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A plane curve given by the parametric equations(1)(2)The plots above show curves for values of from 0 to 7.The teardrop curve has area(3)

Min Max Min Max Re Im Let and be periods of a doubly periodic function, with the half-period ratio a number with . Then Klein's absolute invariant (also called Klein's modular function) is defined as(1)where and are the invariants of the Weierstrass elliptic function with modular discriminant(2)(Klein 1877). If , where is the upper half-plane, then(3)is a function of the ratio only, as are , , and . Furthermore, , , , and are analytic in (Apostol 1997, p. 15).Klein's absolute invariant is implemented in the WolframLanguage as KleinInvariantJ[tau].The function is the same as the j-function, modulo a constant multiplicative factor.Every rational function of is a modular function, and every modular function can be expressed as a rational function of (Apostol 1997, p. 40).Klein's invariant can be given explicitly by(4)(5)(Klein 1878-1879, Cohn 1994), where is the elliptic lambda function(6) is a Jacobi theta function, the are..

The least common multiple of two numbers and , variously denoted (this work; Zwillinger 1996, p. 91; Råde and Westergren 2004, p. 54), (Gellert et al. 1989, p. 25; Graham et al. 1990, p. 103; Bressoud and Wagon 2000, p. 7; D'Angelo and West 2000, p. 135; Yan 2002, p. 31; Bronshtein et al. 2007, pp. 324-325; Wolfram Language), l.c.m. (Andrews 1994, p. 22; Guy 2004, pp. 312-313), or , is the smallest positive number for which there exist positive integers and such that(1)The least common multiple of more than two numbers is similarly defined.The least common multiple of , , ... is implemented in the Wolfram Language as LCM[a, b, ...].The least common multiple of two numbers and can be obtained by finding the prime factorization of each(2)(3)where the s are all prime factors of and , and if does not occur in one factorization, then the corresponding exponent is taken as 0. The least..

The term diamond is another word for a rhombus. The term is also used to denote a square tilted at a angle.The diamond shape is a special case of the superellipse with parameter , giving it implicit Cartesian equation(1)Since the diamond is a rhombus with diagonals and , it has inradius(2)(3)Writing as an algebraic curve gives the quartic curve(4)which is a diamond curve with the diamond edges extended to infinity.When considered as a polyomino, the diamond of order can be considered as the set of squares whose centers satisfy the inequality . There are then squares in the order- diamond, which is precisely the centered square number of order . For , 2, ..., the first few values are 1, 5, 13, 25, 41, 61, 85, 113, 145, ... (OEIS A001844).The diamond is also the name given to the unique 2-polyiamond...

The Kaprekar routine is an algorithm discovered in 1949 by D. R. Kaprekar for 4-digit numbers, but which can be generalized to -digit numbers. To apply the Kaprekar routine to a number , arrange the digits in descending () and ascending () order. Now compute (discarding any initial 0s) and iterate, where is sometimes called the Kaprekar function. The algorithm reaches 0 (a degenerate case), a constant, or a cycle, depending on the number of digits in and the value of . The list of values is sometimes called a Kaprekar sequence, and the result is sometimes called a Kaprekar number (Deutsch and Goldman 2004), though this nomenclature should be deprecated because of confusing with the distinct sort of Kaprekar number.In base-10, the numbers for which are given by 495, 6174, 549945, 631764, ... (OEIS A099009). Similarly, the numbers for which iterating gives a cycle of length are given by 53955, 59994, 61974, 62964, 63954, 71973, ... (OEIS..

An irreducible fraction is a fraction for which , i.e., and are relatively prime. For example, in the complex plane, is reducible, while is not.The figure above shows the irreducible fractions plotted in the complex plane (Pickover 1997; Trott 2004, p. 29).

Beautiful patterns can be created by drawing sets of nested polygons such that the incircle of the th polygon is the circumcircle of the st and successive polygons are rotated one half-turn at each iteration. The results are shown above for nested triangles through heptagons in alternating black and white and in a cyclic rainbow coloring.The animation above shows successive iterations of a nested octagon.The black region of a nested square has areaif the initial square has unit edge length.

In 1704, Sebastien Truchet considered all possible patterns formed by tilings of right triangles oriented at the four corners of a square (Wolfram 2002, p. 875).Truchet's tiles produce beautiful patterns when laid out on a grid, as illustrated by the arrangement of random tiles illustrated above.A modification of Truchet's tiles leads to a single tile consisting of two circular arcs of radius equal to half the tile edge length centered at opposed corners (Pickover 1989). There are two possible orientations of this tile, and tiling the plane using tiles with random orientations gives visually interesting patterns. In fact, these tiles have been used in the construction of various games, including the "black path game" and "meander" (Berlekamp et al. 1982, pp. 682-684).The illustration above shows a Truchet tiling. For random orientations, the fraction of closed circles is approximately 0.054 and the..

