéry's constantA22907410, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ...Catalan's constantA2248192, 107, 1225, 596, 32187, 185043, 20444527, 92589355, 3487283621, ...Champernowne constantA2248961, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ...Copeland-Erdős constantA2248975, 113, 1181, 21670, 263423, 7815547, 35619942, 402720247, 450680638eA2248282, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011Euler-Mascheroni constantA2248265, 139, 163, 10359, 86615, 193446, 236542, 6186099, 36151186Glaisher-Kinkelin constantA2257637,..

The decimal expansion of the natural logarithmof 2 is given by(OEIS A002162). It was computed to decimal digits by S. Kondo on May 14, 2011 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 4, 419, 2114, 3929, 38451, 716837, 6180096, 10680693, 2539803904 (OEIS A228242).-constant primes occur at 321, 466, 1271, 15690, 18872, 89973, ... decimal digits (OEIS A228226).The starting positions of the first occurrence of , 1, ... in the decimal expansion of are 9, 4, 22, 3, 5, 10, 1, 6, 8, ... (OEIS A100077).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 2, 98, 604, 1155, 46847, 175403, ... (OEIS A036901), which end at digits 22, 444, 7655, 98370, 1107795, 12983306, ... (OEIS A036905).The digit string 0123456789 occurs starting at positions 3157027485, 8102152328, ... in the decimal digits of , and 9876543210 occurs starting..

Twin primes are pairs of primes of the form (, ). The term "twin prime" was coined by Paul Stäckel (1862-1919; Tietze 1965, p. 19). The first few twin primes are for , 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, 240, 270, 282, ... (OEIS A014574). Explicitly, these are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ... (OEIS A001359 and A006512).All twin primes except (3, 5) are of the form .It is conjectured that there are an infinite number of twin primes (this is one form of the twin prime conjecture), but proving this remains one of the most elusive open problems in number theory. An important result for twin primes is Brun's theorem, which states that the number obtained by adding the reciprocals of the odd twin primes,(1)converges to a definite number ("Brun's constant"), which expresses the scarcity of twin primes, even if there are infinitely many of them (Ribenboim 1996, p. 201)...

The constant e with decimal expansion(OEIS A001113) can be computed to digits of precision in 10 CPU-minutes on modern hardware. was computed to digits by P. Demichel, and the first have been verified by X. Gourdon on Nov. 21, 1999 (Plouffe). was computed to decimal digits by S. Kondo on Jul. 5, 2010 (Yee).The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 2, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011, ... (OEIS A224828).The starting positions of the first occurrence of in the decimal expansion of (including the initial 2 and counting it as the first digit) are 14, 3, 1, 18, 11, 12, 21, 2, ... (OEIS A088576).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 6, 12, 548, 1769, 92994, ... (OEIS A036900), which end at digits 21, 372, 8092, 102128, ... (OEIS A036904).The digit sequence 0123456789..

A number is -multiperfect (also called a -multiply perfect number or -pluperfect number) iffor some integer , where is the divisor function. The value of is called the class. The special case corresponds to perfect numbers , which are intimately connected with Mersenne primes (OEIS A000396). The number 120 was long known to be 3-multiply perfect () sinceThe following table gives the first few for , 3, ..., 6.2A0003966, 28, 496, 8128, ...3A005820120, 672, 523776, 459818240, 1476304896, 510011801604A02768730240, 32760, 2178540, 23569920, ...5A04606014182439040, 31998395520, 518666803200, ...6A046061154345556085770649600, 9186050031556349952000, ...Lehmer (1900-1901) proved that has at least three distinct prime factors, has at least four, at least six, at least nine, and at least 14, etc.As of 1911, 251 pluperfect numbers were known (Carmichael and Mason 1911). As of 1929, 334 pluperfect numbers were known, many of them found..

In the 1980s, Samuel Yates defined a titanic prime to be a prime number of at least 1000 decimal digits. The smallest titanic prime is . As of 1990, more than 1400 were known (Ribenboim 1990). By 1995, more than were known, and many tens of thousands are known today. The largest prime number known as of December 2018 is the Mersenne prime , which has a whopping decimal digits.

