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For a logarithmic spiral given parametricallyas(1)(2)evolute is given by(3)(4)As first shown by Johann Bernoulli, the evolute of a logarithmic spiral is therefore another logarithmic spiral, having and ,In some cases, the evolute is identical to the original,as can be demonstrated by making the substitution to the new variable(5)Then the above equations become(6)(7)(8)(9)which are equivalent to the form of the original equation if(10)(11)(12)where only solutions with the minus sign in exist. Solving gives the values summarized in the following table.10.2744106319...20.1642700512...30.1218322508...40.0984064967...50.0832810611...60.0725974881...70.0645958183...80.0583494073...90.0533203211...100.0491732529...

According to G. Pólya, the method of finding geometric objects by intersection. 1. For example, the centers of all circles tangent to a straight line at a given point lie on a line that passes through and is perpendicular to . 2. In addition, the circle centered at with radius is the locus of the centers of all circles of radius passing through . The intersection of and consists of two points and which are the centers of two circles of radius tangent to at .Many constructions with straightedge and compass are based on this method, as, for example, the construction of the center of a given circle by means of the perpendicular bisector theorem.

The turning of an object or coordinate system by an angle about a fixed point. A rotation is an orientation-preserving orthogonal transformation. Euler's rotation theorem states that an arbitrary rotation can be parameterized using three parameters. These parameters are commonly taken as the Euler angles. Rotations can be implemented using rotation matrices.Rotation in the plane can be concisely described in the complex plane using multiplication of complex numbers with unit modulus such that the resulting angle is given by . For example, multiplication by represents a rotation to the right by and by represents rotation to the left by . So starting with and rotating left twice gives , which is the same as rotating right twice, , and . For multiplication by multiples of , the possible positions are then concisely represented by , , , and .The rotation symmetry operation for rotation by is denoted "." For periodic arrangements of points..

A paradox mentioned in the Greek work Mechanica, dubiously attributed to Aristotle. Consider the above diagram depicting a wheel consisting of two concentric circles of different diameters (a wheel within a wheel). There is a 1:1 correspondence of points on the large circle with points on the small circle, so the wheel should travel the same distance regardless of whether it is rolled from left to right on the top straight line or on the bottom one. this seems to imply that the two circumferences of different sized circles are equal, which is impossible.The fallacy lies in the assumption that a one-to-one correspondence of points means that two curves must have the same length. In fact, the cardinalities of points in a line segment of any length (or even an infinite line, a plane, a three-dimensional space, or an infinite dimensional Euclidean space) are all the same: (the cardinality of the continuum), so the points of any of these can be put in a one-to-one..

After a half rotation of the coin on the left around the central coin (of the same radius), the coin undergoes a complete rotation. In other words, a coin makes two complete rotations when rolled around the boundary of an identical coin. This fact is readily apparent in the generation of the cardioid as one disk rolling on another.

The equation of the curve of intersection of a torus with a plane perpendicular to both the midplane of the torus and to the plane . (The general intersection of a torus with a plane is called a toric section). Let the tube of a torus have radius , let its midplane lie in the plane, and let the center of the tube lie at a distance from the origin. Now cut the torus with the plane . The equation of the torus with gives the equation(1)(2)(3)The above plots show a series of spiric sections for the ring torus, horn torus, and spindle torus, respectively. When , the curve consists of two circles of radius whose centers are at and . If , the curve consists of one point (the origin), while if , no point lies on the curve.The spiric extensions are an extension of the conic sections constructed by Menaechmus around 150 BC by cutting a cone by a plane, and were first considered around 50 AD by the Greek mathematician Perseus (MacTutor).If , then (3) simplifies to(4)which is the..

Whirls are figures constructed by nesting a sequence of polygons (each having the same number of sides), each slightly smaller and rotated relative to the previous one. The vertices give the path of the mice in the mice problem, and form logarithmic spirals.The square whirl appears on the cover of Freund (1993).

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.

The meeting point of lines that connect corresponding points from homothetic figures. In the above figure, is the homothetic center of the homothetic figures and . For figures which are similar but do not have parallel sides, a similitude center exists (Johnson 1929, pp. 16-20).Given two nonconcentric circles, draw radii parallel and in the same direction. Then the line joining the extremities of the radii passes through a fixed point on the line of centers which divides that line externally in the ratio of radii. This point is called the external homothetic center, or external center of similitude (Johnson 1929, pp. 19-20 and 41).If radii are drawn parallel but instead in opposite directions, the extremities of the radii pass through a fixed point on the line of centers which divides that line internally in the ratio of radii (Johnson 1929, pp. 19-20 and 41). This point is called the internal homothetic center, or internal..