The binomial coefficient is the number of ways of picking unordered outcomes from possibilities, also known as a combination or combinatorial number. The symbols and are used to denote a binomial coefficient, and are sometimes read as " choose ." therefore gives the number of k-subsets possible out of a set of distinct items. For example, The 2-subsets of are the six pairs , , , , , and , so . The number of lattice paths from the origin to a point ) is the binomial coefficient (Hilton and Pedersen 1991).The value of the binomial coefficient for nonnegative and is given explicitly by(1)where denotes a factorial. Writing the factorial as a gamma function allows the binomial coefficient to be generalized to noninteger arguments (including complex and ) as(2)For nonnegative integer arguments, the gamma function reduces to factorials, leading to(3)which is Pascal's triangle. Using the symmetryformula(4)for integer , and complex , this..

A regular polygon is an -sided polygon in which the sides are all the same length and are symmetrically placed about a common center (i.e., the polygon is both equiangular and equilateral). Only certain regular polygons are "constructible" using the classical Greek tools of the compass and straightedge.The terms equilateral triangle and square refer to the regular 3- and 4-polygons, respectively. The words for polygons with sides (e.g., pentagon, hexagon, heptagon, etc.) can refer to either regular or non-regular polygons, although the terms generally refer to regular polygons in the absence of specific wording.A regular -gon is implemented in the Wolfram Language as RegularPolygon[n], or more generally as RegularPolygon[r, n], RegularPolygon[x, y, rspec, n], etc.The sum of perpendiculars from any point to the sides of a regular polygon of sides is times the apothem.Let be the side length, be the inradius, and the circumradius..

The perspective image of an infinite checkerboard. It can be constructed starting from any triangle , where and form the near corner of the floor, and is the horizon (left figure). If is the corner tile, the lines and must be parallel to and respectively. This means that in the drawing they will meet and at the horizon, i.e., at point and point respectively (right figure). This property, of course, extends to the two bunches of perpendicular lines forming the grid.The adjacent tile (left figure) can then be determined by the following conditions: 1. The new vertices and lie on lines and respectively. 2. The diagonal meets the parallel line at the horizon . 3. The line passes through . Similarly, the corner-neighbor of (right figure) can be easily constructed requiring that: 1. Point lie on . 2. Point lie on the common diagonal of the two tiles. 3. Line pass through . Iterating the above procedures will yield the complete picture. This construction shows..

A demiregular tessellation, also called a polymorph tessellation, is a type of tessellation whose definition is somewhat problematical. Some authors define them as orderly compositions of the three regular and eight semiregular tessellations (which is not precise enough to draw any conclusions from), while others defined them as a tessellation having more than one transitivity class of vertices (which leads to an infinite number of possible tilings).The number of demiregular tessellations is commonly given as 14 (Critchlow 1970, pp. 62-67; Ghyka 1977, pp. 78-80; Williams 1979, p. 43; Steinhaus 1999, pp. 79 and 81-82). However, not all sources apparently give the same 14. Caution is therefore needed in attempting to determine what is meant by "demiregular tessellation."A more precise term of demiregular tessellations is 2-uniform tessellations (Grünbaum and Shephard 1986, p. 65)...

Regular tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex are called semiregular tessellations, or sometimes Archimedean tessellations. In the plane, there are eight such tessellations, illustrated above (Ghyka 1977, pp. 76-78; Williams 1979, pp. 37-41; Steinhaus 1999, pp. 78-82; Wells 1991, pp. 226-227). Williams (1979, pp. 37-41) also illustrates the dual tessellations of the semiregular tessellations. The dual tessellation of the tessellation of squares and equilateral triangles is called the Cairo tessellation (Williams 1979, p. 38; Wells 1991, p. 23).

Consider a two-dimensional tessellation with regular -gons at each polygon vertex. In the plane,(1)(2)so(3)(Ball and Coxeter 1987), and the only factorizations are(4)(5)(6)Therefore, there are only three regular tessellations (composed of the hexagon, square, and triangle), illustrated above (Ghyka 1977, p. 76; Williams 1979, p. 36; Wells 1991, p. 213).There do not exist any regular star polygon tessellations in the plane. Regular tessellations of the sphere by spherical triangles are called triangular symmetry groups.

A plane tiling is said to be isohedral if the symmetry group of the tiling acts transitively on the tiles, and -isohedral if the tiles fall into n orbits under the action of the symmetry group of the tiling. A -anisohedral tiling is a tiling which permits no -isohedral tiling with .The numbers of anisohedral polyominoes with , 9, 10, ... are 1, 9, 44, 108, 222, ... (OEIS A075206), the first few of which are illustrated above (Myers).

