Sort by:

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 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..

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 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...

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).

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..

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,..

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.

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..

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..

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).

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.

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..

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 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 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:..

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

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 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 (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..

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..

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 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.

An official chessboard is an board containing squares alternating in color between olive green and buff (where "buff" is a color variously defined as a moderate orange yellow or a light to moderate yellow) on which the game of chess is played. The checkerboard is identical to the chessboard, and in both cases, the squares are referred to as "black" and "white." In chess (as in checkers), the board is oriented so that each player has a black square on his left.It is impossible to cover a chessboard from which two opposite corners have beenremoved with dominoes.The above plot shows a chessboard centered at (0, 0) and its inverse about a small circle also centered at (0, 0) (Gardner 1984, pp. 244-245; Dixon 1991).The illustration above shows an infinite chessboard reflected in a sphere...

Dürer's magic square is a magic square with magic constant 34 used in an engraving entitled Melancholia I by Albrecht Dürer (The British Museum, Burton 1989, Gellert et al. 1989). The engraving shows a disorganized jumble of scientific equipment lying unused while an intellectual sits absorbed in thought. Dürer's magic square is located in the upper right-hand corner of the engraving. The numbers 15 and 14 appear in the middle of the bottom row, indicating the date of the engraving, 1514.Dürer's magic square has the additional property that the sums in any of the four quadrants, as well as the sum of the middle four numbers, are all 34 (Hunter and Madachy 1975, p. 24). It is thus a gnomon magic square. In addition, any pair of numbers symmetrically placed about the center of the square sums to 17, a property making the square even more magical...

A maze, also known as a labyrinth, as is a set of passages (with impermeable walls). The goal of the maze is to start at one given point and find a path through the passages that reaches a second given point.The back of a clay accounting tablet from Pylos, Greece is illustrated above (Wolfram 2002, p. 43). Legend has it that it was the plan for the labyrinth housing the minotaur in the palace at Knossos, Crete, and that it was designed by Daedalus. It is also said that it was a logo for the city of Troy-or perhaps the plan of some of its walls (Wolfram 2002, p. 873).The above pattern (in either its square or rounded form) has appeared with remarkably little variation in a large variety of places all over the world-from Cretan coins, to graffiti at Pompeii, to the floor of the cathedral at Chartres, to carvings in Peru, to logos for aboriginal tribes. For probably three thousand years, it has been the single most common design used for mazes (Wolfram..

A die (plural "dice") is a solid with markings on each of its faces. The faces are usually all the same shape, making Platonic solids and Archimedean duals the obvious choices. The die can be "rolled" by throwing it in the air and allowing it to come to rest on one of its faces. Dice are used in many games of chance as a way of picking random numbers on which to bet, and are used in board or role-playing games to determine the number of spaces to move, results of a conflict, etc. A coin can be viewed as a degenerate 2-sided case of a die.In 1787, Mozart wrote the measures and instructions for a musical composition dice game. The idea is to cut and paste pre-written measures of music together to create a Minuet (Chuang).The most common type of die is a six-sided cube with the numbers 1-6 placed on the faces. The value of the roll is indicated by the number of "spots" showing on the top. For the six-sided die, opposite faces are arranged..

"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 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..

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..

Let one grain of wheat be placed on the first square of a chessboard, two on the second, four on the third, eight on the fourth, etc. How many grains total are placed on an chessboard? Since this is a geometric series, the answer for squares isa Mersenne number. Plugging in then gives .

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)

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 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..

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..

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.

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..

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..

A cubic number is a figurate number of the form with a positive integer. The first few are 1, 8, 27, 64, 125, 216, 343, ... (OEIS A000578). The protagonist Christopher in the novel The Curious Incident of the Dog in the Night-Time recites the cubic numbers to calm himself and prevent himself from wanting to hit someone (Haddon 2003, p. 213).The generating function giving the cubic numbersis(1)The hex pyramidal numbers are equivalent tothe cubic numbers (Conway and Guy 1996).The plots above show the first 255 (top figure) and 511 (bottom figure) cubic numbers represented in binary.Pollock (1843-1850) conjectured that every number is the sum of at most 9 cubic numbers (Dickson 2005, p. 23). As a part of the study of Waring's problem, it is known that every positive integer is a sum of no more than 9 positive cubes (, proved by Dickson, Pillai, and Niven in the early twentieth century), that every "sufficiently large" integer..

A square number, also called a perfect square, is a figurate number of the form , where is an integer. The square numbers for , 1, ... are 0, 1, 4, 9, 16, 25, 36, 49, ... (OEIS A000290).A plot of the first few square numbers represented as a sequence of binary bits is shown above. The top portion shows to , and the bottom shows the next 510 values.The generating function giving the squarenumbers is(1)The st square number is given in terms of the th square number by(2)since(3)which is equivalent to adding a gnomon to the previoussquare, as illustrated above.The th square number is equal to the sum of the st and th triangular numbers,(4)(5)as can seen in the above diagram, in which the st triangular number is represented by the white triangles, the th triangular number is represented by the black triangles, and the total number of triangles is the square number (R. Sobel, pers. comm.).Square numbers can also be generated by taking the product of two consecutive..

The cube is the Platonic solid (also called the regular hexahedron). It is composed of six square faces that meet each other at right angles and has eight vertices and 12 edges. It is also the uniform polyhedron and Wenninger model . It is described by the Schläfli symbol and Wythoff symbol .The cube is illustrated above, together with a wireframe version and a net(top figures). The bottom figures show three symmetric projections of the cube.The cube is implemented in the Wolfram Language as Cube[], and precomputed properties are available as PolyhedronData["Cube"].There are a total of 11 distinct nets for the cube (Turney 1984-85, Buekenhout and Parker 1998, Malkevitch), illustrated above, the same number as the octahedron. Questions of polyhedron coloring of the cube can be addressed using the Pólya enumeration theorem.A cube with unit edge lengths is called a unit cube.The surface area and volume of a cube with edge..

A prime is said to be a Sophie Germain prime if both and are prime. The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, ... (OEIS A005384). It is not known if there are an infinite number of Sophie Germain primes (Hoffman 1998, p. 190).The numbers of Sophie Germain primes less than for , 2, ... are 3, 10, 37, 190, 1171, 7746, 56032, ... (OEIS A092816).The largest known proven Sophie Germain prime pair as of Feb. 29, 2016 is given by where(Caldwell), each of which has decimal digits (PrimeGrid).The definition of Sophie Germain primes and the value of the largest then-known suchprime were mentioned by the characters Hal and Catherine in the 2005 film Proof.Sophie Germain primes of the form correspond to the indices of composite Mersenne numbers .Around 1825, Sophie Germain proved that the first case of Fermat's last theorem is true for such primes, i.e., if is a Sophie Germain prime, then there do not exist integers..

Fermat's last theorem is a theorem first proposed by Fermat in the form of a note scribbled in the margin of his copy of the ancient Greek text Arithmetica by Diophantus. The scribbled note was discovered posthumously, and the original is now lost. However, a copy was preserved in a book published by Fermat's son. In the note, Fermat claimed to have discovered a proof that the Diophantine equation has no integer solutions for and .The full text of Fermat's statement, written in Latin, reads "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet" (Nagell 1951, p. 252). In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number..