A toric section is a curve obtained by slicing a torus (generally a horn torus) with a plane. A spiric section is a special case of a toric section in which the slicing plane is perpendicular to both the midplane of the torus and to the plane .Consider a torus with tube radius . For a cutting plane parallel to the -plane, the toric section is either a single circle (for ) or two concentric circles (for ). For planes containing the z-axis, the section is two equal circles.Toric sections at oblique angles can be more complicated, passing from a crescent shape, through a U-shape, and into two disconnected kidney-shaped curves.

The problem of finding the mean triangle area of a triangle with vertices picked inside a triangle with unit area was proposed by Watson (1865) and solved by Sylvester. It solution is a special case of the general formula for polygon triangle picking.Since the problem is affine, it can be solved by considering for simplicity an isosceles right triangle with unit leg lengths. Integrating the formula for the area of a triangle over the six coordinates of the vertices (and normalizing to the area of the triangle and region of integration by dividing by the integral of unity over the region) gives(1)(2)where(3)is the triangle area of a triangle with vertices , , and .The integral can be solved using computer algebra by breaking up the integration region using cylindrical algebraic decomposition. This results in 62 regions, 30 of which have distinct integrals, each of which can be directly integrated. Combining the results then gives the result(4)(Pfiefer..

Pick three points , , and distributed independently and uniformly in a unit disk (i.e., in the interior of the unit circle). Then the average area of the triangle determined by these points is(1)Using disk point picking, this can be writtenas(2)where(3)A trigonometric substitution can then be used to remove the trigonometric functions and split the integral into(4)where(5)(6)However, the easiest way to evaluate the integral is using Crofton's formula and polar coordinates to yield a mean triangle area(7)for unit-radius disks (OEIS A189511), or(8)for unit-area disks (OEIS A093587; Woolhouse 1867; Solomon 1978; Pfiefer 1989; Zinani 2003). This problem is very closely related to Sylvester's four-point problem, and can be derived as the limit as of the general polygon triangle picking problem.The distribution of areas, illustrated above, is apparently not known exactly.The probability that three random points in a disk form an acute..

The homotopy groups generalize the fundamental group to maps from higher dimensional spheres, instead of from the circle. The th homotopy group of a topological space is the set of homotopy classes of maps from the n-sphere to , with a group structure, and is denoted . The fundamental group is , and, as in the case of , the maps must pass through a basepoint . For , the homotopy group is an Abelian group.The group operations are not as simple as those for the fundamental group. Consider two maps and , which pass through . The product is given by mapping the equator to the basepoint . Then the northern hemisphere is mapped to the sphere by collapsing the equator to a point, and then it is mapped to by . The southern hemisphere is similarly mapped to by . The diagram above shows the product of two spheres.The identity element is represented by the constant map . The choice of direction of a loop in the fundamental group corresponds to a manifold orientation of in a homotopy..

Two mathematical objects are said to be homotopic if one can be continuously deformed into the other. For example, the real line is homotopic to a single point, as is any tree. However, the circle is not contractible, but is homotopic to a solid torus. The basic version of homotopy is between maps. Two maps and are homotopic if there is a continuous mapsuch that and .Whether or not two subsets are homotopic depends on the ambient space. For example, in the plane, the unit circle is homotopic to a point, but not in the punctured plane . The puncture can be thought of as an obstacle.However, there is a way to compare two spaces via homotopy without ambient spaces. Two spaces and are homotopy equivalent if there are maps and such that the composition is homotopic to the identity map of and is homotopic to the identity map of . For example, the circle is not homotopic to a point, for then the constant map would be homotopic to the identity map of a circle, which is impossible..

The stomachion is a 14-piece dissection puzzle similar to tangrams. It is described in fragmentary manuscripts attributed to Archimedes as noted by Magnus Ausonius (310-395 A.D.). The puzzle is also referred to as the "loculus of Archimedes" (Archimedes' box) or "syntemachion" in Latin texts. The word stomachion has as its root the Greek word , meaning "stomach." Note that Ausonius refers to the figure as the "ostomachion," an apparent corruption of the original Greek.The puzzle consists of 14 flat pieces of various shapes arranged in the shape of a square, with the vertices of pieces occurring on a grid. Two pairs of pieces are duplicated. Like tangrams, the object is to rearrange the pieces to form interesting shapes such as the elephant illustrated above (Andrea).Taking the square as having edge lengths 12, the pieces have areas 3, 3, 6, 6, 6, 6, 9, 12, 12, 12, 12, 12, 21, and 24, giving them relative..