A double Mersenne number is a number of the formwhere is a Mersenne number. The first few double Mersenne numbers are 1, 7, 127, 32767, 2147483647, 9223372036854775807, ... (OEIS A077585).A double Mersenne number that is prime is called a double Mersenne prime. Since a Mersenne prime can be prime only for prime , a double Mersenne prime can be prime only for prime , i.e., a Mersenne prime. Double Mersenne numbers are prime for , 3, 5, 7, corresponding to the sequence 7, 127, 2147483647, 170141183460469231731687303715884105727, ... (OEIS A077586).The next four , , , and have known factors summarized in the following table. The status of all other double Mersenne numbers is unknown, with being the smallest unresolved case. Since this number has 694127911065419642 digits, it is much too large for the usual Lucas-Lehmer test to be practical. The only possible method of determining the status of this number is therefore attempting to find small divisors..

Theodorus's constant has decimal expansion(OEIS A002194). It was computed to decimal digits by E. Weisstein on Jul. 23, 2013.The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 27, 215, 1651, 2279, 21640, 176497, 7728291, 77659477, 638679423, ... (OEIS A224874).-constant primes occur at 2, 3, 19, 111, 116, 641, 5411, 170657, ... (OEIS A119344) decimal digits.The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (including the initial 1 and counting it as the first digit) are 5, 1, 4, 3, 23, 6, 12, 2, 8, 18, ... (OEIS A229200).Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 4, 91, 184, 5566, 86134, 35343, ... (OEIS A000000), which end at digits 23, 378, 7862, 77437, 1237533, 16362668, ... (OEIS A000000).The digit sequence 9876543210 does not occur in the first digits of , but 0123456789 does,..

Given the Mertens function defined by(1)where is the Möbius function, Stieltjes claimed in an 1885 letter to Hermite that stays within two fixed bounds, which he suggested could probably be taken to be (Havil 2003, p. 208). In the same year, Stieltjes (1885) claimed that he had a proof of the general result. However, it seems likely that Stieltjes was mistaken in this claim (Derbyshire 2004, pp. 160-161). Mertens (1897) subsequently published a paper opining based on a calculation of that Stieltjes' claim(2)for was "very probable."The Mertens conjecture has important implications, since the truth of any equalityof the form(3)for any fixed (the form of the Mertens conjecture with ) would imply the Riemann hypothesis. In fact, the statement(4)for any is equivalent to the Riemann hypothesis (Derbyshire 2004, p. 251).Mertens (1897) verified the conjecture for , and this was subsequently extended to by..

A Thâbit ibn Kurrah prime, sometimes called a 321-prime, is a Thâbit ibn Kurrah number (i.e., a number of the form for nonnegative integer ) that is prime.The indices for the first few Thâbit ibn Kurrah primes are 0, 1, 2, 3, 4, 6, 7, 11, 18, 34, 38, 43, 55, 64, 76, 94, 103, 143, 206, 216, 306, 324, 391, 458, 470, ... (OEIS A002235), corresponding to the primes 2, 5, 11, 23, 47, 191, 383, 6143, ... (OEIS A007505). Riesel (1969) extended the search to . A search for larger primes was coordinated by P. Underwood. PrimeGrid has continued that search and has checked values of up to as of Nov. 2015 (PrimeGrid). The table below summarizes the largest known Thâbit ibn Kurrah primes.digitsdiscovererPrimeGrid (Dec. 2005; https://primes.utm.edu/primes/page.php?id=76506)PrimeGrid (Mar. 2007; https://primes.utm.edu/primes/page.php?id=79671)PrimeGrid (Apr. 2008; https://primes.utm.edu/primes/page.php?id=84769)PrimeGrid..

A Mersenne prime is a Mersenne number, i.e., anumber of the formthat is prime. In order for to be prime, must itself be prime. This is true since for composite with factors and , . Therefore, can be written as , which is a binomial number that always has a factor .The first few Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (OEIS A000668) corresponding to indices , 3, 5, 7, 13, 17, 19, 31, 61, 89, ... (OEIS A000043).Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number. L. Welsh maintains an extensive bibliography and history of Mersenne numbers.It has been conjectured that there exist an infinite number of Mersenne primes. Fitting a line through the origin to the asymptotic number of Mersenne primes with for the first 51 (known) Mersenne primes gives a best-fit line with , illustrated above. If the line is not restricted to pass through..

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