A geometric sequence is a sequence , , 1, ..., such that each term is given by a multiple of the previous one. Another equivalent definition is that a sequence is geometric iff it has a zero series bias. If the multiplier is , then the th term is given byTaking gives the simple special caseThe Season 1 episode "Identity Crisis" (2005) of the television crime drama NUMB3RS mentions geometric progressions.

The minimum spanning tree of a weighted graph is a set of edges of minimum total weight which form a spanning tree of the graph. When a graph is unweighted, any spanning tree is a minimum spanning tree.The minimum spanning tree can be found in polynomial time. Common algorithms include those due to Prim (1957) and Kruskal's algorithm (Kruskal 1956). The problem can also be formulated using matroids (Papadimitriou and Steiglitz 1982). A minimum spanning tree can be found in the Wolfram Language using the command FindSpanningTree[g].The Season 1 episodes "Vector" and "Man Hunt" (2005) and Season 2 episode "Rampage" (2006) of the television crime drama NUMB3RS feature minimal spanning trees.

A random matrix is a matrix of given type and size whoseentries consist of random numbers from some specified distribution.Random matrix theory is cited as one of the "modern tools" used in Catherine'sproof of an important result in prime number theory in the 2005 film Proof.For a real matrix with elements having a standard normal distribution, the expected number of real eigenvalues is given by(1)(2)where is a hypergeometric function and is a beta function (Edelman et al. 1994, Edelman and Kostlan 1994). has asymptotic behavior(3)Let be the probability that there are exactly real eigenvalues in the complex spectrum of the matrix. Edelman (1997) showed that(4)which is the smallest probability of all s. The entire probability function of the number of expected real eigenvalues in the spectrum of a Gaussian real random matrix was derived by Kanzieper and Akemann (2005) as(5)where(6)(7)In (6), the summation runs over all partitions..

The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk , with hyperbolic metric(1)The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk's boundary (and diameters are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the boundary are a pair of limits rays. The illustration above shows a hyperbolic tessellation similar to M. C. Escher's Circle Limit IV (Heaven and Hell) (Trott 1999, pp. 10 and 83).The endpoints of any arc can be specified by two angles around the disk and . Define(2)(3)Then trigonometry shows that in the above diagram,(4)(5)so the radius of the circle forming the arc is and its center is located at , where(6)The half-angle subtended by the arc is then(7)so(8)The..

Inspired by computer simulations of fossilized worms trails published by Raup and Seilacher (1969), computer scientist Mike Paterson at the University of Warwick and mathematician J. H. Conway created in early 1971 a simple set to rules to study idealized worms traveling along regular grids. Mike Beeler of the MIT Artificial Intelligence Laboratory subsequently published a study Paterson's worms in which he considered paths on a triangular grid (Beeler 1973).The following table summarizes the number of steps required for a number of long-running worms to terminate (Rokicki).patternsteps to terminate10420151042020104202212521211420221infinite14202241450221infinite14502241525115201414221451422454142Paterson's worms are featured in the 2003 Stephen Low IMAX film Volcanoesof the Deep Sea...

Rule 110 is one of the elementary cellular automaton rules introduced by Stephen Wolfram in 1983 (Wolfram 1983, 2002). It specifies the next color in a cell, depending on its color and its immediate neighbors. Its rule outcomes are encoded in the binary representation . This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (OEIS A075437; Wolfram 2002, p. 55).250 iterations of rule 110 are illustrated above.The mirror image is rule 124, the complement is rule 137, and the mirrored complement is rule 193.Starting with a single black cell, successive generations are given by interpreting the numbers 1, 6, 28, 104, 496, 1568, 7360, 27520, ... (OEIS A117999) in binary. Omitting trailing zeros (since the right cells in step of the triangle are always 0) gives the sequence 1, 3, 7, 13, 31, 49, 115, 215, 509, 775, 1805, ... (OEIS A006978), which are simply the previous numbers divided by , and the..

A cellular automaton is a collection of "colored" cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells. The rules are then applied iteratively for as many time steps as desired. von Neumann was one of the first people to consider such a model, and incorporated a cellular model into his "universal constructor." Cellular automata were studied in the early 1950s as a possible model for biological systems (Wolfram 2002, p. 48). Comprehensive studies of cellular automata have been performed by S. Wolfram starting in the 1980s, and Wolfram's fundamental research in the field culminated in the publication of his book A New Kind of Science (Wolfram 2002) in which Wolfram presents a gigantic collection of results concerning automata, among which are a number of groundbreaking new discoveries.The Season 2 episode..

A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequenceThis commutative diagram shows how the first commutative diagram (shown here in blue) can be modified to exhibit the long exact sequence (shown here in red) explicitly. The map is called a connecting homomorphism and describes a curve from the end of the upper row () to the beginning of the lower row (), which suggested the name given to this lemma.The snake lemma is explained in the first scene of Claudia Weill's film Itis My Turn (1980), starring Jill Clayburgh and Michael Douglas.

Two trains are on the same track a distance 100 km apart heading towards one another, each at a speed of 50 km/h. A fly starting out at the front of one train, flies towards the other at a speed of 75 km/h. Upon reaching the other train, the fly turns around and continues towards the first train. How many kilometers does the fly travel before getting squashed in the collision of the two trains?Now, the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time. The position of the fly at time is plotted above.However, a brute force method instead solves for the position of the fly along each traversal between the trains. For example, the fly reaches the second train when(1)or h, at which point..

A fast boat is overtaking a slower one when fog suddenly sets in. At this point, the boat being pursued changes course, but not speed, and proceeds straight in a new direction which is not known to the fast boat. How should the pursuing vessel proceed in order to be sure of catching the other boat?The amazing answer is that the pursuing boat should continue to the point where the slow boat would be if it had set its course directly for the pursuing boat when the fog set in. If the boat is not there, it should proceed in a spiral whose origin is the point where the slow boat was when the fog set in. The spiral must be constructed in such a way that, while circling the origin, the fast boat's distance from it increases at the same rate as the boat being pursued. The two courses must therefore intersect before the fast boat has completed one circuit. In order to make the problem reasonably practical, the fast boat should be capable of maintaining a speed four or five times..

If moves along a known curve, then describes a pursuit curve if is always directed toward and and move with uniform velocities. Pursuit curves were considered in general by the French scientist Pierre Bouguer in 1732, and subsequently by the English mathematician Boole.Under the name "path minimization," pursuit curves are alluded to by math genius Charlie Eppes in the Season 2 episode "Dark Matter" of the television crime drama NUMB3RS when considering the actions of the mysterious third shooter.The equations of pursuit are given by(1)which specifies that the tangent vector at point is always parallel to the line connecting and , combined with(2)which specifies that the point moves with constant speed (without loss of generality, taken as unity above). Plugging (2) into (1) therefore gives(3)The case restricting to a straight line was studied by Arthur Bernhart (MacTutor Archive). Taking the parametric equation..

A complete bipartite graph, sometimes also called a complete bicolored graph (Erdős et al. 1965) or complete bigraph, is a bipartite graph (i.e., a set of graph vertices decomposed into two disjoint sets such that no two graph vertices within the same set are adjacent) such that every pair of graph vertices in the two sets are adjacent. If there are and graph vertices in the two sets, the complete bipartite graph is denoted . The above figures show and . is also known as the utility graph (and is the circulant graph ), and is the unique 4-cage graph. is a Cayley graph. A complete bipartite graph is a circulant graph (Skiena 1990, p. 99), specifically , where is the floor function.Special cases of are summarized in the table below.path graph path graph claw graphstar graph square graph utility graphThe numbers of (directed) Hamiltonian cycles for the graph with , 2, ... are 0, 2, 12, 144, 2880, 86400, 3628800, 203212800, ... (OEIS A143248), where..

