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The salinon is the figure illustrated above formed from four connected semicircles. The word salinon is Greek for "salt cellar," which the figure resembles. If the radius of the large enclosing circle is and the radius of the small central circle is , then the radii of the two small side circles are .In his Book of Lemmas, Archimedes proved that the salinon has an area equal to the circle having the line segment joining the top and bottom points as its diameter (Wells 1991), namely

The term "vesica piscis," meaning "fish bladder" in Latin, is used for the particular symmetric lens formed by the intersection of two equal circles whose centers are offset by a distance equal to the circle radii (Pedoe 1995, p. xii). The height of the lens is given by letting in the equation for a circle-circle intersection(1)giving(2)The vesica piscis therefore has two equilateral triangles inscribed in it as illustrated above.The area of the vesica piscis is given by plugging into the circle-circle intersection area equation with ,(3)giving(4)(5)(OEIS A093731). Since each arcof the lens is precisely 1/3 of a circle, perimeter is given by(6)Renaissance artists frequently surrounded images of Jesus with the vesica piscis (Pedoe 1995, p. xii; Rawles 1997).

The symbol , used for the union of sets, and, sometimes, also for the logical connective OR instead of the symbol (vee). In fact, for any two sets and and this equivalence demonstrates the connection between the set-theoretical and the logical meaning.

If is a class of recursively enumerable sets, then the set of Gödel numbers of functions whose domains belong to is called its index set. If the index set of is a recursive set, then either is empty or contains all recursively enumerable sets.Rice's theorem is an important result for computer science because it sets up boundaries for research in that area. It basically states that only trivial properties of programs are algorithmically decidable.

Assume that numbered pancakes are stacked, and that a spatula can be used to reverse the order of the top pancakes for . Then the pancake sorting problem asks how many such "prefix reversals" are sufficient to sort an arbitrary stack (Skiena 1990, p. 48).The maximum numbers of flips needed to sort a random stack of , 2, 3, ... pancakes are 0, 1, 3, 4, 5, 7, 8, 9, 10, 11, 13, ... (OEIS A058986), with the number of maximal stacks for , 3, ... being 1, 1, 3, 20, 2, 35, 455, ... (OEIS A067607).The following table (OEIS A092113) gives the numbers of stacks of pancakes requiring flips. A flattened version is shown above as a binary plot.0123456781121131221413611351412354820615207919928113327163014954313571903101635For example, the three stacks of four pancakes requiring the maximum of four flips are , , and , which can be ordered using the flip sequences , , and , respectively (illustrated above). Similarly, the two stacks of six pancakes..

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

The fish curve is a term coined in this work for the ellipse negative pedal curve with pedal point at the focus for the special case of the eccentricity . For an ellipse with parametric equations(1)(2)the corresponding fish curve has parametric equations(3)(4)The Cartesian equation is(5)which, when the origin is translated to the node, canbe written(6)(Lockwood 1957).The interior of the curve is not consistently oriented in the above parametrization, with the fish's head being on the left of the curve and the tail on the right as the curve is traversed. Treating the two pieces separately then gives the areas of the tail and head as(7)(8)giving an overall area for the fish as(9)(Lockwood 1957).The arc length of the curve is given by(10)(11)(12)(Lockwood 1957).The curvature and tangentialangle are given by(13)(14)where is the complex argument.The Tschirnhausen cubic, illustrated above,also resembles a fish, as does the trefoil curve...

A barrel solid of revolution composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical about the midplane.The term also has a technical meaning in functional analysis. In particular, a subset of a topological linear space is a barrel if it is absorbing, closed, and absolutely convex (Taylor and Lay 1980, p. 111). (A subset of a topological linear space is absorbing if for each there is an such that is in if for each such that . A subset of a topological linear space is absolutely convex if for each and in , is in if .)When buying supplies for his second wedding, the great astronomer Johannes Kepler became unhappy about the inexact methods used by the merchants to estimate the liquid contents of a wine barrel. Kepler therefore investigated the properties of nearly 100 solids of revolution generated by rotation of conic sections about non-principal axes (Kepler, MacDonnell, Shechter, Tikhomirov..