Consider the plane quartic curve defined bywhere homogeneous coordinates have been used here so that can be considered a parameter (the plot above shows the curve for a number of values of between and 2), over a field of characteristic 3. Hartshorne (1977, p. 305) terms this "a funny curve" since it is nonsingular, every point is an inflection point, and the dual curve is isomorphic to but the natural map is purely inseparable.The surface in complex projective coordinates (Levy 1999, p. ix; left figure), and with the ideal surface determined by the equation(Thurston 1999, p. 3; right figure) is more properly known as the Klein quarticor Klein curve. It has constant zero Gaussian curvature.Klein (1879; translation reprinted in 1999) discovered that this surface has a number of remarkable properties, including an incredible 336-fold symmetry when mirror reflections are allowed (Levy 1999, p. ix; Thurston..

A dragon curve is a recursive nonintersecting curve whose name derives from its resemblance to a certain mythical creature.The curve can be constructed by representing a left turn by 1 and a right turn by 0. The first-order curve is then denoted 1. For higher order curves, append a 1 to the end, then append the string of preceding digits with its middle digit complemented. For example, the second-order curve is generated as follows: , and the third as .Continuing gives 110110011100100... (OEIS A014577), which is sometimes known as the regular paperfolding sequence and written with s instead of 0s (Allouche and Shallit 2003, p. 155). A recurrence plot of the limiting value of this sequence is illustrated above.Representing the sequence of binary digits 1, 110, 1101100, 110110011100100, ... in octal gives 1, 6, 154, 66344, ...(OEIS A003460; Gardner 1978, p. 216).This procedure is equivalent to drawing a right angle and subsequently..

A matrix with elements(1)for , 2, ..., . Hilbert matrices are implemented in the Wolfram Language by HilbertMatrix[m, n]. The figure above shows a plot of the Hilbert matrix with elements colored according to their values.Hilbert matrices whose entries are specified as machine-precision numbers are difficult to invert using numerical techniques.The determinants for the first few values of for , 2, ... are given by one divided by 1, 12, 2160, 6048000, 266716800000, ... (OEIS A005249). The terms of sequence have the closed form(2)(3)(4)where is the Glaisher-Kinkelin constant and is the Barnes G-function. The numerical values are given in the following table.det()1123456The elements of the matrix inverse of the Hilbert matrix are given analytically by(5)(Choi 1983, Richardson 1999).

The truncated icosahedron is the 32-faced Archimedean solid corresponding to the facial arrangement . It is the shape used in the construction of soccer balls, and it was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in the Fat Man atomic bomb (Rhodes 1996, p. 195). The truncated icosahedron has 60 vertices, and is also the structure of pure carbon known as buckyballs (a.k.a. fullerenes).The truncated icosahedron is uniform polyhedron and Wenninger model . It has Schläfli symbol t and Wythoff symbol .It is implemented in the Wolfram Languageas PolyhedronData["TruncatedIcosahedron"].Several symmetrical projections of the truncated icosahedron are illustrated above.The dual polyhedron of the truncated icosahedron is the pentakis dodecahedron. The inradius of the dual, midradius of the solid and dual, and circumradius of the solid for are(1)(2)(3)The distances..

Guilloché patterns are spirograph-like curves that frame a curve within an inner and outer envelope curve. They are used on banknotes, securities, and passports worldwide for added security against counterfeiting. For currency, the precise techniques used by the governments of Russia, the United States, Brazil, the European Union, Madagascar, Egypt, and all other countries are likely quite different. The figures above show the same guilloche pattern plotted in polar and Cartesian coordinates generated by a series of nested additions and multiplications of sinusoids of various periods.Guilloché machines (alternately called geometric lathes, rose machines, engine-turners, and cycloidal engines) were first used for a watch casing dated 1624, and consist of myriad gears and settings that can produce many different patterns. Many goldsmiths, including Fabergè, employed guilloché machines.The..

A superellipse is a curve with Cartesian equation(1)first discussed in 1818 by Lamé. A superellipse may be described parametrically by(2)(3)The restriction to is sometimes made.Superellipses with are also known as Lamé curves or Lamé ovals, and the case with is sometimes known as the squircle. By analogy, the superellipse with and might be termed the rectellipse.A range of superellipses are shown above, with special cases , 1, and 2 illustrated right above. The following table summarizes a few special cases. Piet Hein used with a number of different ratios for various of his projects. For example, he used for Sergels Torg (Sergel's Square) in Stockholm, Sweden (Vestergaard), and for his table.curve(squashed) astroid1(squashed) diamond2ellipsePiet Hein's "superellipse"4rectellipseIf is a rational, then a superellipse is algebraic. However, for irrational , it is transcendental. For even integers..

A hypotrochoid generated by a fixed point on a circle rolling inside a fixed circle. The curves above correspond to values of , 0.2, ..., 1.0.Additional attractive designs such as the one above can also be made by superposing individual spirographs.The Season 1 episode "Counterfeit Reality" (2005) of the television crime drama NUMB3RS features spirographs when discussing Guilloché patterns.