Dijkstra's algorithm is an algorithm for finding a graph geodesic, i.e., the shortest path between two graph vertices in a graph. It functions by constructing a shortest-path tree from the initial vertex to every other vertex in the graph. The algorithm is implemented in the Wolfram Language as FineShortestPath[g, Method -> "Dijkstra"].The worst-case running time for the Dijkstra algorithm on a graph with nodes and edges is because it allows for directed cycles. It even finds the shortest paths from a source node to all other nodes in the graph. This is basically for node selection and for distance updates. While is the best possible complexity for dense graphs, the complexity can be improved significantly for sparse graphs.With slight modifications, Dijkstra's algorithm can be used as a reverse algorithm that maintains minimum spanning trees for the sink node. With further modifications, it can be extended to become bidirectional.The..

Taking a connected graph or network with a high graph diameter and adding a very small number of edges randomly, the diameter tends to drop drastically. This is known as the small world phenomenon. It is sometimes also known as "six degrees of separation" since, in the social network of the world, any person turns out to be linked to any other person by roughly six connections.Short-term memory uses small world networks between neurons to remember this sentence.In modern mathematics, the center of the network of coauthorship is considered to be P. Erdős, resulting in the so-called Erdős number. In movies, Kevin Bacon is often mentioned as the center of the movie universe, but a recent study (Reynolds) has shown Christopher Lee to be the actual center. Both actors have co-starred with Julius LeFlore, so the Lee-Bacon distance is two...

Min Max Re Im A special function which is given by the logarithmic derivative of the gamma function (or, depending on the definition, the logarithmic derivative of the factorial).Because of this ambiguity, two different notations are sometimes (but not always) used, with(1)defined as the logarithmic derivative of the gamma function , and(2)defined as the logarithmic derivative of the factorial function. The two are connected by the relationship(3)The th derivative of is called the polygamma function, denoted . The notation(4)is therefore frequently used for the digamma function itself, and Erdélyi et al. (1981) use the notation for . The digamma function is returned by the function PolyGamma[z] or PolyGamma[0, z] in the Wolfram Language, and typeset using the notation .The digamma function arises in simple sums such as(5)(6)where is a Lerch transcendent.Special cases are given by(7)(8)(9)(10)Gauss's digamma theorem states..

The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323). The strip bearing his name was invented by Möbius in 1858, although it was independently discovered by Listing, who published it, while Möbius did not (Derbyshire 2004, p. 381). Like the cylinder, it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110).The Möbius strip has Euler characteristic (Dodson and Parker 1997, p. 125).According to Madachy (1979), the B. F. Goodrich Company patented a conveyor belt in the form of a Möbius strip which lasts twice as long as conventional belts. M. C. Escher was fond of portraying Möbius strips, and..

A curve on the unit sphere is an eversion if it has no corners or cusps (but it may be self-intersecting). These properties are guaranteed by requiring that the curve's velocity never vanishes. A mapping forms an immersion of the circle into the sphere iff, for all ,Smale (1958) showed it is possible to turn a sphere insideout (sphere eversion) using eversion.The Season 1 episode "Sniper Zero" (2005) of the television crime drama NUMB3RS mentions eversion.

Smale (1958) proved that it is mathematically possible to turn a sphere inside-out without introducing a sharp crease at any point. This means there is a regular homotopy from the standard embedding of the 2-sphere in Euclidean three-space to the mirror-reflection embedding such that at every stage in the homotopy, the sphere is being immersed in Euclidean space. This result is so counterintuitive and the proof so technical that the result remained controversial for a number of years.In 1961, Arnold Shapiro devised an explicit eversion but did not publicize it. Phillips (1966) heard of the result and, in trying to reproduce it, actually devised an independent method of his own. Yet another eversion was devised by Morin, which became the basis for the movie by Max (1977). Morin's eversion also produced explicit algebraic equations describing the process. The original method of Shapiro was subsequently published by Francis and Morin (1979).The..

Sudoku (literally, "single number"), sometimes also is a pencil-and-paper logic puzzle whose goal is to complete a grid satisfying various constraints. In the "classic" Sudoku, a square is divided into "regions", with various squares filled with "givens." Valid solutions use each of the numbers 1-9 exactly once within each row, column and region. This kind of sudoku is therefore a particular case of a Latin square.Under the U.S.-only trademarked name "Number Place," Sudoku was first published anonymously by Garns (1979) for Dell Pencil Puzzles. In 1984, the puzzle was used by Nikoli with the Japan-only trademarked name Sudoku (Su = number, Doku = single). Due to the trademark issues, in Japan, the puzzle became well-known as nanpure, or Number Place, often using the English name. Outside Japan, the Japanese name predominates.The puzzle received a large amount of attention in the..

666 is the occult "number of the beast," also called the "sign of the devil" (Wang 1994), associated in the Bible with the Antichrist. It has figured in many numerological studies. It is mentioned in Revelation 13:18: "Here is wisdom. Let him that hath understanding count the number of the beast: for it is the number of a man; and his number is 666." The origin of this number is not entirely clear, although it may be as simple as the number containing the concatenation of one symbol of each type (excluding ) in Roman numerals: (Wells 1986).The first few numbers containing the beast number in their digits are 666, 1666,2666, 3666, 4666, 5666, 6660, ... (OEIS A051003)."666" is the combination of the mysterious suitcase retrieved by Vincent Vega (John Travolta) and Jules Winnfield (Samuel L. Jackson) in Quentin Tarantino's 1994 film Pulp Fiction. Various conspiracy theories, including the novel..

The necker cube is an illusion in which a two-dimensional drawing of an array of cubes appears to simultaneously protrude from and intrude into the page.A Necker cube appears on the banner shown in Escher's lithographs "Metamorphosis I" (Bool et al. 1982, p. 271; Forty 2003, p. 39), "Cycle" (Bool et al. 1982, p. 274), and "Convex and Concave". It is also the basis for the arcade game Q*bert.Depending on the view point chosen for projection, the cubes may be composed of one,two, or three types of rhombi.The Necker cube is also a tiling that was used in ancient times, including as a mosaic on the floor of one of the houses in Pompeii, as illustrated in the photograph above (courtesy of S. Jaskulowski).The image above shows a Necker cube pattern emblazoned on a quilt created by Janice Ewing using a pattern by Karen Combs. ..

A sequence whose terms are integers. The most complete printed references for such sequences are Sloane (1973) and its update, Sloane and Plouffe (1995). Neil Sloane maintains the sequences from both these works in a vastly expanded on-line encyclopedia known as the On-Line Encyclopedia of Integer Sequences (https://www.research.att.com/~njas/sequences/). In this listing, sequences are identified by a unique 6-digit A-number. Sequences appearing in Sloane and Plouffe (1995) are ordered lexicographically and identified with a 4-digit M-number, and those appearing in Sloane (1973) are identified with a 4-digit N-number. To look up sequences by e-mail, send a message to either mailto:[email protected] or mailto:[email protected] containing lines of the form lookup 5 14 42 132 ... (note that spaces must be used instead of commas).Integer sequences can be analyzed by a variety of techniques (Sloane and Plouffe..