Consider the average length of a line segment determined by two points picked at random in the interior of an arbitrary triangle. This problem is not affine, so a simple formula in terms of the area or linear properties of the original triangle apparently does not exist.However, if the original triangle is chosen to be an isosceles right triangle with unit legs, then the average length of a line with endpoints chosen at random inside it is given by(1)(2)(3)(OEIS A093063; M. Trott, pers. comm., Mar. 10, 2004), which is numerically surprisingly close to .Similarly, if the original triangle is chosen to be an equilateral triangle with unit side lengths, then the average length of a line with endpoints chosen at random inside it is given by(4)(5)The integrand can be split up into the four pieces(6)(7)(8)(9)As illustrated above, symmetry immediately gives and , so(10)With some effort, the integrals and can be done analytically to give..

Square triangle picking is the selection of triples of points (corresponding to endpoints of a triangle) randomly placed inside a square. random triangles can be picked in a unit square in the Wolfram Language using the function RandomPoint[Rectangle[], n, 3].Given three points chosen at random inside a unit square, the average area of the triangle determined by these points is given analytically by the multiple integrals(1)(2)Here, represent the polygon vertices of the triangle for , 2, 3, and the (signed) area of these triangles is given by the determinant(3)(4)The solution was first given by Woolhouse (1867). Since attempting to do the integrals by brute force result in intractable integrands, the best approach using computer algebra is to divide the six-dimensional region of integration into subregions using cylindrical algebraic decomposition such that the sign of does not change, do the integral in each region directly, and then..

Using disk point picking,(1)(2)for , , choose two points at random in a unit disk and find the distribution of distances between the two points. Without loss of generality, take the first point as and the second point as . Then(3)(4)(5)(6)(OEIS A093070; Uspensky 1937, p. 258;Solomon 1978, p. 36).This is a special case of ball line picking with , so the full probability function for a disk of radius is(7)(Solomon 1978, p. 129).The raw moments of the distribution of line lengthsare given by(8)(9)where is the gamma function and . The expected value of is given by , giving(10)(Solomon 1978, p. 36). The first few moments are then(11)(12)(13)(14)(15)(16)(OEIS A093526 and A093527 and OEIS A093528 and A093529). The moments that are integers occur at , 2, 6, 15, 20, 28, 42, 45, 66, ... (OEIS A014847), which rather amazingly are exactly the values of such that , where is a Catalan number (E. Weisstein, Mar. 30, 2004)...

The combination of a central dilation and a rotation about the same center. However, the combination of a central dilation and a rotation whose centers are distinct is also a spiral symmetry. In fact, any two directly similar figures are related either by a translation or by a spiral symmetry (Coxeter and Greitzer 1967, p. 97).A spiral similarity tessellation of any ordinary tessellation can be constructed by placing a series of polygonal tiles of decreasing size on an equilateral spiral.

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 piecewise polynomial function that can have a locally very simple form, yet at the same time be globally flexible and smooth. Splines are very useful for modeling arbitrary functions, and are used extensively in computer graphics.Cubic splines are implemented in the Wolfram Language as BSplineCurve[pts, SplineDegree -> 3] (red), Bézier curves as BezierCurve[pts] (blue), and B-splines as BSplineCurve[pts].

The curve traced by a fixed point on a closed convex curve as that curve rolls without slipping along a second curve. The roulettes described by the foci of conics when rolled upon a line are sections of minimal surfaces (i.e., they yield minimal surfaces when revolved about the line) known as unduloids.A particularly interesting case of a roulette is a regular -gon rolling on a "road" composed of a sequence of truncated catenaries, as illustrated above. This motion is smooth in the sense that the geometric centroid follows a straight line, although in the case of the rolling equilateral triangle, a physical model would be impossible to construct because the vertices of the triangles would get "stuck" in the ruts (Wagon 2000). For the rolling square, the shape of the road is the catenary truncated at (Wagon 2000). For a regular -gon, the Cartesian equation of the corresponding catenary iswhereThe roulette consisting of a square..

The curve produced by fixed point on the circumference of a small circle of radius rolling around the inside of a large circle of radius . A hypocycloid is therefore a hypotrochoid with .To derive the equations of the hypocycloid, call the angle by which a point on the small circle rotates about its center , and the angle from the center of the large circle to that of the small circle . Then(1)so(2)Call . If , then the first point is at minimum radius, and the Cartesian parametric equations of the hypocycloid are(3)(4)(5)(6)If instead so the first point is at maximum radius (on the circle), then the equations of the hypocycloid are(7)(8)The curvature, arc length, and tangential angle of a hypocycloid are given by(9)(10)(11)An -cusped hypocycloid has . For an integer and with , the equations of the hypocycloid therefore become(12)(13)and the arc length and area are therefore(14)(15)A 2-cusped hypocycloid is a line segment (Steinhaus 1999, p. 145;..