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

Ede (1958) enumerates 13 basic series of stellations of the rhombic triacontahedron, the total number of which is extremely large. Pawley (1973) gave a set of restrictions upon which a complete enumeration of stellations can be achieved (Wenninger 1983, p. 36). Messer (1995) describes 227 stellations (including the original solid in the count as usual), some of which are illustrated above.The Great Stella stellation software reproduces Messer's 227 fully supported stellations. Using Miller's rules gives 358833098 stellations, 84959 of them reflexible and 358748139 of them chiral.The convex hull of the dodecadodecahedron is an icosidodecahedron and the dual of the icosidodecahedron is the rhombic triacontahedron, so the dual of the dodecadodecahedron (the medial rhombic triacontahedron) is one of the rhombic triacontahedron stellations (Wenninger 1983, p. 41). Others include the great rhombic triacontahedron,..

A string figure is any pattern produced when a looped string is spanned between two hands and is twisted and woven in various manners around the fingers and the wrists. The combinations of crossings which can be realized in this way can be studied using knot theory.The string figure above is known as the Apache door (Jayne 1975, pp. 12-15, Fig. 21) or tent flap (Ball 1971, p. 5, Fig. 2).The string figure illustrated above is known as "Jacob's ladder," Osage diamonds (Jayne 1975, pp. 24-27, Fig. 50), the fishing net, or quadruple diamonds (Ball 1971, p. 19, Fig. 7).String figures, which belong to the ancient traditions of many peoples around the world, and are even present in primitive cultures, are nowadays considered as a recreational activity in mathematics education. In English-speaking countries they are also known as the children's game called "cat's cradle."..

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

The length of the polygonal spiral is found by noting that the ratio of inradius to circumradius of a regular polygon of sides is(1)The total length of the spiral for an -gon with side length is therefore(2)(3)Consider the solid region obtained by filling in subsequent triangles which the spiral encloses. The area of this region, illustrated above for -gons of side length , is(4)The shaded triangular polygonal spiral is a rep-4-tile.

The Borromean rings, also called the Borromean links (Livingston 1993, p. 10) are three mutually interlocked rings (left figure), named after the Italian Renaissance family who used them on their coat of arms. The configuration of rings is also known as a "Ballantine," and a brand of beer (right figure; Falstaff Brewing Corporation) has been brewed under this name. In the Borromean rings, no two rings are linked, so if any one of the rings is cut, all three rings fall apart. Any number of rings can be linked in an analogous manner (Steinhaus 1999, Wells 1991).The Borromean rings are a prime link. They have link symbol 06-0302, braid word , and are also the simplest Brunnian link.It turns out that rigid Borromean rings composed of real (finite thickness) tubes cannot be physically constructed using three circular rings of either equal or differing radii. However, they can be made from three congruent elliptical rings...

The continuous Fourier transform is definedas(1)(2)Now consider generalization to the case of a discrete function, by letting , where , with , ..., . Writing this out gives the discrete Fourier transform as(3)The inverse transform is then(4)Discrete Fourier transforms (DFTs) are extremely useful because they reveal periodicities in input data as well as the relative strengths of any periodic components. There are however a few subtleties in the interpretation of discrete Fourier transforms. In general, the discrete Fourier transform of a real sequence of numbers will be a sequence of complex numbers of the same length. In particular, if are real, then and are related by(5)for , 1, ..., , where denotes the complex conjugate. This means that the component is always real for real data.As a result of the above relation, a periodic function will contain transformed peaks in not one, but two places. This happens because the periods of the input data..

The Jacobi theta functions are the elliptic analogs of the exponential function, and may be used to express the Jacobi elliptic functions. The theta functions are quasi-doubly periodic, and are most commonly denoted in modern texts, although the notations and (Borwein and Borwein 1987) are sometimes also used. Whittaker and Watson (1990, p. 487) gives a table summarizing notations used by various earlier writers.The theta functions are given in the Wolfram Language by EllipticTheta[n, z, q], and their derivatives are given by EllipticThetaPrime[n, z, q].The translational partition function for an ideal gas can be derived using elliptic theta functions (Golden 1961, pp. 119 and 133; Melzak 1973, p. 122; Levine 2002, p. 838).The theta functions may be expressed in terms of the nome , denoted , or the half-period ratio , denoted , where and and are related by(1)Let the multivalued function be interpreted to stand..

A Pythagorean triple is a triple of positive integers , , and such that a right triangle exists with legs and hypotenuse . By the Pythagorean theorem, this is equivalent to finding positive integers , , and satisfying(1)The smallest and best-known Pythagorean triple is . The right triangle having these side lengths is sometimes called the 3, 4, 5 triangle.Plots of points in the -plane such that is a Pythagorean triple are shown above for successively larger bounds. These plots include negative values of and , and are therefore symmetric about both the x- and y-axes.Similarly, plots of points in the -plane such that is a Pythagorean triple are shown above for successively larger bounds.It is usual to consider only primitive Pythagorean triples (also called "reduced"triples) in which and are relatively prime, since other solutions can be generated trivially from the primitive ones. The primitive triples are illustrated above, and..