If is an arbitrary integer relatively prime to and is a primitive root of , then there exists among the numbers 0, 1, 2, ..., , where is the totient function, exactly one number such thatThe number is then called the discrete logarithm of with respect to the base modulo and is denotedThe term "discrete logarithm" is most commonly used in cryptography, although the term "generalized multiplicative order" is sometimes used as well (Schneier 1996, p. 501). In number theory, the term "index" is generally used instead (Gauss 1801; Nagell 1951, p. 112).For example, the number 7 is a positive primitive root of (in fact, the set of primitive roots of 41 is given by 6, 7, 11, 12, 13, 15, 17, 19, 22, 24, 26, 28, 29, 30, 34, 35), and since , the number 15 has multiplicative order 3 with respect to base 7 (modulo 41) (Nagell 1951, p. 112). The generalized multiplicative order is implemented in the Wolfram Language..

A right triangle is triangle with an angle of ( radians). The sides , , and of such a triangle satisfy the Pythagorean theorem(1)where the largest side is conventionally denoted and is called the hypotenuse. The other two sides of lengths and are called legs, or sometimes catheti.The favorite A-level math exam question of the protagonist Christopher in the novel The Curious Incident of the Dog in the Night-Time asks for proof that a triangle with sides of the form , , and where is a right triangle, and that the converse does not hold (Haddon 2003, pp. 214 and 223-226).The side lengths of a right triangle form a so-called Pythagorean triple. A triangle that is not a right triangle is sometimes called an oblique triangle. Special cases of the right triangle include the isosceles right triangle (middle figure) and 30-60-90 triangle (right figure).For any three similar shapes of area on the sides of a right triangle,(2)which is equivalent to the..

The perpendicular bisector of a linesegment is the locus of all points that are equidistant from its endpoints.This theorem can be applied to determine the center of a given circle with straightedge and compass. Pick three points , and on the circle. Since the center is equidistant from all of them, it lies on the bisector of segment and also on the bisector of segment , i.e., it is the intersection point of the two bisectors. This construction is shown on a window pane by tutor Justin McLeod (Mel Gibson) to his pupil Chuck Norstadt (Nick Stahl) in the 1993 film The Man Without a Face.

In 1803, Malfatti posed the problem of determining the three circular columns of marble of possibly different sizes which, when carved out of a right triangular prism, would have the largest possible total cross section. This is equivalent to finding the maximum total area of three circles which can be packed inside a right triangle of any shape without overlapping. This problem is now known as the marble problem (Martin 1998, p. 92). Malfatti gave the solution as three circles (the Malfatti circles) tangent to each other and to two sides of the triangle. In 1930, it was shown that the Malfatti circles were not always the best solution. Then Goldberg (1967) showed that, even worse, they are never the best solution (Ogilvy 1990, pp. 145-147). Ogilvy (1990, pp. 146-147) and Wells (1991) illustrate specific cases where alternative solutions are clearly optimal.The general Malfatti problem on an arbitrary triangle was actually..

The wallpaper groups are the 17 possible plane symmetry groups. They are commonly represented using Hermann-Mauguin-like symbols or in orbifold notation (Zwillinger 1995, p. 260).orbifold notationHermann-Mauguin symbolop12222p2**pmxxpg*2222pmm22*pmg22xpggx*cm2*22cmm442p4*442p4m4*2p4g333p3*333p3ml3*3p3lm632p6*632p6mPatterns created with Artlandia SymmetryWorks for each of these groups are illustrated above.Beautiful patterns can be created by repeating geometric and artistic motifs according to the symmetry of the wallpaper groups, as exemplified in works by M. C. Escher and in the patterns created by I. Bakshee in the Wolfram Language using Artlandia, illustrated above.For a description of the symmetry elements present in each space group, see Coxeter (1969, p. 413)...

A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting of the entire original group. Simple groups include the infinite families of alternating groups of degree , cyclic groups of prime order, Lie-type groups, and the 26 sporadic groups.Since all subgroups of an Abelian group are normal and all cyclic groups are Abelian, the only simple cyclic groups are those which have no subgroups other than the trivial subgroup and the improper subgroup consisting of the entire original group. And since cyclic groups of composite order can be written as a group direct product of factor groups, this means that only prime cyclic groups lack nontrivial subgroups. Therefore, the only simple cyclic groups are the prime cyclic groups. Furthermore, these are the only Abelian simple groups.In fact, the classification theorem of finite groups states that such groups can be classified completely..

The group is the unique group of group order 2. is both Abelian and cyclic. Examples include the point groups , , and , the integers modulo 2 under addition (), and the modulo multiplication groups , , and (which are the only modulo multiplication groups isomorphic to ).The group is also trivially simple, and forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University mathematics department a capella group "The Klein Four."The cycle graph is shown above, and the cycleindex isThe elements satisfy , where 1 is the identity element.Its multiplication table is illustrated aboveand enumerated below. 1111The conjugacy classes are and . The only subgroups of are the trivial group and entire group , both of which are trivially normal.The irreducible representation for the group is ...

A finite group is a group having finite group order. Examples of finite groups are the modulo multiplication groups, point groups, cyclic groups, dihedral groups, symmetric groups, alternating groups, and so on.Properties of finite groups are implemented in the Wolfram Language as FiniteGroupData[group, prop].The classification theorem of finite groups states that the finite simple groups can be classified completely into one of five types.A convenient way to visualize groups is using so-called cycle graphs, which show the cycle structure of a given abstract group. For example, cycle graphs of the 5 nonisomorphic groups of order 8 are illustrated above (Shanks 1993, p. 85).Frucht's theorem states that every finite group is the graph automorphism group of a finite undirected graph.The finite (cyclic) group forms the subject for the humorous a capella song "Finite Simple Group (of Order 2)" by the Northwestern University..

A problem in game theory first discussed by A. Tucker. Suppose each of two prisoners and , who are not allowed to communicate with each other, is offered to be set free if he implicates the other. If neither implicates the other, both will receive the usual sentence. However, if the prisoners implicate each other, then both are presumed guilty and granted harsh sentences.A dilemma arises in deciding the best course of action in the absence of knowledge of the other prisoner's decision. Each prisoner's best strategy would appear to be to turn the other in (since if makes the worst-case assumption that will turn him in, then will walk free and will be stuck in jail if he remains silent). However, if the prisoners turn each other in, they obtain the worst possible outcome for both.Mosteller (1987) describes a different problem he terms "the prisoner's dilemma." In this problem, three prisoners , , and with apparently equally good records..

There are several closely related results that are variously known as the binomial theorem depending on the source. Even more confusingly a number of these (and other) related results are variously known as the binomial formula, binomial expansion, and binomial identity, and the identity itself is sometimes simply called the "binomial series" rather than "binomial theorem."The most general case of the binomial theorem is the binomialseries identity(1)where is a binomial coefficient and is a real number. This series converges for an integer, or . This general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with the binomial theorem.When is a positive integer , the series terminates at and can be written in the form(2)This form of the identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).The differing terminologies are..

Although Descartes originally used the term "imaginary number" to refer to what is today known as a complex number, in standard usage today, "imaginary number" means a complex number that has zero real part (i.e., such that ). For clarity, such numbers are perhaps best referred to as purely imaginary numbers.A (purely) imaginary number can be written as a real number multiplied by the "imaginary unit" i (equal to the square root ), i.e., in the form .In the novel The Da Vinci Code, the character Robert Langdon jokes that character Sophie Neveu "believes in the imaginary number because it helps her break code" (Brown 2003, p. 351). In Isaac Asimov's short story "The Imaginary" (1942), eccentric psychologist Tan Porus explains the behavior of a mysterious species of squid by using imaginary numbers in the equations which describe its psychology. The anthology Imaginary Numbers:..