The path traced out by a point on the edge of a circle of radius rolling on the outside of a circle of radius . An epicycloid is therefore an epitrochoid with . Epicycloids are given by the parametric equations(1)(2)A polar equation can be derived by computing(3)(4)so(5)But(6)so(7)(8)Note that is the parameter here, not the polar angle. The polar angle from the center is(9)To get cusps in the epicycloid, , because then rotations of bring the point on the edge back to its starting position.(10)(11)(12)(13)so(14)(15)An epicycloid with one cusp is called a cardioid, one with two cusps is called a nephroid, and one with five cusps is called a ranunculoid.Epicycloids can also be constructed by beginning with the diameter of a circle and offsetting one end by a series of steps of equal arc length along the circumference while at the same time offsetting the other end along the circumference by steps times as large. After traveling around the circle once,..

Given the closed interval with , let one-dimensional "cars" of unit length be parked randomly on the interval. The mean number of cars which can fit (without overlapping!) satisfies(1)The mean density of the cars for large is(2)(3)(4)(OEIS A050996). While the inner integral canbe done analytically,(5)(6)where is the Euler-Mascheroni constant and is the incomplete gamma function, it is not known how to do the outer one(7)(8)(9)where is the exponential integral. The slowly converging series expansion for the integrand is given by(10)(OEIS A050994 and A050995).In addition,(11)for all (Rényi 1958), which was strengthened by Dvoretzky and Robbins (1964) to(12)Dvoretzky and Robbins (1964) also proved that(13)Let be the variance of the number of cars, then Dvoretzky and Robbins (1964) and Mannion (1964) showed that(14)(15)(16)(OEIS A086245), where(17)(18)and the numerical value is due to Blaisdell and Solomon..

The curve given by the polar equation(1)sometimes also written(2)where .The cardioid has Cartesian equation(3)and the parametric equations(4)(5)The cardioid is a degenerate case of the limaçon. It is also a 1-cusped epicycloid (with ) and is the catacaustic formed by rays originating at a point on the circumference of a circle and reflected by the circle.The cardioid has a cusp at the origin.The name cardioid was first used by de Castillon in Philosophical Transactions of the Royal Society in 1741. Its arc length was found by la Hire in 1708. There are exactly three parallel tangents to the cardioid with any given gradient. Also, the tangents at the ends of any chord through the cusp point are at right angles. The length of any chord through the cusp point is .The cardioid may also be generated as follows. Draw a circle and fix a point on it. Now draw a set of circles centered on the circumference of and passing through . The envelope of these circles..

The 2-cusped epicycloid is called a nephroid. The name nephroid means "kidney shaped" and was first used for the two-cusped epicycloid by Proctor in 1878 (MacTutor Archive).The nephroid is the catacaustic for rays originating at the cusp of a cardioid and reflected by it. In addition, Huygens showed in 1678 that the nephroid is the catacaustic of a circle when the light source is at infinity, an observation which he published in his Traité de la luminère in 1690 (MacTutor Archive). (Trott 2004, p. 17, mistakenly states that the catacaustic for parallel light falling on any concave mirror is a nephroid.) The shape of the "flat visor curve" produced by a pop-up card dubbed the "knight's visor" is half a nephroid (Jakus and O'Rourke 2012).Since the nephroid has cusps, , and the equation for in terms of the parameter is given by epicycloid equation(1)with ,(2)where(3)This can be written(4)The..

The trefoil knot , also called the threefoil knot or overhand knot, is the unique prime knot with three crossings. It is a (3, 2)-torus knot and has braid word . The trefoil and its mirror image are not equivalent, as first proved by Dehn (1914). In other words, the trefoil knot is not amphichiral. It is, however, invertible, and has Arf invariant 1.Its laevo form is implemented in the WolframLanguage, as illustrated above, as KnotData["Trefoil"].M. C. Escher's woodcut "Knots" (Bool et al. 1982, pp. 128 and 325; Forty 2003, Plate 71) depicts three trefoil knots composed of differing types of strands. A preliminary study (Bool et al. 1982, p. 123) depicts another trefoil.The animation above shows a series of gears arranged along a Möbiusstrip trefoil knot (M. Trott).The bracket polynomial can be computed as follows.(1)(2)Plugging in(3)(4)gives(5)The corresponding Kauffman polynomial..

The three circumcircles through the triangle centroid of a given triangle and the pairs of the vertices of the second Brocard triangle are called the McCay circles (Johnson 1929, p. 306).The circumcircle of their centers (i.e., of the second Brocard triangle) is therefore the Brocard circle.The -McCay circle has center functionand radius,1/3 that of the Neuberg circle, where is the Brocard angle (Johnson 1929, p. 307).If the polygon vertex of a triangle describes a Neuberg circle , then its triangle centroid describes one of the McCay circles (Johnson 1929, p. 290). In the above figure, the inner triangle is the second Brocard triangle of , whose two indicated edges are concyclic with on the McCay circle.