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

The simplest class of one-dimensional cellular automata. Elementary cellular automata have two possible values for each cell (0 or 1), and rules that depend only on nearest neighbor values. As a result, the evolution of an elementary cellular automaton can completely be described by a table specifying the state a given cell will have in the next generation based on the value of the cell to its left, the value the cell itself, and the value of the cell to its right. Since there are possible binary states for the three cells neighboring a given cell, there are a total of elementary cellular automata, each of which can be indexed with an 8-bit binary number (Wolfram 1983, 2002). For example, the table giving the evolution of rule 30 () is illustrated above. In this diagram, the possible values of the three neighboring cells are shown in the top row of each panel, and the resulting value the central cell takes in the next generation is shown below in the center...

A totalistic cellular automaton is a cellular automata in which the rules depend only on the total (or equivalently, the average) of the values of the cells in a neighborhood. These automata were introduced by Wolfram in 1983. Like an elementary cellular automaton, the evolution of a one-dimensional totalistic cellular automaton can completely be described by a table specifying the state a given cell will have in the next generation based on the average value of the three cells consisting of the cell to its left, the value the cell itself, and the value of the cell to its right.For a -color one-dimensional totalistic automaton, there are possible states for the average of three cells neighboring a given cell, and a total of -color totalistic cellular automata, each of which can be indexed with an -digit -ary number, known as a "code." For example, the table giving the evolution of the 3-color code is illustrated above. In this diagram,..

The Chebyshev polynomials of the first kind are a set of orthogonal polynomials defined as the solutions to the Chebyshev differential equation and denoted . They are used as an approximation to a least squares fit, and are a special case of the Gegenbauer polynomial with . They are also intimately connected with trigonometric multiple-angle formulas. The Chebyshev polynomials of the first kind are denoted , and are implemented in the Wolfram Language as ChebyshevT[n, x]. They are normalized such that . The first few polynomials are illustrated above for and , 2, ..., 5.The Chebyshev polynomial of the first kind can be defined by the contour integral(1)where the contour encloses the origin and is traversed in a counterclockwise direction (Arfken 1985, p. 416).The first few Chebyshev polynomials of the first kind are(2)(3)(4)(5)(6)(7)(8)When ordered from smallest to largest powers, the triangle of nonzero coefficients is 1; 1; , 2;..

The Pochhammer symbol(1)(2)(Abramowitz and Stegun 1972, p. 256; Spanier 1987; Koepf 1998, p. 5) for is an unfortunate notation used in the theory of special functions for the rising factorial, also known as the rising factorial power (Graham et al. 1994, p. 48) or ascending Factorial (Boros and Moll 2004, p. 16). The Pochhammer symbol is implemented in the Wolfram Language as Pochhammer[x, n].In combinatorics, the notation (Roman 1984, p. 5), (Comtet 1974, p. 6), or (Graham et al. 1994, p. 48) is used for the rising factorial, while or denotes the falling factorial (Graham et al. 1994, p. 48). Extreme caution is therefore needed in interpreting the notations and .The first few values of for nonnegative integers are(3)(4)(5)(6)(7)(OEIS A054654).In closed form, can be written(8)where is a Stirling number of the first kind.The Pochhammer symbol satisfies(9)the dimidiation formulas(10)(11)and..

The curlicue fractal is a figure obtained by the following procedure. Let be an irrational number. Begin with a line segment of unit length, which makes an angle to the horizontal. Then define iteratively bywith . To the end of the previous line segment, draw a line segment of unit length which makes an angleto the horizontal (Pickover 1995ab). The result is a fractal, and the above figures correspond to the curlicue fractals with points for the golden ratio , , , , the Euler-Mascheroni constant , , and the Feigenbaum constant .The temperature of these curves is given in the followingtable.constanttemperaturegolden ratio 46515858Euler-Mascheroni constant 6390Feigenbaum constant 92

The above topological structure, composed of a countable union of compact sets, is called Alexander's horned sphere. It is homeomorphic with the ball , and its boundary is therefore a sphere. It is therefore an example of a wild embedding in . The outer complement of the solid is not simply connected, and its fundamental group is not finitely generated. Furthermore, the set of nonlocally flat ("bad") points of Alexander's horned sphere is a Cantor set.The horned sphere as originally drawn by Alexander (1924) is illustrated above.The complement in of the bad points for Alexander's horned sphere is simply connected, making it inequivalent to Antoine's horned sphere. Alexander's horned sphere has an uncountable infinity of wild points, which are the limits of the sequences of the horned sphere's branch points (roughly, the "ends" of the horns), since any neighborhood of a limit contains a horned complex.A humorous drawing..