The Leibniz integral rule gives a formula for differentiation of a definiteintegral whose limits are functions of the differential variable,(1)It is sometimes known as differentiation under the integral sign.This rule can be used to evaluate certain unusual definite integrals such as(2)(3)for (Woods 1926).Feynman (1997, pp. 69-72) recalled seeing the method in Woods (1926) and remarked "So because I was self-taught using that book, I had peculiar methods for doing integrals," and "I used that one damn tool again and again."

Gaussian primes are Gaussian integers satisfying one of the following properties. 1. If both and are nonzero then, is a Gaussian prime iff is an ordinary prime. 2. If , then is a Gaussian prime iff is an ordinary prime and . 3. If , then is a Gaussian prime iff is an ordinary prime and . The above plot of the complex plane shows the Gaussianprimes as filled squares.The primes which are also Gaussian primes are 3, 7, 11, 19, 23, 31, 43, ... (OEIS A002145). The Gaussian primes with are given by , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 3, , , , , , , , , .The numbers of Gaussian primes with complex modulus (where the definition has been used) for , 1, ... are 0, 100, 4928, 313752, ... (OEIS A091134).The cover of Bressoud and Wagon (2000) shows an illustration of the distributionof Gaussian primes in the complex plane.As of 2009, the largest known Gaussian prime, found in Sep. 2006, is , whose real and imaginary parts both have decimal digits and whose squared..

There are many mathematical and recreational problems related to folding. Origami,the Japanese art of paper folding, is one well-known example.It is possible to make a surprising variety of shapes by folding a piece of paper multiple times, making one complete straight cut, then unfolding. For example, a five-pointed star can be produced after four folds (Demaine and Demaine 2004, p. 23), as can a polygonal swan, butterfly, and angelfish (Demaine and Demaine 2004, p. 29). Amazingly, every polygonal shape can be produced this way, as can any disconnected combination of polygonal shapes (Demaine and Demaine 2004, p. 25). Furthermore, algorithms for determining the patterns of folds for a given shape have been devised by Bern et al. (2001) and Demaine et al. (1998, 1999).Wells (1986, p. 37; Wells 1991) and Gurkewitz and Arnstein (2003, pp. 49-59) illustrate the construction of the equilateral triangle, regular..

A theorem is a statement that can be demonstrated to be true by accepted mathematical operations and arguments. In general, a theorem is an embodiment of some general principle that makes it part of a larger theory. The process of showing a theorem to be correct is called a proof.Although not absolutely standard, the Greeks distinguished between "problems" (roughly, the construction of various figures) and "theorems" (establishing the properties of said figures; Heath 1956, pp. 252, 262, and 264).According to the Nobel Prize-winning physicist Richard Feynman (1985), any theorem, no matter how difficult to prove in the first place, is viewed as "trivial" by mathematicians once it has been proven. Therefore, there are exactly two types of mathematical objects: trivial ones, and those which have not yet been proven.The late mathematician P. Erdős has often been associated with the observation..

A strange loop is a phenomenon in which, whenever movement is made upwards or downwards through the levels of some hierarchical system, the system unexpectedly arrives back where it started. Hofstadter (1989) uses the strange loop as a paradigm in which to interpret paradoxes in logic (such as Grelling's paradox, the liar's paradox, and Russell's antinomy) and calls a system in which a strange loop appears a tangled hierarchy.Canon 5 from Bach's Musical Offering (sometimes known as Bach's endlessly rising canon) is a musical piece that continues to rise in key, modulating through the entire chromatic scale until it ends in the same key in which it began. This is the first example cited by Hofstadter (1989) as a strange loop.Other examples include the endlessly rising stairs in M. C. Escher 1960 lithograph Ascending and Descending, the endlessly falling waterfall in his 1961 lithograph Waterfall, and the pair of hands drawing each..

The "Foxtrot series" is a mathematical sum that appeared in the June 2, 1996 comic strip FoxTrot by Bill Amend (Amend 1998, p. 19; Mitchell 2006/2007). It arose from a convergence testing problem in a calculus book by Anton, but was inadvertently converted into a summation problem on an alleged final exam by the strip's author:(1)The sum can be done using partial fraction decomposition to obtain(2)(3)(4)(5)(OEIS A127198), where and the last sums have been done in terms of the digamma function and symbolically simplified.

The Penrose triangle, also called the tribar (Cerf), tri-bar (Ernst 1987), impossible tribar (Pappas 1989, p. 13), or impossible triangle, is an impossible figure published by Penrose and Penrose (1958). Penrose triangles appear prominently in the works of Escher, who not only inspired creation of this object (Escher 1954, Penrose and Penrose 1958), but also subsequently publicized it.The Penrose triangle can be extended to -gonal barred objects (Cerf, Elber), including the so-called tribox.The figure was drawn earlier by artist Oscar Reutersvärd in 1934 during a "long lecture." For this, he was honored with a stamp by the government of Sweden in 1982 (Miller).The Penrose triangle appears on the cover of Raghavachary (2004).Henderson (2006) offers an impossible triangle net...

An impossible figure in which a stairway in the shape of a square appears to circulate indefinitely while still possessing normal steps (Penrose and Penrose 1958). The Dutch artist M. C. Escher included a Penrose stairway in his mind-bending illustration "Ascending and Descending" (Bool et al. 1982, p. 321; Forty 2003, Plate 68). Distorted variations of the stairway are also depicted in Escher's "House of Stairs" (Bool et al. 1982, p. 301; Forty 2003, Plate 40).In the 1998 film The Avengers, Uma Thurman is shown walking down a Penrose stairway and ending up back where she began.

The café wall illusion, sometimes also called the Münsterberg illusion (Ashton Raggatt McDougall 2006), is an optical illusion produced by a black and white rectangular tessellation when the tiles are shifted in a zigzag pattern, as illustrated above. While the pattern seems to diverge towards the upper and lower right corners in the upper figure, the gray lines are actually parallel. Interestingly, the illusion greatly diminishes if black lines are used instead of gray.Gregory and Heard (1979) first noticed the illusion on the wall decoration of a café in Bristol, England. The café wall illusion is only one among many visual distortion effects involving parallel lines. The most famous example of this kind is Zöllner's illusion.The image above shows a picture of a building in Melbourne, Australia designed to exhibit this illusion (C. L. Taylor, pers. comm., Aug. 5, 2006). The building,..

Cluster analysis is a technique used for classification of data in which data elements are partitioned into groups called clusters that represent collections of data elements that are proximate based on a distance or dissimilarity function.Cluster analysis is implemented as FindClusters[data] or FindClusters[data, n].The Season 1 pilot (2005) and Season 2 episode "Dark Matter" of the television crime drama NUMB3RS feature clusters and cluster analysis. In "Dark Matter," math genius Charlie Eppes runs a cluster analysis to find connections between the students that seemed to be systematically singled out by the anomalous third shooter. In Season 4 episode"Black Swan," characters Charles Eppes and Amita Ramanujan adjust cluster radii in their attempt to do a time series analysis of overlapping Voronoi regions to track the movements of a suspect. ..