For a unit circle with parametricequations(1)(2)the negative pedal curve with respect to the pedal point is(3)(4)Therefore if the point is inside the circle (), the negative pedal is an ellipse, if , it is a single point, if the point is outside the circle (), the negative pedal is a hyperbola.

Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality, the first point can be taken as , and the second by , with (by symmetry, the range can be limited to instead of ). The distance between the two points is then(1)The average distance is then given by(2)The probability density function is obtained from(3)The raw moments are then(4)(5)(6)giving the first few as(7)(8)(9)(10)(11)(OEIS A000984 and OEIS A093581 and A001803), where the numerators of the odd terms are 4 times OEIS A061549.The central moments are(12)(13)(14)giving the skewness and kurtosisexcess as(15)(16)Bertrand's problem asks for the probability that a chord drawn at random on a circle of radius has length .

The probability that random arcs of angular size cover the circumference of a circle completely (for a circle with unit circumference) iswhere is the floor function (Solomon 1978, p. 75). This was first given correctly by Stevens (1939), although partial results were obtains by Whitworth (1897), Baticle (1935), Garwood (1940), Darling (1953), and Shepp (1972).The probability that arcs leave exactly gaps is given by(Stevens 1939; Solomon 1978, pp. 75-76).

Given a parabola with parametricequations(1)(2)the negative pedal curve for a pedal point has equation(3)(4)Taking the pedal point at the origin gives(5)(6)which is a semicubical parabola. Similarly, taking gives(7)(8)which is a Tschirnhausen cubic.

A curve of constant width constructed by drawing arcs from each polygon vertex of an equilateral triangle between the other two vertices. The Reuleaux triangle has the smallest area for a given width of any curve of constant width. Let the arc radius be . Since the area of each meniscus-shaped portion of the Reuleaux triangle is a circular segment with opening angle ,(1)(2)But the area of the central equilateral triangle with is(3)so the total area is then(4)Because it can be rotated inside a square, as illustratedabove, it is the basis for the Harry Watt square drill bit.When rotated inside a square of side length 2 having corners at ), the envelope of the Reuleaux triangle is a region of the square with rounded corners. At the corner , the envelope of the boundary is given by the segment of the ellipse with parametric equations(5)(6)for , extending a distance from the corner (Gleißner and Zeitler 2000). The ellipse has center , semimajor axis..

A convolution is an integral that expresses the amount of overlap of one function as it is shifted over another function . It therefore "blends" one function with another. For example, in synthesis imaging, the measured dirty map is a convolution of the "true" CLEAN map with the dirty beam (the Fourier transform of the sampling distribution). The convolution is sometimes also known by its German name, faltung ("folding").Convolution is implemented in the Wolfram Language as Convolve[f, g, x, y] and DiscreteConvolve[f, g, n, m].Abstractly, a convolution is defined as a product of functions and that are objects in the algebra of Schwartz functions in . Convolution of two functions and over a finite range is given by(1)where the symbol denotes convolution of and .Convolution is more often taken over an infinite range,(2)(3)(Bracewell 1965, p. 25) with the variable (in this case ) implied, and also occasionally..

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..

Given two circles with one interior to the other, if small tangent circles can be inscribed around the region between the two circles such that the final circle is tangent to the first, the circles form a Steiner chain.The simplest way to construct a Steiner chain is to perform an inversion on a symmetrical arrangement on circles packed between a central circle of radius and an outer concentric circle of radius (Wells 1991). In this arrangement,(1)so the ratio of the radii for the small and large circles is(2)In addition, the radii of the circles in the ring are(3)and their centers are located at a distance(4)from the origin.To transform the symmetrical arrangement into a Steiner chain, take an inversion center which is a distance from the center of the symmetrical figure. Then the radii and of the outer and center circles become(5)(6)respectively. Equivalently, a Steiner chain results whenever the inversivedistance between the two original..

Clean tile is a game investigated by Buffon (1777) in which players bet on the number of different tiles a thrown coin will partially cover on a floor that is regularly tiled. Buffon investigated the probabilities on a triangular grid, square grid, hexagonal grid, and grid composed of rhombi. Assume that the side length of the tile is greater than the coin diameter . Then, on a square grid, it is possible for a coin to land so that it partially covers 1, 2, 3, or 4 tiles. On a triangular grid, it can land on 1, 2, 3, 4, or 6 tiles. On a hexagonal grid, it can land on 1, 2, or 3 tiles.Special cases of this game give the Buffon-Laplace needle problem (for a square grid) and Buffon's needle problem (for infinite equally spaced parallel lines).As shown in the figure above, on a square grid with tile edge length , the probability that a coin of diameter will lie entirely on a single tile (indicated by yellow disks in the figure) is given by(1)since the shortening of the side..