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

A quartic algebraic curve also called the peg-top curve and given by the Cartesian equation(1)and the parametric curves(2)(3)for . It was studied by G. de Longchamps in 1886.The area of the piriform is(4)which is exactly the same as the ellipse with semiaxes and .The curvature of the piriform is given by(5)

A necessary and sufficient condition that there should exist at least one nondecreasing function such thatfor , 1, 2, ..., with all the integrals converging, is that sequence is positive definite (Widder 1941, p. 129).

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:..

For some range of , the Mandelbrot set lemniscate in the iteration towards the Mandelbrot set is a pear-shaped curve. In Cartesian coordinates with a constant , the equation is given byThe plots above show the resulting curve for (left figure) and for a range of between 0 and 2 (right figure).

The thin plate spline is the two-dimensional analog of the cubic spline in one dimension. It is the fundamental solution to the biharmonic equation, and has the formGiven a set of data points, a weighted combination of thin plate splines centered about each data point gives the interpolation function that passes through the points exactly while minimizing the so-called "bending energy." Bending energy is defined here as the integral over of the squares of the second derivatives,Regularization may be used to relax the requirement that the interpolant pass through the data points exactly.The name "thin plate spline" refers to a physical analogy involving the bending of a thin sheet of metal. In the physical setting, the deflection is in the direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the or coordinates..

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..

A McNugget number is a positive integer that can be obtained by adding together orders of McDonald's® Chicken McNuggetsTM (prior to consuming any), which originally came in boxes of 6, 9, and 20 (Vardi 1991, pp. 19-20 and 233-234; Wah and Picciotto 1994, p. 186). All integers are McNugget numbers except 1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 22, 23, 25, 28, 31, 34, 37, and 43. The value 43 therefore corresponds to the Frobenius number of .Since the Happy MealTM-sized nugget box (4 to a box) can now be purchased separately, the modern McNugget numbers are linear combinations of 4, 6, 9, and 20. These new-fangled numbers are much less interesting than before, with only 1, 2, 3, 5, 7, and 11 remaining as non-McNugget numbers. The value 11 therefore corresponds to the Frobenius number of .The greedy algorithm can be used to find a McNugget expansion of a given integer . This can also be done in the Wolfram Language using FrobeniusSolve[6,..

where is a modified Bessel function of the first kind and . For a derivation, see Papoulis (1962). For = 0, this reduces to the Rayleigh distribution.

A sieving procedure that can be used in conjunction with Dixon's factorization method to factor large numbers . Pick values of given by(1)where , 2, ... and is the floor function. We are then looking for factors such that(2)which means that only numbers with Legendre symbol (less than for trial divisor , where is the prime counting function) need be considered. The set of primes for which this is true is known as the factor base. Next, the congruences(3)must be solved for each in the factor base. Finally, a sieve is applied to find values of which can be factored completely using only the factor base. Gaussian elimination is then used as in Dixon's factorization method in order to find a product of the s, yielding a perfect square.The method requires about steps, improving on the continued fraction factorization algorithm by removing the 2 under the square root (Pomerance 1996). The use of multiple polynomials gives a better chance of factorization,..

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 .

A surface of revolution defined by Kepler. It consists of less than half of a circular arc rotated about an axis passing through the endpoints of the arc. The equations of the upper and lower boundaries in the plane are(1)for and . The cross section of a lemon is a lens. The lemon is the inside surface of a spindle torus. The American football is shaped like a lemon.Two other lemon-shaped surfaces are given by the sexticsurface(2)called the "citrus" (or zitrus) surface by Hauser (left figure), and thesextic surface(3)whose upper and lower portions resemble two halves of a lemon, called the limão surface by Hauser (right figure).The citrus surface had bounding box , centroid at , volume(4)and a moment of inertia tensor(5)for a uniform density solid citrus with mass .

The kiss surface is the quintic surfaceof revolution given by the equation(1)that is closely related to the ding-dong surface. It is so named because the shape of the lower portion resembles that of a Hershey's Chocolate Kiss.It can be represented parametrically as(2)(3)(4)The coefficients of the first fundamental formare(5)(6)(7)and of the second fundamental form are(8)(9)(10)The Gaussian and meancurvatures are given by(11)(12)The Gaussian curvature can be given implicitlyby(13)The surface area and volumeenclosed of the top teardrop are given by(14)(15)Its centroid is at and the moment of inertia tensor is(16)for a solid kiss with uniform density and mass .

