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 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..
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,..
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..
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 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..
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..
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...
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..
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...
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..
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.
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 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 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."..
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 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 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,..