A point lattice is a regularly spaced array of points.In the plane, point lattices can be constructed having unit cells in the shape of a square, rectangle, hexagon, etc. Unless otherwise specified, point lattices may be taken to refer to points in a square array, i.e., points with coordinates , where , , ... are integers. Such an array is often called a grid or a mesh.Point lattices are frequently simply called "lattices," which unfortunately conflicts with the same term applied to ordered sets treated in lattice theory. Every "point lattice" is a lattice under the ordering inherited from the plane, although a point lattice may not be a sublattice of the plane, since the infimum operation in the plane need not agree with the infimum operation in the point lattice. On the other hand, many lattices are not point lattices.Properties of lattice are implemented in the Wolfram Language as LatticeData[lattice, prop].Formally,..

The game of life is the best-known two-dimensional cellular automaton, invented by John H. Conway and popularized in Martin Gardner's Scientific American column starting in October 1970. The game of life was originally played (i.e., successive generations were produced) by hand with counters, but implementation on a computer greatly increased the ease of exploring patterns.The life cellular automaton is run by placing a number of filled cells on a two-dimensional grid. Each generation then switches cells on or off depending on the state of the cells that surround it. The rules are defined as follows. All eight of the cells surrounding the current one are checked to see if they are on or not. Any cells that are on are counted, and this count is then used to determine what will happen to the current cell. 1. Death: if the count is less than 2 or greater than 3, the current cell is switched off. 2. Survival: if (a) the count is exactly 2, or (b) the count..

Rule 102 is one of the elementary cellular automaton rules introduced by Stephen Wolfram in 1983 (Wolfram 1983, 2002). It specifies the next color in a cell, depending on its color and its immediate neighbors. Its rule outcomes are encoded in the binary representation . This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (OEIS A075439; Wolfram 2002, p. 55).Starting with a single black cell, successive generations are given by interpreting the numbers 1, 6, 20, 120, 272, 1632, 5440, 32640, ... (OEIS A117998) in binary. Omitting trailing zeros (the right cells in step of the triangle are always 0) gives 1, 3, 5, 15, 17, 51, 85, 255, 257, 771, ... (OEIS A001317), which are simply the previous numbers divided by and which have binary representation 1, 11; 101, 1111, 10001, ... (OEIS A047999). Surprisingly, this is precisely the Sierpiński sieve.The mirror image is rule 60, the complement..

Rule 90 is one of the elementary cellular automaton rules introduced by Stephen Wolfram in 1983 (Wolfram 1983, 2002). It specifies the next color in a cell, depending on its color and its immediate neighbors. Its rule outcomes are encoded in the binary representation . This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (Wolfram 2002, p. 55).Starting with a single black cell, successive generations are given by interpreting the numbers 1, 5, 17, 85, 257, 1285, 4369, 21845, ... (OEIS A038183) in binary, namely as 1, 101, 10001, 1010101, 100000001, ... (OEIS A070886).Rule 90 is amphichiral, and its complement is rule165.The fractal produced by this rule was described by Sierpiński in 1915 and appearing in Italian art from the 13th century (Wolfram 2002, p. 43). It is therefore also known as the Sierpiński sieve, Sierpiński gasket, or Sierpiński triangle...

Rule 60 is one of the elementary cellular automaton rules introduced by Stephen Wolfram in 1983 (Wolfram 1983, 2002). It specifies the next color in a cell, depending on its color and its immediate neighbors. Its rule outcomes are encoded in the binary representation . This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (OEIS A075438; Wolfram 2002, p. 55).Starting with a single black cell, successive generations are given by interpreting the numbers 1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, ... (OEIS A001317) in binary (where left cells in step of the triangle are always 0), namely 1, 11, 101, 1111, 10001; ... (OEIS A047999).The mirror image is rule 102, the complement is rule 195, and the mirrored complement is rule 153.Rule 60 is one of the eight additive elementary cellular automata (Wolfram 2002, p. 952)...

A cellular automaton is a collection of "colored" cells on a grid of specified shape that evolves through a number of discrete time steps according to a set of rules based on the states of neighboring cells. The rules are then applied iteratively for as many time steps as desired. von Neumann was one of the first people to consider such a model, and incorporated a cellular model into his "universal constructor." Cellular automata were studied in the early 1950s as a possible model for biological systems (Wolfram 2002, p. 48). Comprehensive studies of cellular automata have been performed by S. Wolfram starting in the 1980s, and Wolfram's fundamental research in the field culminated in the publication of his book A New Kind of Science (Wolfram 2002) in which Wolfram presents a gigantic collection of results concerning automata, among which are a number of groundbreaking new discoveries.The Season 2 episode..

