Universality is the property of being able to perform different tasks with the same underlying construction just by being programmed in a different way. Universal systems are effectively capable of emulating any other system. Digital computers are universal, but proving that idealized computational systems are universal can be extremely difficult and technical. Nonetheless, examples have been found in many systems, and any system that can be translated into another system known to be universal must itself be universal. Specific universal Turing machines, universal cellular automata (in both one and two dimensions), and universal cyclic tag systems are known, although the smallest universal example is known only in the case of elementary cellular automata (Wolfram 2002, Cook 2004).
Find the maximum number of bishops that can be placed on an chessboard such that no two attack each other. The answer is (Dudeney 1970, Madachy 1979), giving the sequence 2, 4, 6, 8, ... (the even numbers) for , 3, .... One maximal solution for is illustrated above. The numbers of distinct maximal arrangements for , 2, ... bishops are 1, 4, 26, 260, 3368, ... (OEIS A002465). The numbers of rotationally and reflectively distinct solutions on an board for is(1)for (Dudeney 1970, p. 96; Madachy 1979, p. 45; Pickover 1995). An equivalent formula also valid for is(2)where is the floor function, giving the sequence for , 2, ... as 1, 1, 2, 3, 6, 10, 20, 36, ... (OEIS A005418).The minimum number of bishops needed to occupy or attack all squares on an chessboard is , arranged as illustrated above...
Solomon's seal knot is the prime (5,2)-torus knot with braid word . It is also known as the cinquefoil knot (a name derived from certain herbs and shrubs of the rose family which have five-lobed leaves and five-petaled flowers) or the double overhand knot. It has Arf invariant 1 and is not amphichiral, although it is invertible.The knot group of Solomon's seal knot is(1)(Livingston 1993, p. 127).The Alexander polynomial , BLM/Ho polynomial , Conway polynomial , HOMFLY polynomial , Jones polynomial , and Kauffman polynomial F of the Solomon's seal knot are(2)(3)(4)(5)(6)(7)Surprisingly, the knot 10-132 shares the same Alexander polynomial and Jones polynomial with the Solomon's seal knot. However, no knots on 10 or fewer crossings share the same BLM/Ho polynomial with it.
The singleton graph is the graph consisting of a single isolated node with no edges. It is therefore the empty graph on one node. It is commonly denoted (i.e., the complete graph on one node).By convention, the singleton graph is considered to be Hamiltonian(B. McKay, pers. comm., Mar. 22, 2007).
A transformation (a.k.a., map, function) over a domain takes the elements to elements , where the range (a.k.a., image) of is defined asNote that when transformations are specified with respect to a coordinate system, it is important to specify whether the rotation takes place on the coordinate system, with space and objects embedded in it being viewed as fixed (a so-called alias transformation), or on the space itself relative to a fixed coordinate system (a so-called alibi transformation).Examples of transformations are summarized in the following table.TransformationCharacterizationdilationcenter of dilation, scale decrease factorexpansioncenter of expansion, scale increase factorreflectionmirror line or planerotationcenter of rotation, rotation angleshearinvariant line and shear factorstretch (1-way)invariant line and scale factorstretch (2-way)invariant lines and scale factorstranslationdisplacement..
Gabriel's horn, also called Torricelli's trumpet, is the surface of revolution of the function about the x-axis for . It is therefore given by parametric equations(1)(2)(3)The surprising thing about this surface is that it (taking for convenience here) has finite volume(4)(5)(6)but infinite surface area,since(7)(8)(9)(10)(11)(12)This leads to the paradoxical consequence that while Gabriel's horn can be filled up with cubic units of paint, an infinite number of square units of paint are needed to cover its surface!The coefficients of the first fundamental formare,(13)(14)(15)and of the second fundamental form are(16)(17)(18)The Gaussian and meancurvatures are(19)(20)The Gaussian curvature can be expressed implicitly as(21)
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 Maltese cross is a symbol identified with the Christian warrior whose outward points form an octagon (left figure). Another class of cross sometimes (incorrectly) known as the Maltese cross is the cross pattée (from the French word meaning "paw," which each arm of the cross resembles). The TeX macro gives the form of the cross pattée illustrated in the middle figure. Around 1901, Dudeney published a seven-piece dissection of what he termed a "Maltese cross" (but which is actually a variant of the cross pattée) to a square (right figure) due to A. E. Hill (Gardner 1991, p. 46).