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)

A surface of revolution defined by Kepler. It consists of more than half of a circular arc rotated about an axis passing through the endpoints of the arc. The equations of the upper and lower boundaries in the - plane are(1)for and . It is the outside surface of a spindle torus.It is also a quartic surface given by Cartesianequation(2)or(3)

An equilateral parallelogram whose acute angles are . Sometimes, the restriction to is dropped, and it is required only that two opposite angles are acute and the other two obtuse. The term rhombus is commonly used for an arbitrary equilateral parallelogram.The area of a lozenge of side length is(1)its diagonals have lengths(2)(3)and it has inradius(4)

The Lovász number of a graph satisfieswhere is the clique number, is the chromatic number of , and is the graph complement of . Furthermore, can be computed efficiently despite the fact that the computation of the two numbers it lies between is an NP-hard problem.The squeezing theorem is also sometimes knownas the sandwich theorem.

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

Fermat's sandwich theorem states that 26 is the only number sandwiched between a perfect square number ( and a perfect cubic number (). According to Singh (1997), after challenging other mathematicians to establish this result while not revealing his own proof, Fermat took particular delight in taunting the English mathematicians Wallis and Digby with their inability to prove the result.

A Diophantine problem (i.e., one whose solution must be given in terms of integers) which seeks a solution to the following problem. Given men and a pile of coconuts, each man in sequence takes th of the coconuts left after the previous man removed his (i.e., for the first man, , for the second, ..., for the last) and gives coconuts (specified in the problem to be the same number for each man) which do not divide equally to a monkey. When all men have so divided, they divide the remaining coconuts ways (i.e., taking an additional coconuts each), and give the coconuts which are left over to the monkey. If is the same at each division, then how many coconuts were there originally? The solution is equivalent to solving the Diophantine equations(1)(2)(3)(4)(5)which can be rewritten as(6)(7)(8)(9)(10)(11)Since there are equations in the unknowns , , ..., , , and , the solutions span a one-dimensional space (i.e., there is an infinite family of solution parameterized..

There are at least two theorems known as Salmon's theorem. This first states that if and are two points, and are the perpendiculars from and to the polars of and , respectively, with respect to a circle with center , then (Durell 1928; Salmon 1954, §101, p. 93).The second Salmon's theorem states that, given a track bounded by two confocal ellipses, if a ball is rolled so that its trajectory is tangent to the inner ellipse, the ball's trajectory will be tangent to the inner ellipse following all subsequent caroms as well (Salmon 1954, §189, pp. 181-182).

The cup product is a product on cohomology classes. In the case of de Rham cohomology, a cohomology class can be represented by a closed form. The cup product of and is represented by the closed form , where is the wedge product of differential forms. It is the dual operation to intersection in homology.In general, the cup product is a mapwhich satisfies , where is the th cohomology group.

Home plate in the game of baseball is an irregular pentagon with two parallel sides, each perpendicular to a base. It seems reasonable to dub such a figure (i.e., a rectangle with a coincident isosceles triangle placed on one side) a "isosceles right pentagon."However, specification of the shape of home plate, illustrated above, as specified by both the Major League Baseball Official Rules and the Little League rulebook (Kreutzer and Kerley 1990) is not physically realizable, since it requires the existence of a (12, 12, 17) right triangle, whereas(Bradley 1996). More specifically, the standards require the existence of an isosceles right triangle with side lengths 8.5 inches and a hypotenuse of length 12 inches, which does not satisfy the Pythagorean theorem.

The term "real line" has a number of different meanings in mathematics.Most commonly, "real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. The generalization of the real line to two dimensions is called the complex plane.The term "real line" is also used to distinguish an ordinary line from a so-called imaginary line which can arise in algebraic geometry.Renteln and Dundes (2005) give the following (bad) mathematical jokes about the real line:Q: What is green and homeomorphic to the open unit interval?A: The real lime.

Bezdek and Kuperberg (1991) have constructed packings of identical ellipsoidsof densities arbitrarily close to(OEIS A093824), greater than the maximum density of (OEIS A093825) that is possible for sphere packing (Sloane 1998), as established by proof of the Kepler conjecture. Furthermore, J. Wills has modified the ellipsoid packing to yield an even higher density of (Bezdek and Kuperberg 1991).Donev et al. (2004) showed that a maximally random jammed state of M&Ms chocolate candies has a packing density of about 68%, or 4% greater than spheres. Furthermore, Donev et al. (2004) also showed by computer simulations other ellipsoid packings resulted in random packing densities approaching that of the densest sphere packings, i.e., filling nearly 74% of space.