A two-dimensional binary () totalistic cellular automaton with a von Neumann neighborhood of range . It has a birth rule that at least 2 of its 4 neighbors are alive, and a survival rule that all cells survive. steps of bootstrap percolation on an grid with random initial condition of density can be implemented in the Wolfram Language asWith[{n = 10, p = 0.1, s = 20}, CellularAutomaton[ {1018, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, Table[If[Random[Real] < p, 1, 0], {s}, {s}], n ]]If the initial condition consists of a random sparse arrangement of cells with density , then the system seems to quickly converge to a steady state of rectangular islands of live cells surrounded by a sea of dead cells. However, as crosses some threshold on finite-sized grids, the behavior appears to change so that every cell becomes live. Several examples are shown above on three grids with random initial conditions and different starting densities.However, this..

Rule 150 is one of the elementary cellular automaton rules introduced by Stephen Wolfram in 1983 (Wolfram 1983, 2002). It specifies the next color in a cell, depending on its color and its immediate neighbors. Its rule outcomes are encoded in the binary representation . This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (Wolfram 2002, p. 55).Starting with a single black cell, successive generations are given by interpreting the numbers 1, 7, 21, 107, 273, 1911, 5189, ... (OEIS A038184) in binary, namely 1, 111, 10101, 1101011, 100010001, ... (OEIS A118110).Rule 150 is one of the eight additive elementary cellular automata (Wolfram 2002, p. 952).

An additive cellular automaton is a cellular automaton whose rule is compatible with an addition of states. Typically, this addition is derived from modular arithmetic. Additive rules allow the evolution for different initial conditions to be computed independently, then the results combined by simply adding. The results for arbitrary starting conditions can therefore be computed very efficiently by convolving the evolution of a single cell with an appropriate convolution kernel (which, in the case of two-color automata, would correspond to the set of initially "active" cells).A simple example of an additive cellular automaton is provided by the rule 90 elementary cellular automaton. As can be seen from the graphical representation of this rule, the rule as a function of left, central, and right neighbors is simply given by the sum of the rules for the left and right neighbors taken modulo 2, where white cells are assigned the..

From the point of view of coordinate charts, the notion of tangent space is quite simple. The tangent space consists of all directions, or velocities, a particle can take. In an open set in there are no constraints, so the tangent space at a point is another copy of . The set could be a coordinate chart for an -dimensional manifold.The tangent space at , denoted , is the set of possible velocity vectors of paths through . Hence there is a canonical vector basis: if are the coordinates, then are a basis for the tangent space, where is the velocity vector of a particle with unit speed moving inward along the coordinate . The collection of all tangent vectors to every point on the manifold, called the tangent bundle, is the phase space of a single particle moving in the manifold .It seems as if the tangent space at is the same as the tangent space at all other points in the chart . However, while they do share the same dimension and are isomorphic, in a change of coordinates,..

The idea of a velocity vector comes from classical physics. By representing the position and motion of a single particle using vectors, the equations for motion are simpler and more intuitive. Suppose the position of a particle at time is given by the position vector . Then the velocity vector is the derivative of the position,For example, suppose a particle is confined to the plane and its position is given by . Then it travels along the unit circle at constant speed. Its velocity vector is . In a diagram, it makes sense to translate the velocity vector so it originates at . In particular, it is drawn as an arrow from to .Another example is a particle traveling along a hyperbola specified parametrically by . Its velocity vector is then given by , illustrated above.Travel down the same path, but using a different function is called a reparameterization, and the chain rule describes the change in velocity. For example, the hyperbola can also be parametrized..

A particle is said to be undergoing uniform circular motion if its radius vector in appropriate coordinates has the form , where(1)(2)Geometrically, uniform circular motions means that moves in a circle in the -plane with some radius at constant speed. The quantity is called the angular velocity of . The speed of is(3)and the acceleration of P has constant magnitude(4)and is directed toward the center of the circle traced by . This is called centripetal acceleration.Ignoring the ellipticity of their orbits, planet show nearly uniform circular motion about the Sun. (Although due to orbital inclinations, the orbital planes of the different planets are not necessarily coplanar.)

Two trains are on the same track a distance 100 km apart heading towards one another, each at a speed of 50 km/h. A fly starting out at the front of one train, flies towards the other at a speed of 75 km/h. Upon reaching the other train, the fly turns around and continues towards the first train. How many kilometers does the fly travel before getting squashed in the collision of the two trains?Now, the trains take one hour to collide (their relative speed is 100 km/h and they are 100 km apart initially). Since the fly is traveling at 75 km/h and flies continuously until it is squashed (which it is to be supposed occurs a split second before the two oncoming trains squash one another), it must therefore travel 75 km in the hour's time. The position of the fly at time is plotted above.However, a brute force method instead solves for the position of the fly along each traversal between the trains. For example, the fly reaches the second train when(1)or h, at which point..