An irregular dodecagonal cross in the shape of a dagger . The six faces of a cube can be cut along seven edges and unfolded into a Latin cross (i.e., the Latin cross is the net of the cube). Similarly, eight hypersurfaces of a hypercube can be cut along 17 squares and unfolded to form a three-dimensional Latin cross.Another cross also called the Latin cross is illustrated above. It is a Greekcross with flared ends, and is also known as the crux immissa or cross patée.
The Christmas stocking theorem, also known as the hockey stick theorem, states that the sum of a diagonal string of numbers in Pascal's triangle starting at the th entry from the top (where the apex has ) on left edge and continuing down rows is equal to the number to the left and below (the "toe") bottom of the diagonal (the "heel"; Butterworth 2002). This follows from the identitywhere is a binomial coefficient.
As originally stated by Gould (1972),(1)where GCD is the greatest common divisor and is a binomial coefficient. This was subsequently extended by D. Singmaster to(2)(Sato 1975), and generalized by Sato (1975) to(3)An even larger generalization was obtained by Hitotumatu and Sato (1975), who defined(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)with(16)and showed that each of the twelve binomial coefficients , , , , , , , , , , , and has equal greatest common divisor.A second star of David theorem states that if two triangles are drawn centered on a given element of Pascal's triangle as illustrated above, then the products of the three numbers in the associated points of each of the two stars are the same (Butterworth 2002). This follows from the fact that(17)(18)(19)The second star of David theorem holds true not only for the usual binomial coefficients, but also for q-binomial coefficients, where the common product is given by(20)In..
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).
Let and be any sets, and let be the cardinal number of a set . Then cardinal exponentiation is defined by(Ciesielski 1997, p. 68; Dauben 1990, p. 174; Moore 1982, p. 37; Rubin 1967, p. 275, Suppes 1972, p. 116).It is easy to show that the cardinal number of the power set of is , since and there is a natural bijection between the subsets of and the functions from into .
For any sets and , their cardinal numbers satisfy iff there is a one-to-one function from into (Rubin 1967, p. 266; Suppes 1972, pp. 94 and 116). It is easy to show this satisfies the reflexive and transitive axioms of a partial order. However, it is difficult to show the antisymmetry property, whose proof is known as the Schröder-Bernstein theorem. To show the trichotomy property, one must use the axiom of choice.Although an order type can be defined similarly, it does not seem usual to do so.
Let and be any sets with empty intersection, and let denote the cardinal number of a set . Then(Ciesielski 1997, p. 68; Dauben 1990, p. 173; Rubin 1967, p. 274; Suppes 1972, pp. 112-113).It is an interesting exercise to show that cardinal addition is well-defined. The main steps are to show that for any cardinal numbers and , there exist disjoint sets and with cardinal numbers and , and to show that if and are disjoint and and disjoint with and then . The second of these is easy. The first is a little tricky and requires an appeal to the axioms of set theory. Also, one needs to restrict the definition of cardinal to guarantee if is a cardinal, then there is a set satisfying .
The Mephisto waltz sequence is defined by beginning with 0 and then iterating the maps and . This gives 0, 001, 001001110, 001001110001001110110110001, ... (OEIS A064990). These words are fourth power-free (Allouche and Shallit 2003, p. 25).The numbers of 0s and 1s in step , 1, ... are given by 1, 2, 5, 14, 41, 122, ... (OEIS A007051) and 0, 1, 4, 13, 40, 121, ... (OEIS A003462), respectively, which are given in closed form by and , respectively.A recurrence plot of the Mephisto waltz sequenceis illustrated above.