In dimensions for the arrangement of hyperspheres whose convex hull has minimal content is always a "sausage" (a set of hyperspheres arranged with centers along a line), independent of the number of -spheres. The conjecture was proposed by Fejes Tóth, and solved for dimensions by Betke et al. (1994) and Betke and Henk (1998).

The dihedral group is the symmetry group of an -sided regular polygon for . The group order of is . Dihedral groups are non-Abelian permutation groups for .The th dihedral group is represented in the Wolfram Language as DihedralGroup[n].One group presentation for the dihedral group is .A reducible two-dimensional representation of using real matrices has generators given by and , where is a rotation by radians about an axis passing through the center of a regular -gon and one of its vertices and is a rotation by about the center of the -gon (Arfken 1985, p. 250).Dihedral groups all have the same multiplication table structure. The table for is illustrated above.The cycle index (in variables , ..., ) for the dihedral group is given by(1)where(2)is the cycle index for the cyclic group , means divides , and is the totient function (Harary 1994, p. 184). The cycle indices for the first few are(3)(4)(5)(6)(7)Renteln and Dundes (2005) give..

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..

A fractal curve created from the base curve and motifillustrated above (Lauwerier 1991, p. 37).As illustrated above, the number of segments after the th iteration is(1)and the length of each segment is given by(2)so the capacity dimension is(3)(4)(5)(6)(Mandelbrot 1983, p. 48).The term Minkowski sausage is also used to refer to the Minkowskicover of a curve.

A compact set with areacreated by punching a square hole of length in the center of a square. In each of the eight squares remaining, punch out another hole of length , and so on.

The Menger sponge is a fractal which is the three-dimensionalanalog of the Sierpiński carpet. The th iteration of the Menger sponge is implemented in the Wolfram Language as MengerMesh[n, 3].Let be the number of filled boxes, the length of a side of a hole, and the fractional volume after the th iteration, then(1)(2)(3)The capacity dimension is therefore(4)(5)(6)(7)(OEIS A102447).The Menger sponge, in addition to being a fractal, is also a super-object for all compact one-dimensional objects, i.e., the topological equivalent of all one-dimensional objects can be found in a Menger sponge (Peitgen et al. 1992).The image above shows a metal print of the Menger sponge created by digital sculptorBathsheba Grossman (https://www.bathsheba.com/).

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 short theorem used in proving a larger theorem. Related concepts are the axiom, porism, postulate, principle, and theorem.The late mathematician P. Erdős has often been associated with the observation that "a mathematician is a machine for converting coffee into theorems" (e.g., Hoffman 1998, p. 7). However, this characterization appears to be due to his friend, Alfred Rényi (MacTutor, Malkevitch). This thought was developed further by Erdős' friend and Hungarian mathematician Paul Turán, who suggested that weak coffee was suitable "only for lemmas" (MacTutor, Malkevitch).

The impossible fork (Seckel 2002, p. 151), also known as the devil's pitchfork (Singmaster), blivet, or poiuyt, is a classic impossible figure originally due to Schuster (1964). While each prong of the fork (or, in the original work, "clevis") appears normal, attempting to determine their manner of attachment shows that something is seriously out of whack. The second figure above shows three impossible figures: the ambihelical hexnut in the lower left-hand corner, tribox in the middle, and impossible fork in the upper right.About the time of the impossible fork's discovery by Schuster (1964), it was used by Mad Magazine as a recurring theme. Their term for it was "poiuyt," which corresponds to the third row of a standard keyboard typed from right to left. The "poiuyt" was commonly used in Mad throughout the 1960s indicating absurdity or impossibility.Hayward incorporated this figure into a picture..

Arithmetic is the branch of mathematics dealing with integers or, more generally, numerical computation. Arithmetical operations include addition, congruence calculation, division, factorization, multiplication, power computation, root extraction, and subtraction. Arithmetic was part of the quadrivium taught in medieval universities. A mnemonic for the spelling of "arithmetic" is "a rat in the house may eat the ice cream."The branch of mathematics known as number theoryis sometimes known as higher arithmetic.Modular arithmetic is the arithmetic of congruences.Floating-point arithmetic is the arithmetic performed on real numbers by computers or other automated devices using a fixed number of bits.The fundamental theorem of arithmetic, also called the unique factorization theorem, states that any positive integer can be represented in exactly one way as a product of primes.The Löwenheim-Skolem..