The golden ratio, also known as the divine proportion, golden mean, or golden section, is a number often encountered when taking the ratios of distances in simple geometric figures such as the pentagon, pentagram, decagon and dodecahedron. It is denoted , or sometimes .The designations "phi" (for the golden ratio conjugate ) and "Phi" (for the larger quantity ) are sometimes also used (Knott), although this usage is not necessarily recommended.The term "golden section" (in German, goldener Schnitt or der goldene Schnitt) seems to first have been used by Martin Ohm in the 1835 2nd edition of his textbook Die Reine Elementar-Mathematik (Livio 2002, p. 6). The first known use of this term in English is in James Sulley's 1875 article on aesthetics in the 9th edition of the Encyclopedia Britannica. The symbol ("phi") was apparently first used by Mark Barr at the beginning of the 20th century in commemoration..

A prime number (or prime integer, often simply called a "prime" for short) is a positive integer that has no positive integer divisors other than 1 and itself. More concisely, a prime number is a positive integer having exactly one positive divisor other than 1, meaning it is a number that cannot be factored. For example, the only divisors of 13 are 1 and 13, making 13 a prime number, while the number 24 has divisors 1, 2, 3, 4, 6, 8, 12, and 24 (corresponding to the factorization ), making 24 not a prime number. Positive integers other than 1 which are not prime are called composite numbers.While the term "prime number" commonly refers to prime positive integers, other types of primes are also defined, such as the Gaussian primes.The number 1 is a special case which is considered neither prime nor composite (Wells 1986, p. 31). Although the number 1 used to be considered a prime (Goldbach 1742; Lehmer 1909, 1914; Hardy and Wright..

The derivative of a function represents an infinitesimalchange in the function with respect to one of its variables.The "simple" derivative of a function with respect to a variable is denoted either or(1)often written in-line as . When derivatives are taken with respect to time, they are often denoted using Newton's overdot notation for fluxions,(2)The "d-ism" of Leibniz's eventually won the notation battle against the "dotage" of Newton's fluxion notation (P. Ion, pers. comm., Aug. 18, 2006).When a derivative is taken times, the notation or(3)is used, with(4)etc., the corresponding fluxion notation.When a function depends on more than one variable, a partial derivative(5)can be used to specify the derivative with respect to one or more variables.The derivative of a function with respect to the variable is defined as(6)but may also be calculated more symmetrically as(7)provided the..

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

A definite integral is an integral(1)with upper and lower limits. If is restricted to lie on the real line, the definite integral is known as a Riemann integral (which is the usual definition encountered in elementary textbooks). However, a general definite integral is taken in the complex plane, resulting in the contour integral(2)with , , and in general being complex numbers and the path of integration from to known as a contour.The first fundamental theorem of calculus allows definite integrals to be computed in terms of indefinite integrals, since if is the indefinite integral for a continuous function , then(3)This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral. Definite integrals may be evaluated in the Wolfram Language using Integrate[f, x, a, b].The question of which definite..

has decimal expansion given by(1)(OEIS A000796). The following table summarizes some record computations of the digits of .1999Kanada, Ushio and KurodaDec. 2002Kanada, Ushio and Kuroda (Peterson 2002, Kanada 2003)Aug. 2012A. J. Yee (Yee)Aug. 2012S. Kondo and A. J. Yee (Yee)Dec. 2013A. J. Yee and S. Kondo (Yee)The calculation of the digits of has occupied mathematicians since the day of the Rhind papyrus (1500 BC). Ludolph van Ceulen spent much of his life calculating to 35 places. Although he did not live to publish his result, it was inscribed on his gravestone. Wells (1986, p. 48) discusses a number of other calculations. The calculation of also figures in the Season 2 Star Trek episode "Wolf in the Fold" (1967), in which Captain Kirk and Mr. Spock force an evil entity (composed of pure energy and which feeds on fear) out of the starship..

The constant pi, denoted , is a real number defined as the ratio of a circle's circumference to its diameter ,(1)(2) has decimal expansion given by(3)(OEIS A000796). Pi's digits have many interesting properties, although not very much is known about their analytic properties. However, spigot (Rabinowitz and Wagon 1995; Arndt and Haenel 2001; Borwein and Bailey 2003, pp. 140-141) and digit-extraction algorithms (the BBP formula) are known for .A brief history of notation for pi is given by Castellanos (1988ab). is sometimes known as Archimedes' constant or Ludolph's constant after Ludolph van Ceulen (1539-1610), a Dutch calculator. The symbol was first used by Welsh mathematician William Jones in 1706, and subsequently adopted by Euler. In Measurement of a Circle, Archimedes (ca. 225 BC) obtained the first rigorous approximation by inscribing and circumscribing -gons on a circle using the Archimedes algorithm. Using (a 96-gon),..

Given three jugs with pints in the first, in the second, and in the third, obtain a desired amount in one of the vessels by completely filling up and/or emptying vessels into others. This problem can be solved with the aid of trilinear coordinates (Tweedie 1939).A variant of this problem asks to obtain a fixed quantity of liquid using only two initially empty buckets of capacities and and a well containing an inexhaustible supply of water.This two bucket variant is used in the film Die Hard: With a Vengeance (1995). The characters John McClane and Zeus Carver (played by Bruce Willis and Samuel L. Jackson) solve the two bucket variant with two jugs and water from a public fountain in order to try to prevent a bomb from exploding by obtaining 4 gallons of water using only 5-gallon and 3-gallon jugs.General problems of this type are sometimes collectively known as "decanting problems."..

Find the figure bounded by a line which has the maximum area for a given perimeter. The solution is a semicircle. The problem is based on a passage from Virgil's Aeneid:"The Kingdom you see is Carthage, the Tyrians, the town of Agenor;But the country around is Libya, no folk to meet in war.Dido, who left the city of Tyre to escape her brother,Rules here--a long and labyrinthine tale of wrongIs hers, but I will touch on its salient points in order....Dido, in great disquiet, organised her friends for escape.They met together, all those who harshly hated the tyrantOr keenly feared him: they seized some ships which chanced to be ready...They came to this spot, where to-day you can behold the mightyBattlements and the rising citadel of New Carthage,And purchased a site, which was named 'Bull's Hide' after the bargainBy which they should get as much land as they could enclose with a bull's hide."..

Game theory is a branch of mathematics that deals with the analysis of games (i.e., situations involving parties with conflicting interests). In addition to the mathematical elegance and complete "solution" which is possible for simple games, the principles of game theory also find applications to complicated games such as cards, checkers, and chess, as well as real-world problems as diverse as economics, property division, politics, and warfare.Game theory has two distinct branches: combinatorialgame theory and classical game theory.Combinatorial game theory covers two-player games of perfect knowledge such as go, chess, or checkers. Notably, combinatorial games have no chance element, and players take turns.In classical game theory, players move, bet, or strategize simultaneously. Both hidden information and chance elements are frequent features in this branch of game theory, which is also a branch of economics.The..

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

A quadratic equation is a second-order polynomial equation in a single variable (1)with . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex.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 quadratic equations 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 the hypotenuse."The..

For , the Riemann zeta function is given by(1)(2)where is the th prime. This is Euler's product (Whittaker and Watson 1990), called by Havil (2003, p. 61) the "all-important formula" and by Derbyshire (2004, pp. 104-106) the "golden key."This can be proved by expanding the product, writing each term as a geometricseries, expanding, multiplying, and rearranging terms,(3)Here, the rearrangement leading to equation (1) follows from the fundamental theorem of arithmetic, since each product of prime powers appears in exactly one denominator and each positive integer equals exactly one product of prime powers.This product is related to the Möbius function via(4)which can be seen by expanding the product to obtain(5)(6)(7)(8)(9), but the finite product exists, giving(10)For upper limits , 1, 2, ..., the products are given by 1, 2, 3, 15/4, 35/8, 77/16, 1001/192, 17017/3072, ... (OEIS A060753 and..