In general, a cross is a figure formed by two intersecting line segments. In linear algebra, a cross is defined as a set of mutually perpendicular pairs of vectors of equal magnitude from a fixed origin in Euclidean -space.The word "cross" is also used to denote the operation of the cross product, so would be pronounced " cross ."
Let be a function and let , and define the cardinal series of with respect to the interval as the formal serieswhere is the sinc function. If this series converges, it is known as the cardinal function (or Whittaker cardinal function) of , denoted (McNamee et al. 1971).
The surface given by the parametricequations(1)(2)(3)For , the coefficients of the first fundamental form are(4)(5)(6)and of the second fundamental form are(7)(8)(9)The Gaussian and meancurvatures are given by(10)(11)and the principal curvatures are(12)(13)
The 27 real or imaginary lines which lie on the general cubic surface and the 45 triple tangent planes to the surface. All are related to the 28 bitangents of the general quartic curve.Schoute (1910) showed that the 27 lines can be put into a one-to-one correspondence with the vertices of a particular polytope in six-dimensional space in such a manner that all incidence relations between the lines are mirrored in the connectivity of the polytope and conversely (Du Val 1933). A similar correspondence can be made between the 28 bitangents and a seven-dimensional polytope (Coxeter 1928) and between the tritangent planes of the canonical curve of genus four and an eight-dimensional polytope (Du Val 1933).
The lituus is an Archimedean spiral with , having polar equation(1)Lituus means a "crook," in the sense of a bishop's crosier. The lituus curve originated with Cotes in 1722. Maclaurin used the term lituus in his book Harmonia Mensurarum in 1722 (MacTutor Archive). The lituus is the locus of the point moving such that the area of a circular sector remains constant.The arc length, curvature,and tangential angle are given by(2)(3)(4)where the arc length is measured from .
In common usage, a cardinal number is a number used in counting (a countingnumber), such as 1, 2, 3, ....In formal set theory, a cardinal number (also called "the cardinality") is a type of number defined in such a way that any method of counting sets using it gives the same result. (This is not true for the ordinal numbers.) In fact, the cardinal numbers are obtained by collecting all ordinal numbers which are obtainable by counting a given set. A set has (aleph-0) members if it can be put into a one-to-one correspondence with the finite ordinal numbers. The cardinality of a set is also frequently referred to as the "power" of a set (Moore 1982, Dauben 1990, Suppes 1972).In Georg Cantor's original notation, the symbol for a set annotated with a single overbar indicated stripped of any structure besides order, hence it represented the order type of the set. A double overbar then indicated stripping the order from the set and thus indicated..
Let and be any sets. Then the product of and is defined as the Cartesian product(Ciesielski 1997, p. 68; Dauben 1990, p. 173; Moore 1982, p. 37; Rubin 1967, p. 274; Suppes 1972, pp. 114-115).
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 number in which the first decimal digits of the fractional part sum to 666 is known as an evil number (Pegg and Lomont 2004).However, the term "evil" is also used to denote nonnegative integers that have an even number of 1s in their binary expansions, the first few of which are 0, 3, 5, 6, 9, 10, 12, 15, 17, 18, 20, ... (OEIS A001969), illustrated above as a binary plot. Numbers that are not evil are then known as odious numbers.Returning to Pegg's definition of evil, the fact that is evil was noted by Keith, while I. Honig (pers. comm., May 9, 2004) noted that the golden ratio is also evil. The following table gives a list of some common evil numbers (Pegg and Lomont 2004).Ramanujan constant 132hard hexagon entropy constant 137139140Stieltjes constant 142pi 144golden ratio 146146151Glaisher-Kinkelin constant 153cube line picking average length155Delian constant 156The probability of the digits of a given real number summing..