The volumes of any -dimensional solids can always be simultaneously bisected by a -dimensional hyperplane. Proving the theorem for (where it is known as the pancake theorem) is simple and can be found in Courant and Robbins (1978).The proof is more involved for (Hunter and Madachy 1975, p. 69), but an intuitive proof can be obtained by the following argument due to G. Beck (pers. comm., Feb. 18, 2005). Note that given any direction , the volume of a solid can be bisected by a plane with normal . To see this, start with a plane that has all of the solid on one side and move it parallel to itself until the solid is completely on its other side. There must have been an intermediate position where the plane bisected the solid.Now take a sphere centered at the origin large enough to contain the three solids. Each point on the surface of the sphere indicates a direction. For any direction and each solid, find a plane that bisects the solid with that..

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."..

If a circular pizza is divided into 8, 12, 16, ... slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal.There is also a second pizza theorem. This one gives the volume of a pizza of thickness and radius :

The maximum number of regions that can be created by cuts using space division by planes, cube division by planes, cylinder cutting, etc., is given by(Yaglom and Yaglom 1987, pp. 102-106). For , 2, ... planes, this gives the values 2, 4, 8, 15, 26, 42, ... (OEIS A000125), a sequence whose values are sometimes called the cake numbers.

The longstanding conjecture that the nonimaginary solutions of(1)where is the Riemann zeta function, are the eigenvalues of an "appropriate" Hermitian operator . Berry and Keating (1999) further conjecture that this operator is(2)(3)where and are the position and conjugate momentum operators, respectively, and multiplication is noncommutative. Note that is symmetric but might have nontrivial deficiency indices, so while physicists define this operator to be Hermitian, mathematicians do not.

The Sierpiński sieve is a fractal described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. 43). It is also called the Sierpiński gasket or Sierpiński triangle. The curve can be written as a Lindenmayer system with initial string "FXF--FF--FF", string rewriting rules "F" -> "FF", "X" -> "--FXF++FXF++FXF--", and angle .The th iteration of the Sierpiński sieve is implemented in the Wolfram Language as SierpinskiMesh[n].Let be the number of black triangles after iteration , the length of a side of a triangle, and the fractional area which is black after the th iteration. Then(1)(2)(3)The capacity dimension is therefore(4)(5)(6)(7)(OEIS A020857; Wolfram 1984; Borwein and Bailey2003, p. 46).The Sierpiński sieve is produced by the beautiful recurrenceequation(8)where denote bitwise..

The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathological continuous function which is nowhere differentiable. Its name derives from the resemblance of its first iteration to the shape of the dessert commonly made with milk or cream and sugar thickened with gelatin.The iterations towards the continuous function are batrachions resembling the Hofstadter-Conway $10,000 sequence. The first six iterations are illustrated below. The th iteration contains points, where , and can be obtained by setting , lettingand looping over to 1 by steps of and to by steps of .

An Abelian group is a group for which the elements commute (i.e., for all elements and ). Abelian groups therefore correspond to groups with symmetric multiplication tables.All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.In the Wolfram Language, the function AbelianGroup[n1, n2, ...] represents the direct product of the cyclic groups of degrees , , ....No general formula is known for giving the number of nonisomorphic finite groups of a given group order. However, the number of nonisomorphic Abelian finite groups of any given group order is given by writing as(1)where the are distinct prime factors, then(2)where is the partition function, which is implemented in the Wolfram Language as FiniteAbelianGroupCount[n]...

It is always possible to "fairly" divide a cake among people using only vertical cuts. Furthermore, it is possible to cut and divide a cake such that each person believes that everyone has received of the cake according to his own measure (Steinhaus 1999, pp. 65-71). Finally, if there is some piece on which two people disagree, then there is a way of partitioning and dividing a cake such that each participant believes that he has obtained more than of the cake according to his own measure.There are also similar methods of dividing collections of individually indivisible objects among two or more people when cash payments are used to even up the final division (Steinhaus 1999, pp. 67-68).Ignoring the height of the cake, the cake-cutting problem is really a question of fairly dividing a circle into equal area pieces using cuts in its plane. One method of proving fair cake cutting to always be possible relies on the Frobenius-König..

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