The line in the complex plane on which the Riemann hypothesis asserts that all nontrivial (complex) Riemann zeta function zeros lie. The plot above shows the first few zeros of the Riemann zeta function, with the critical line shown in red. The zeros with and that do not line on the critical line are the trivial zeros of at , , .... Although it is known that an infinite number of zeros lie on the critical line and that these comprise at least 40% of all zeros, the Riemann hypothesis is still unproven.An attractive poster plotting the Riemann zeta function zeros on the critical line together with annotations for relevant historical information, illustrated above, was created by Wolfram Research (1995).In the Season 1 episode "Prime Suspect" (2005) of the television crime drama NUMB3RS, math genius Charlie Eppes discusses the critical line after realizing that character Ethan's daughter has been kidnapped because he is close to solving..

The Hurwitz zeta function is a generalization of the Riemann zeta function that is also known as the generalized zeta function. It is classically defined by the formula(1)for and by analytic continuation to other , where any term with is excluded. It is implemented in this form in the Wolfram Language as HurwitzZeta[s, a].The slightly different form(2)is implemented in the Wolfram Language as Zeta[s, a]. Note that the two are identical only for .The plot above shows for real and , with the zero contour indicated in black.For , a globally convergent series for (which, for fixed , gives an analytic continuation of to the entire complex -plane except the point ) is given by(3)(Hasse 1930).The Hurwitz zeta function is implemented in the Wolfram Language as Zeta[s, a].For , reduces to the Riemann zeta function ,(4)If the singular term is excluded from the sum definition of , then as well.The Hurwitz zeta function is given by the integral(5)for and .The..

The Einstein field equations are the 16 coupled hyperbolic-elliptic nonlinear partial differential equations that describe the gravitational effects produced by a given mass in general relativity. As result of the symmetry of and , the actual number of equations reduces to 10, although there are an additional four differential identities (the Bianchi identities) satisfied by , one for each coordinate.The Einstein field equations state thatwhere is the stress-energy tensor, andis the Einstein tensor, with the Ricci curvature tensor and the scalar curvature.The opening sequence of the 2003 French film Les Triplettes de Belleville (The Triplets of Belleville) features the Einstein field equations.

The Haferman carpet is the beautiful fractal constructed using string rewriting beginning with a cell [1] and iterating the rules(1)(Allouche and Shallit 2003, p. 407).Taking five iterations gives the beautiful pattern illustrated above.This fractal also appears on the cover of Allouche and Shallit (2003).Let be the number of black boxes, the length of a side of a white box, and the fractional area of black boxes after the th iteration. Then(2)(3)The numbers of black cells after , 1, 2, ... iterations are therefore 1, 4, 61, 424, 4441, 36844, ... (OEIS A118005). The capacity dimension is therefore(4)(5)

Consider an arbitrary one-dimensional map(1)(with implicit parameter ) at the onset of chaos. After a suitable rescaling, the Feigenbaum function(2)is obtained. This function satisfies(3)with .Proofs for the existence of an even analytic solution to this equation, sometimes called the Feigenbaum-Cvitanović functional equation, have been given by Campanino and Epstein (1981), Campanino et al. (1982), and Lanford (1982, 1984).The picture above illustrate the Feigenbaum function for the logistic map with ,(4)along the real axis (M. Trott, pers. comm., Sept. 9, 2003).The images above show two views of a sculpture presented by Stephen Wolfram to Mitchell Feigenbaum on the occasion of his 60th birthday that depicts the Feigenbaum function in the complex plane. The sculpture (photos courtesy of A. Young) was designed by M. Trott and laser-etched into a block of glass by Bathsheba Grossman (https://www.bathsheba.com/)...

Actuarial science is the study of risk through the use of mathematics, probability, and statistics. A person who performs risk assessment is known as an actuary. Actuaries typically are employed in financial, insurance, pensions, and other related sectors.Actuarial science is similar to medicine in that a lot of time must be taken for schooling and taking examinations, but salaries are typically rather high.The Season 1 episode "Sacrifice" (2005) of the television crime drama NUMB3RS mentions actuarial science.

Reversion to the mean, also called regression to the mean, is the statistical phenomenon stating that the greater the deviation of a random variate from its mean, the greater the probability that the next measured variate will deviate less far. In other words, an extreme event is likely to be followed by a less extreme event.Although this phenomenon appears to violate the definition of independent events, it simply reflects the fact that the probability density function of any random variable , by definition, is nonnegative over every interval and integrates to one over the interval . Thus, as you move away from the mean, the proportion of the distribution that lies closer to the mean than you do increases continuously. Formally,for .The Season 1 episode "Sniper Zero" (2005) of the television crime drama NUMB3RS mentions regression to the mean. ..

A phenomenological law also called the first digit law, first digit phenomenon, or leading digit phenomenon. Benford's law states that in listings, tables of statistics, etc., the digit 1 tends to occur with probability , much greater than the expected 11.1% (i.e., one digit out of 9). Benford's law can be observed, for instance, by examining tables of logarithms and noting that the first pages are much more worn and smudged than later pages (Newcomb 1881). While Benford's law unquestionably applies to many situations in the real world, a satisfactory explanation has been given only recently through the work of Hill (1998).Benford's law was used by the character Charlie Eppes as an analogy to help solve a series of high burglaries in the Season 2 "The Running Man" episode (2006) of the television crime drama NUMB3RS.Benford's law applies to data that are not dimensionless, so the numerical values of the data depend on the units. If there..

Bayesian analysis is a statistical procedure which endeavors to estimate parameters of an underlying distribution based on the observed distribution. Begin with a "prior distribution" which may be based on anything, including an assessment of the relative likelihoods of parameters or the results of non-Bayesian observations. In practice, it is common to assume a uniform distribution over the appropriate range of values for the prior distribution.Given the prior distribution, collect data to obtain the observed distribution. Then calculate the likelihood of the observed distribution as a function of parameter values, multiply this likelihood function by the prior distribution, and normalize to obtain a unit probability over all possible values. This is called the posterior distribution. The mode of the distribution is then the parameter estimate, and "probability intervals" (the Bayesian analog of confidence..

The sequence of numbers giving the number of digits in the three-fold power tower . The values of for , 2, ... are 1, 16, 7625597484987, ... (OEIS A002488; Rossier 1948), so the Joyce sequence is 1, 2, 13, 155, 2185, 36306, ... (OEIS A054382). Laisant (1906) found the term , and Uhler (1947) published the logarithm of this number to 250 decimal places (Wells 1986, p. 208).The sequence is named in honor of the following excerpt from the "Ithaca" chapter of James Joyce's Ulysses: "Because some years previously in 1886 when occupied with the problem of the quadrature of the circle he had learned of the existence of a number computed to a relative degree of accuracy to be of such magnitude and of so many places, e.g., the 9th power of the 9th power of 9, that, the result having been obtained, 33 closely printed volumes of 1000 pages each of innumerable quires and reams of India paper would have to be requisitioned in order to contain the complete..