Archimedes' cattle problem, also called the bovinum problema, or Archimedes' reverse, is stated as follows: "The sun god had a herd of cattle consisting of bulls and cows, one part of which was white, a second black, a third spotted, and a fourth brown. Among the bulls, the number of white ones was one half plus one third the number of the black greater than the brown; the number of the black, one quarter plus one fifth the number of the spotted greater than the brown; the number of the spotted, one sixth and one seventh the number of the white greater than the brown. Among the cows, the number of white ones was one third plus one quarter of the total black cattle; the number of the black, one quarter plus one fifth the total of the spotted cattle; the number of spotted, one fifth plus one sixth the total of the brown cattle; the number of the brown, one sixth plus one seventh the total of the white cattle. What was the composition of the herd?"Solution consists..
A topological space that contains a homeomorphicimage of every topological space of a certain class.A metric space is said to be universal for a family of metric spaces if any space from is isometrically embeddable in . Fréchet (1910) proved that , the space of all bounded sequences of real numbers endowed with a supremum norm, is a universal space for the family of all separable metric spaces. Holsztynski (1978) proved that there exists a metric on , inducing the usual topology, such that every finite metric space embeds in (Ovchinnikov 2000).
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..
Let denote the mean of a set of quantities , then the absolute deviation is defined by
Consider a game, first proposed by Nicolaus Bernoulli, in which a player bets on how many tosses of a coin will be needed before it first turns up heads. The player pays a fixed amount initially, and then receives dollars if the coin comes up heads on the th toss. The expectation value of the gain is then(1)dollars, so any finite amount of money can be wagered and the player will still come out ahead on average.Feller (1968) discusses a modified version of the game in which the player receives nothing if a trial takes more than a fixed number of tosses. The classical theory of this modified game concluded that is a fair entrance fee, but Feller notes that "the modern student will hardly understand the mysterious discussions of this 'paradox.' "In another modified version of the game, the player bets $2 that heads will turn up on the first throw, $4 that heads will turn up on the second throw (if it did not turn up on the first), $8 that heads will turn..
A real number that is -normal for every base 2, 3, 4, ... is said to be absolutely normal. As proved by Borel (1922, p. 198), almost all real numbers in are absolutely normal (Niven 1956, p. 103; Stoneham 1970; Kuipers and Niederreiter 1974, p. 71; Bailey and Crandall 2002).The first specific construction of an absolutely normal number was by Sierpiński (1917), with another method presented by Schmidt (1962). These results were both obtained by complex constructive devices (Stoneham 1970), and are by no means easy to construct (Stoneham 1970, Sierpiński and Schinzel 1988).
One of the numbers ..., , , 0, 1, 2, .... The set of integers forms a ring that is denoted . A given integer may be negative (), nonnegative (), zero (), or positive (). The set of integers is, not surprisingly, called Integers in the Wolfram Language, and a number can be tested to see if it is a member of the integers using the command Element[x, Integers]. The command IntegerQ[x] returns True if has function head Integer in the Wolfram Language.Numbers that are integers are sometimes described as "integral" (instead of integer-valued), but this practice may lead to unnecessary confusions with the integrals of integral calculus.The ring of integers has cardinal number of aleph0. The generating function for the nonnegative integers isThere are several symbols used to perform operations having to do with conversion between real numbers and integers. The symbol ("floor ") means "the largest integer not greater than ,"..
A plot of the map winding number resulting from mode locking as a function of for the circle map(1)with . (Since the circle map becomes mode-locked, the map winding number is independent of the initial starting argument .) At each value of , the map winding number is some rational number. The result is a monotonic increasing "staircase" for which the simplest rational numbers have the largest steps. The Devil's staircase continuously maps the interval onto , but is constant almost everywhere (i.e., except on a Cantor set).For , the measure of quasiperiodic states ( irrational) on the -axis has become zero, and the measure of mode-locked state has become 1. The dimension of the Devil's staircase .Another type of devil's staircase occurs for the sum(2)for , where is the floor function (Böhmer 1926ab; Kuipers and Niederreiter 1974, p. 10; Danilov 1974; Adams 1977; Davison 1977; Bowman 1988; Borwein and Borwein 1993; Bowman..
Let be the volume of a ball of radius in a complete -dimensional Riemannian manifold with Ricci curvature tensor . Then , where is the volume of a ball in a space having constant sectional curvature. In addition, if equality holds for some ball, then this ball is isometric to the ball of radius in the space of constant sectional curvature .