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

The greatest common divisor, sometimes also called the highest common divisor (Hardy and Wright 1979, p. 20), of two positive integers and is the largest divisor common to and . For example, , , and . The greatest common divisor can also be defined for three or more positive integers as the largest divisor shared by all of them. Two or more positive integers that have greatest common divisor 1 are said to be relatively prime to one another, often simply just referred to as being "relatively prime."Various notational conventions are summarized in the following table.notationsourcethis work, Zwillinger (1996, p. 91), Råde and Westergren (2004, p. 54)Gellert et al. (1989, p. 25), D'Angelo and West (1990, p. 13), Graham et al. (1990, p. 103), Bressoud and Wagon (2000, p. 7), Yan (2002, p. 30), Bronshtein et al. (2007, pp. 323-324), Wolfram Languageg.c.d.Andrews 1994,..

Inversion is the process of transforming points to a corresponding set of points known as their inverse points. Two points and are said to be inverses with respect to an inversion circle having inversion center and inversion radius if is the perpendicular foot of the altitude of , where is a point on the circle such that .The analogous notation of inversion can be performed in three-dimensional space withrespect to an inversion sphere.If and are inverse points, then the line through and perpendicular to is sometimes called a "polar" with respect to point , known as the "inversion pole". In addition, the curve to which a given curve is transformed under inversion is called its inverse curve (or more simply, its "inverse"). This sort of inversion was first systematically investigated by Jakob Steiner.From similar triangles, it immediately follows that the inverse points and obey(1)or(2)(Coxeter 1969, p. 78),..

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

A root of a polynomial is a number such that . The fundamental theorem of algebra states that a polynomial of degree has roots, some of which may be degenerate. For example, the roots of the polynomial(1)are , 1, and 2. Finding roots of a polynomial is therefore equivalent to polynomial factorization into factors of degree 1.Any polynomial can be numerically factored, althoughdifferent algorithms have different strengths and weaknesses.The roots of a polynomial equation may be found exactly in the Wolfram Language using Roots[lhs==rhs, var], or numerically using NRoots[lhs==rhs, var]. In general, a given root of a polynomial is represented as Root[#^n+a[n-1]#^(n-1)+...+a[0]&, k], where , 2, ..., is an index identifying the particular root and the pure function polynomial is irreducible. Note that in the Wolfram Language, the ordering of roots is different in each of the commands Roots, NRoots, and Table[Root[p, k], k, n].In the Wolfram..

Solving the nome for the parameter gives(1)(2)where is a Jacobi theta function, is the Dedekind eta function, and is the nome.The inverse nome function is essentially the same as the elliptic lambda function, the difference being that elliptic lambda function is a function of the half-period ratio , while the inverse nome is a function of the nome , where is itself a function of .The inverse nome is implemented as InverseEllipticNomeQ[q]in the Wolfram Language.As a rule, inverse and direct functions satisfy the relation -for example, . The inverse nome is an exception to this rule due to a historical mistake made more a century ago. In particular, the inverse nome and nome itself are connected by the opposite relation .Special values include(3)(4)(5)although strictly speaking, is not defined at 1 because is a modular function, therefore has a dense set of singularities on the unit circle, and is therefore only defined strictly inside the unit..

Let and be real numbers (usually taken as and ). The Dirichlet function is defined by(1)and is discontinuous everywhere. The Dirichletfunction can be written analytically as(2)Because the Dirichlet function cannot be plotted without producing a solid blend of lines, a modified version, sometimes itself known as the Dirichlet function (Bruckner et al. 2008), Thomae function (Beanland et al. 2009), or small Riemann function (Ballone 2010, p. 11), can be defined as(3)(Dixon 1991), illustrated above. This function is continuous at irrational and discontinuous at rational (although a small interval around an irrational point contains infinitely many rational points, these rationals will have very large denominators). When viewed from a corner along the line in normal perspective, a quadrant of Euclid's orchard turns into the modified Dirichlet function (Gosper)...

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/).

The tetrix is the three-dimensional analog of the Sierpiński sieve illustrated above, also called the Sierpiński sponge or Sierpiński tetrahedron.The th iteration of the tetrix is implemented in the Wolfram Language as SierpinskiMesh[n, 3].Let be the number of tetrahedra, the length of a side, and the fractional volume of tetrahedra after the th iteration. Then(1)(2)(3)The capacity dimension is therefore(4)(5)so the tetrix has an integer capacity dimension (which is one less than the dimension of the three-dimensional tetrahedra from which it is built), despite the fact that it is a fractal.The following illustrations demonstrate how the dimension of the tetrix can be the same as that of the plane by showing three stages of the rotation of a tetrix, viewed along one of its edges. In the last frame, the tetrix "looks" like the two-dimensional plane. ..

