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 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 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 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..
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 positive integers 216 and appear in an obscure passage in Plato's The Republic. In this passage, Plato alludes to the fact that 216 is equal to , where 6 is one of the numbers representing marriage since it is the product to the female 2 and the male 3. Plato was also aware of the fact the sum of the cubes of the 3-4-5 Pythagorean triple is equal to (Livio 2002, p. 66).In Laws, Plato suggests that is the optimal number of citizens in a state because 1. It is the product of 12, 20, and 21. 2. The 12th part of it can still be divided by 12. 3. It has 59 proper divisors, including all numbers for 1 to 12 except 11, and 5038--which is very close to 5040--is divisible by 11 (Livio 2002, p. 65).
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).
The Skewes number (or first Skewes number) is the number above which must fail (assuming that the Riemann hypothesis is true), where is the prime counting function and is the logarithmic integral.Isaac Asimov featured the Skewes number in his science fact article "Skewered!"(1974).In 1912, Littlewood proved that exists (Hardy 1999, p. 17), and the upper boundwas subsequently found by Skewes (1933). The Skewes number has since been reduced to by Lehman in 1966 (Conway and Guy 1996; Derbyshire 2004, p. 237), by te Riele (1987), and less than (Bays and Hudson 2000; Granville 2002; Borwein and Bailey 2003, p. 65; Havil 2003, p. 200; Derbyshire 2004, p. 237). The results of Bays and Hudson left open the possibility that the inequality could fail around , and thus established a large range of violation around (Derbyshire 2004, p. 237). More recent work by Demichel establishes that the first crossover..
An algorithm for making tables of primes. Sequentially write down the integers from 2 to the highest number you wish to include in the table. Cross out all numbers which are divisible by 2 (every second number). Find the smallest remaining number . It is 3. So cross out all numbers which are divisible by 3 (every third number). Find the smallest remaining number . It is 5. So cross out all numbers which are divisible by 5 (every fifth number).Continue until you have crossed out all numbers divisible by , where is the floor function. The numbers remaining are prime. This procedure is illustrated in the above diagram which sieves up to 50, and therefore crosses out composite numbers up to . If the procedure is then continued up to , then the number of cross-outs gives the number of distinct prime factors of each number.The sieve of Eratosthenes can be used to compute the primecounting function aswhich is essentially an application of the inclusion-exclusionprinciple..
The 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..
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..
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 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..
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..
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..
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."
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 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),..
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."..
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 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,..