The number (the cube root of 2) which is to be constructed in the cube duplication problem. This number is not a Euclidean number although it is an algebraic number of third degree.It has decimal digits 1.25992104989... (OEIS A002580).Its continued fraction is [1, 3, 1, 5, 1, 1,4, 1, 1, 8, 1, 14, 1, ...] (OEIS A002945).
The function from a given nonempty set to the power set that maps every element of to the set .
Min Max The absolute value of a real number is denoted and defined as the "unsigned" portion of ,(1)(2)where is the sign function. The absolute value is therefore always greater than or equal to 0. The absolute value of for real is plotted above. Min Max Re Im The absolute value of a complex number , also called the complex modulus, is defined as(3)This form is implemented in the Wolfram Language as Abs[z] and is illustrated above for complex .Note that the derivative (read: complex derivative) does not exist because at every point in the complex plane, the value of the derivative of depends on the direction in which the derivative is taken (so the Cauchy-Riemann equations cannot and do not hold). However, the real derivative (i.e., restricting the derivative to directions along the real axis) can be defined for points other than as(4)As a result of the fact that computer algebra languages such as the Wolfram Language generically deal with..
An element of an extension field of a field which is not algebraic over . A transcendental number is a complex number which is transcendental over the field of rational numbers.
A series which is not convergent. Series may diverge by marching off to infinity or by oscillating. Divergent series have some curious properties. For example, rearranging the terms of gives both and .The Riemann series theorem states that, by a suitable rearrangement of terms, a conditionally convergent series may be made to converge to any desired value, or to diverge.No less an authority than N. H. Abel wrote "The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever" (Gardner 1984, p. 171; Hoffman 1998, p. 218). However, divergent series can actually be "summed" rigorously by using extensions to the usual summation rules (e.g., so-called Abel and Cesàro sums). For example, the divergent series has both Abel and Cesàro sums of 1/2...
The difference between the measured or inferred value of a quantity and its actual value , given by(sometimes with the absolute value taken) is called the absolute error. The absolute error of the sum or difference of a number of quantities is less than or equal to the sum of their absolute errors.
An extension field of a field that is not algebraic over , i.e., an extension field that has at least one element that is transcendental over .For example, the field of rational functions in the variable is a transcendental extension of since is transcendental over . The field of real numbers is a transcendental extension of the field of rational numbers, since is transcendental over .
The transcendence degree of , sometimes called the transcendental degree, is one because it is generated by one extra element. In contrast, (which is the same field) also has transcendence degree one because is algebraic over . In general, the transcendence degree of an extension field over a field is the smallest number elements of which are not algebraic over , but needed to generate . If the smallest set of transcendental elements needed to generate is infinite, then the transcendence degree is the cardinal number of that set.For instance, the transcendence degree of over is one. The transcendence degree of over is an infinite cardinal number. There are many open questions about the traditional constants in mathematics, such as the transcendence degree of .
The Maltese cross curve is the cubic algebraiccurve with Cartesian equation(1)and polar equation(2)(Cundy and Rollett 1989, p. 71), so named for its resemblance to the Maltesecross.It has curvature and tangentialangle given by(3)(4)
The devil's curve was studied by G. Cramer in 1750 and Lacroix in 1810 (MacTutor Archive). It appeared in Nouvelles Annales in 1858. The Cartesian equation is(1)equivalent to(2)the polar equation is(3)and the parametric equations are(4)(5)The curve illustrated above corresponds to parameters and .It has a crunode at the origin.For , the cental hourglass is horizontal, for , it is vertical, and as it passes through , the curve changes to a circle.A special case of the Devil's curve is the so-called "electric motor curve":(6)(Cundy and Rollett 1989).
A plane quartic curve also called the cross curve or policeman on point duty curve (Cundy and Rollett 1989). It is given by the implicit equation(1)which is equivalent to(2)and(3)In parametric form,(4)(5)The curvature is given by(6)(7)which, in the special case , reduces to(8)