The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable."The" Mandelbrot set is the set obtained from the quadraticrecurrence equation(1)with , where points in the complex plane for which the orbit of does not tend to infinity are in the set. Setting equal to any point in the set that is not a periodic point gives the same result. The Mandelbrot set was originally called a molecule by Mandelbrot. J. Hubbard and A. Douady proved that the Mandelbrot set is connected.A plot of the Mandelbrot set is shown above in which values of in the complex plane are colored according to the number of steps required to reach . The kidney bean-shaped portion of the Mandelbrot set turns out to be bordered by a cardioid with equations(2)(3)The..

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..

The tribox, also called the Penrose rectangle or Penrose square, is an impossible figure that is the generalization of the Penrose triangle from a triangle to a square. Similar -gonal figures can also be constructed for (Elber).The 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.

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

The Gauss map is a function from an oriented surface in Euclidean space to the unit sphere in . It associates to every point on the surface its oriented unit normal vector. Since the tangent space at a point on is parallel to the tangent space at its image point on the sphere, the differential can be considered as a map of the tangent space at into itself. The determinant of this map is the Gaussian curvature, and negative one-half of the trace is the mean curvature.Another meaning of the Gauss map is the function(Trott 2004, p. 44), where is the floor function, plotted above on the real line and in the complex plane.The related function is plotted above, where is the fractional part.The plots above show blowups of the absolute values of these functions (a version of the left figure appears in Trott 2004, p. 44)...

Newton's method, also called the Newton-Raphson method, is a root-finding algorithm that uses the first few terms of the Taylor series of a function in the vicinity of a suspected root. Newton's method is sometimes also known as Newton's iteration, although in this work the latter term is reserved to the application of Newton's method for computing square roots.For a polynomial, Newton's method is essentially the same as Horner's method.The Taylor series of about the point is given by(1)Keeping terms only to first order,(2)Equation (2) is the equation of the tangent line to the curve at , so is the place where that tangent line intersects the -axis. A graph can therefore give a good intuitive idea of why Newton's method works at a well-chosen starting point and why it might diverge with a poorly-chosen starting point.This expression above can be used to estimate the amount of offset needed to land closer to the root starting from an initial guess..

The Lorenz attractor is an attractor that arises in a simplified system of equations describing the two-dimensional flow of fluid of uniform depth , with an imposed temperature difference , under gravity , with buoyancy , thermal diffusivity , and kinematic viscosity . The full equations are(1)(2)Here, is a stream function, defined such that the velocity components of the fluid motion are(3)(4)(Tabor 1989, p. 205).In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified system, periodic solutions of the form(5)(6)grew for Rayleigh numbers larger than the critical value, . Furthermore, vastly different results were obtained for very small changes in the initial values, representing one of the earliest discoveries of the so-called butterfly effect.Lorenz included the terms(7)(8)(9)where is proportional to convective intensity, to the temperature difference..

There are at least two maps known as the Hénon map.The first is the two-dimensional dissipative quadraticmap given by the coupled equations(1)(2)(Hénon 1976).The strange attractor illustrated above is obtained for and .The illustration above shows two regions of space for the map with and colored according to the number of iterations required to escape (Michelitsch and Rössler 1989).The plots above show evolution of the point for parameters (left) and (right).The Hénon map has correlation exponent (Grassberger and Procaccia 1983) and capacity dimension (Russell et al. 1980). Hitzl and Zele (1985) give conditions for the existence of periods 1 to 6.A second Hénon map is the quadratic area-preserving map(3)(4)(Hénon 1969), which is one of the simplest two-dimensional invertible maps...

The fractal-like two-dimensional functionThe function is named for the appearance of a butterfly-like pattern centered around the origin (left figure). In the above illustration, the left plot runs from to 5 and the right plot runs from to 20.

Let be a rational function(1)where , is the Riemann sphere , and and are polynomials without common divisors. The "filled-in" Julia set is the set of points which do not approach infinity after is repeatedly applied (corresponding to a strange attractor). The true Julia set is the boundary of the filled-in set (the set of "exceptional points"). There are two types of Julia sets: connected sets (Fatou set) and Cantor sets (Fatou dust).Quadratic Julia sets are generated by the quadratic mapping(2)for fixed . For almost every , this transformation generates a fractal. Examples are shown above for various values of . The resulting object is not a fractal for (Dufner et al. 1998, pp. 224-226) and (Dufner et al. 1998, pp. 125-126), although it does not seem to be known if these two are the only such exceptional values.The special case of on the boundary of the Mandelbrot set is called a dendrite fractal (top left figure),..

Consider the sequence defined by and , where denotes the reverse of a sequence . The first few terms are then 01, 010110, 010110010110011010, .... All words are cubefree (Allouche and Shallit 2003, p. 28, Ex. 1.49). Iterating gives the sequence 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, ... (OEIS A118006)Plotting (mod 2), where denotes the th digit of the infinitely iterated sequence, gives the beautiful pattern shown above, known as Reverend Back's abbey floor (Wegner 1982; Siromoney and Subramanian 1983; Allouche and Shallit 2003, pp. 410-411). Note that this plot is identical to the recurrence plot (mod 2).