The term limit comes about relative to a number of topics from several different branches of mathematics.A sequence of elements in a topological space is said to have limit provided that for each neighborhood of , there exists a natural number so that for all . This very general definition can be specialized in the event that is a metric space, whence one says that a sequence in has limit if for all , there exists a natural number so that(1)for all . In many commonly-encountered scenarios, limits are unique, whereby one says that is the limit of and writes(2)On the other hand, a sequence of elements from an metric space may have several - even infinitely many - different limits provided that is equipped with a topology which fails to be T2. One reads the expression in (1) as "the limit as approaches infinity of is ."The topological notion of convergence can be rewritten to accommodate a wider array of topological spaces by utilizing the language..

Origami is the Japanese art of paper folding. In traditional origami, constructions are done using a single sheet of colored paper that is often, though not always, square. In modular origami, a number of individual "units," each folded from a single sheet of paper, are combined to form a compound structure. Origami is an extremely rich art form, and constructions for thousands of objects, from dragons to buildings to vegetables have been devised. Many mathematical shapes can also be constructed, especially using modular origami. The images above show a number of modular polyhedral origami, together with an animated crane constructed in the Wolfram Language by L. Zamiatina.To distinguish the two directions in which paper can be folded, the notations illustrated above are conventionally used in origami. A "mountain fold" is a fold in which a peak is formed, whereas a "valley fold" is a fold forming..

In the early 1950s, Ernst Straus asked 1. Is every region illuminable from every point in the region? 2. Is every region illuminable from at least one point in the region? Here, illuminable means that there is a path from every point to every other by repeated reflections.In 1958, a young Roger Penrose used the properties of the ellipse to describe a room with curved walls that would always have dark (unilluminated) regions, regardless of the position of the candle. Penrose's room, illustrated above, consists of two half-ellipses at the top and bottom and two mushroom-shaped protuberances (which are in turn built up from straight line segments and smaller half-ellipses) on the left and right sides. The ellipses and mushrooms are strategically placed as shown, with the red points being the foci of the half-ellipses. There are essentially three possible configurations of illumination. In this figure, lit regions are indicated in white, unilluminated..

The Fibonacci numbers are the sequence of numbers defined by the linear recurrence equation(1)with . As a result of the definition (1), it is conventional to define .The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ... (OEIS A000045).Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials with .Fibonacci numbers are implemented in the WolframLanguage as Fibonacci[n].The Fibonacci numbers are also a Lucas sequence , and are companions to the Lucas numbers (which satisfy the same recurrence equation).The above cartoon (Amend 2005) shows an unconventional sports application of the Fibonacci numbers (left two panels). (The right panel instead applies the Perrin sequence).A scrambled version 13, 3, 2, 21, 1, 1, 8, 5 (OEIS A117540) of the first eight Fibonacci numbers appear as one of the clues left by murdered museum curator Jacque Saunière in D. Brown's novel The Da Vinci Code (Brown 2003, pp. 43,..

The integer sequence defined by the recurrence(1)with the initial conditions , , . This recurrence relation is the same as that for the Padovan sequence but with different initial conditions. The first few terms for , 1, ..., are 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, ... (OEIS A001608).The above cartoon (Amend 2005) shows an unconventional sports application of the Perrin sequence (right panel). (The left two panels instead apply the Fibonacci numbers). is the solution of a third-order linear homogeneous recurrence equation having characteristic equation(2)Denoting the roots of this equation by , , and , with the unique real root, the solution is then(3)Here,(4)is the plastic constant , which is also given by the limit(5)The asymptotic behavior of is(6)The first few primes in this sequence are 2, 3, 2, 5, 5, 7, 17, 29, 277, 367, 853, ... (OEIS A074788), which occur for terms , 3, 4, 5, 6, 7, 10, 12, 20, 21, 24, 34, 38, 75, 122, 166, 236, 355, 356, 930, 1042,..

An SIR model is an epidemiological model that computes the theoretical number of people infected with a contagious illness in a closed population over time. The name of this class of models derives from the fact that they involve coupled equations relating the number of susceptible people , number of people infected , and number of people who have recovered . One of the simplest SIR models is the Kermack-McKendrick model.The Season 1 episode "Vector" (2005) of the television crime drama NUMB3RS features SIR models.

The traveling salesman problem is a problem in graph theory requiring the most efficient (i.e., least total distance) Hamiltonian cycle a salesman can take through each of cities. No general method of solution is known, and the problem is NP-hard.The Wolfram Language command FindShortestTour[g] attempts to find a shortest tour, which is a Hamiltonian cycle (with initial vertex repeated at the end) for a Hamiltonian graph if it returns a list with first element equal to the vertex count of .The traveling salesman problem is mentioned by the character Larry Fleinhardt in the Season 2 episode "Rampage" (2006) of the television crime drama NUMB3RS.

Linear programming, sometimes known as linear optimization, is the problem of maximizing or minimizing a linear function over a convex polyhedron specified by linear and non-negativity constraints. Simplistically, linear programming is the optimization of an outcome based on some set of constraints using a linear mathematical model.Linear programming is implemented in the Wolfram Language as LinearProgramming[c, m, b], which finds a vector which minimizes the quantity subject to the constraints and for .Linear programming theory falls within convex optimization theory and is also considered to be an important part of operations research. Linear programming is extensively used in business and economics, but may also be used to solve certain engineering problems.Examples from economics include Leontief's input-output model, the determination of shadow prices, etc., an example of a business application would be maximizing..

Set covering deployment (sometimes written "set-covering deployment" and abbreviated SCDP for "set covering deployment problem") seeks an optimal stationing of troops in a set of regions so that a relatively small number of troop units can control a large geographic region. ReVelle and Rosing (2000) first described this in a study of Emperor Constantine the Great's mobile field army placements to secure the Roman Empire. Set covering deployment can be mathematically formulated as a (0,1)-integer programming problem.To formulate the Roman domination problem, consider the eight provinces of the Constantinian Roman Empire illustrated above. Each region is represented as a white disk, and the red lines indicate region connections. Call a region secured if one or more field armies are stationed in that region, and call a region securable if a field army can be deployed to that area from an adjacent area. In addition,..

Wavelets are a class of a functions used to localize a given function in both space and scaling. A family of wavelets can be constructed from a function , sometimes known as a "mother wavelet," which is confined in a finite interval. "Daughter wavelets" are then formed by translation () and contraction (). Wavelets are especially useful for compressing image data, since a wavelet transform has properties which are in some ways superior to a conventional Fourier transform.An individual wavelet can be defined by(1)Then(2)and Calderón's formula gives(3)A common type of wavelet is defined using Haar functions.The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features wavelets.

A Gröbner basis for a system of polynomials is an equivalence system that possesses useful properties, for example, that another polynomial is a combination of those in iff the remainder of with respect to is 0. (Here, the division algorithm requires an order of a certain type on the monomials.) Furthermore, the set of polynomials in a Gröbner basis have the same collection of roots as the original polynomials. For linear functions in any number of variables, a Gröbner basis is equivalent to Gaussian elimination.The algorithm for computing Gröbner bases is known as Buchberger's algorithm. Calculating a Gröbner basis is typically a very time-consuming process for large polynomial systems (Trott 2006, p. 37).Gröbner bases are pervasive in the construction of symbolic algebra algorithms, and Gröbner bases with respect to lexicographic order are very useful for solving equations and for elimination..

Check the price

for your project

for your project

we accept

Money back

guarantee

guarantee

Price calculator

We've got the best prices, check out yourself!