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

A Julia set fractal obtainedby iterating the functionwhere is the sign function and is the real part of . The plot above sets and uses a maximum of 50 iterations with escape radius 2.

The pentaflake is a fractal with 5-fold symmetry. As illustrated above, five pentagons can be arranged around an identical pentagon to form the first iteration of the pentaflake. This cluster of six pentagons has the shape of a pentagon with five triangular wedges removed. This construction was first noticed by Albrecht Dürer (Dixon 1991).For a pentagon of side length 1, the first ring of pentagons has centers at radius(1)where is the golden ratio. The inradius and circumradius are related by(2)and these are related to the side length by(3)The height is(4)giving a radius of the second ring as(5)Continuing, the th pentagon ring is located at(6)Now, the length of the side of the first pentagon compound is given by(7)so the ratio of side lengths of the original pentagon to that of the compound is(8)We can now calculate the dimension of the pentaflake fractal. Let be the number of black pentagons and the length of side of a pentagon after the iteration,(9)(10)The..

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

Two integers are relatively prime if they share no common positive factors (divisors) except 1. Using the notation to denote the greatest common divisor, two integers and are relatively prime if . Relatively prime integers are sometimes also called strangers or coprime and are denoted . The plot above plots and along the two axes and colors a square black if and white otherwise (left figure) and simply colored according to (right figure).Two numbers can be tested to see if they are relatively prime in the Wolfram Language using CoprimeQ[m, n].Two distinct primes and are always relatively prime, , as are any positive integer powers of distinct primes and , .Relative primality is not transitive. For example, and , but .The probability that two integers and picked at random are relatively prime is(1)(OEIS A059956; Cesàro and Sylvester 1883; Lehmer 1900; Sylvester 1909; Nymann 1972; Wells 1986, p. 28; Borwein and Bailey 2003, p. 139;..

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

The Delannoy numbers are the number of lattice paths from to in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed (i.e., , , and ). They are given by the recurrence relation(1)with . The are also given by the sums(2)(3)(4)where is a hypergeometric function.A table for values for the Delannoy numbers is given by(5)(OEIS A008288) for , 1, ... increasing from left to right and , 1, ... increasing from top to bottom.They have the generating function(6)(Comtet 1974, p. 81).Taking gives the central Delannoy numbers , which are the number of "king walks" from the corner of an square to the upper right corner . These are given by(7)where is a Legendre polynomial (Moser 1955; Comtet 1974, p. 81; Vardi 1991). Another expression is(8)(9)(10)where is a binomial coefficient and is a hypergeometric function. These numbers have a surprising connection with the Cantor set (E. W. Weisstein, Apr. 9,..

There are two sorts of transforms known as the fractional Fourier transform.The linear fractional Fourier transform is a discrete Fourier transform in which the exponent is modified by the addition of a factor ,However, such transforms may not be consistent with their inverses unless is an integer relatively prime to so that . Fractional fourier transforms are implemented in the Wolfram Language as Fourier[list, FourierParameters -> a, b], where is an additional scaling parameter. For example, the plots above show 2-dimensional fractional Fourier transforms of the function for parameter ranging from 1 to 6.The quadratic fractional Fourier transform is defined in signal processing and optics. Here, the fractional powers of the ordinary Fourier transform operation correspond to rotation by angles in the time-frequency or space-frequency plane (phase space). So-called fractional Fourier domains correspond to oblique axes in..

A recurrence plot is defined as a plot of the quantitywhere is the Heaviside step function and denotes a norm. A recurrence plot is therefore a binary plot. The figure above shows a recurrence plot for the Lorenz attractor with , , , , , , and .Recurrence plots were initially used to graphically display nonstationarity in time series (Eckmann et al. 1987, Gao and Cai 2000), but are also useful for visualizing functions.A so-called global recurrence plot or unthresholded recurrence plot of a function is a plot of (or ) in the - plane. Recurrence plots for a number of common functions are illustrated above.

Strang's strange figures are the figures produced by plotting a periodic function as a function of an integer argument for , 2, .... Unexpected patterns and periodicities result from near-commensurabilities of certain rational numbers with the period (Richert 1992). Strang figures are shown above for a number of common functions.

A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated structures and interconnections (Knopp 1996, pp. 98-99). Riemann surfaces are one way of representing multiple-valued functions; another is branch cuts. The above plot shows Riemann surfaces for solutions of the equationwith , 3, 4, and 5, where is the Lambert W-function (M. Trott).The Riemann surface of the function field is the set of nontrivial discrete valuations on . Here, the set corresponds to the ideals of the ring of integers of over . ( consists of the elements of that are roots of monic polynomials over .) Riemann surfaces provide a geometric visualization of functions elements and their analytic continuations.Schwarz proved at the end of nineteenth century that the automorphism group of a compact Riemann surface of genus is finite,..

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