If moves along a known curve, then describes a pursuit curve if is always directed toward and and move with uniform velocities. Pursuit curves were considered in general by the French scientist Pierre Bouguer in 1732, and subsequently by the English mathematician Boole.Under the name "path minimization," pursuit curves are alluded to by math genius Charlie Eppes in the Season 2 episode "Dark Matter" of the television crime drama NUMB3RS when considering the actions of the mysterious third shooter.The equations of pursuit are given by(1)which specifies that the tangent vector at point is always parallel to the line connecting and , combined with(2)which specifies that the point moves with constant speed (without loss of generality, taken as unity above). Plugging (2) into (1) therefore gives(3)The case restricting to a straight line was studied by Arthur Bernhart (MacTutor Archive). Taking the parametric equation..

In the mice problem, also called the beetle problem, mice start at the corners of a regular -gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise direction at constant speed. The mice each trace out a logarithmic spiral, meet in the center of the polygon, and travel a distanceThe first few values for , 3, ..., aregiving the numerical values 0.5, 0.666667, 1, 1.44721, 2, 2.65597, 3.41421, 4.27432, 5.23607, .... The curve formed by connecting the mice at regular intervals of time is an attractive figure called a whirl.The problem is also variously known as the (three, four, etc.) (bug, dog, etc.) problem. It can be generalized to irregular polygons and mice traveling at differing speeds (Bernhart 1959). Miller (1871) considered three mice in general positions with speeds adjusted to keep paths similar and the triangle similar to the original...

The osculating circle of a curve at a given point is the circle that has the same tangent as at point as well as the same curvature. Just as the tangent line is the line best approximating a curve at a point , the osculating circle is the best circle that approximates the curve at (Gray 1997, p. 111).Ignoring degenerate curves such as straight lines, the osculating circle of a given curve at a given point is unique.Given a plane curve with parametric equations and parameterized by a variable , the radius of the osculating circle is simply the radius of curvature(1)where is the curvature, and the center is just the point on the evolute corresponding to ,(2)(3)Here, derivatives are taken with respect to the parameter .In addition, let denote the circle passing through three points on a curve with . Then the osculating circle is given by(4)(Gray 1997)...

For every dimension , the orthogonal group is the group of orthogonal matrices. These matrices form a group because they are closed under multiplication and taking inverses.Thinking of a matrix as given by coordinate functions, the set of matrices is identified with . The orthogonal matrices are the solutions to the equations(1)where is the identity matrix, which are redundant. Only of these are independent, leaving "free variables." In fact, the orthogonal group is a smooth -dimensional submanifold.Because the orthogonal group is a group and a manifold, it is a Lie group. has a submanifold tangent space at the identity that is the Lie algebra of antisymmetric matrices . In fact, the orthogonal group is a compact Lie group.The determinant of an orthogonal matrix is either 1 or , and so the orthogonal group has two components. The component containing the identity is the special orthogonal group . For example, the group has group action..

The Möbius strip, also called the twisted cylinder (Henle 1994, p. 110), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (right figure; Gray 1997, pp. 322-323). The strip bearing his name was invented by Möbius in 1858, although it was independently discovered by Listing, who published it, while Möbius did not (Derbyshire 2004, p. 381). Like the cylinder, it is not a true surface, but rather a surface with boundary (Henle 1994, p. 110).The Möbius strip has Euler characteristic (Dodson and Parker 1997, p. 125).According to Madachy (1979), the B. F. Goodrich Company patented a conveyor belt in the form of a Möbius strip which lasts twice as long as conventional belts. M. C. Escher was fond of portraying Möbius strips, and..

The tower of Hanoi (commonly also known as the "towers of Hanoi"), is a puzzle invented by E. Lucas in 1883. It is also known as the Tower of Brahma puzzle and appeared as an intelligence test for apes in the film Rise of the Planet of the Apes (2011) under the name "Lucas Tower."Given a stack of disks arranged from largest on the bottom to smallest on top placed on a rod, together with two empty rods, the tower of Hanoi puzzle asks for the minimum number of moves required to move the stack from one rod to another, where moves are allowed only if they place smaller disks on top of larger disks. The puzzle with pegs and disks is sometimes known as Reve's puzzle.The problem is isomorphic to finding a Hamiltonian path on an -hypercube (Gardner 1957, 1959).Given three rods and disks, the sequence giving the number of the disk ( to ) to be moved at the th step is given by the remarkably simple recursive procedure of starting with the list for..

Let be a group, and let be a set of group elements such that the identity element . The Cayley graph associated with is then defined as the directed graph having one vertex associated with each group element and directed edges whenever . The Cayley graph may depend on the choice of a generating set, and is connected iff generates (i.e., the set are group generators of ).Care is needed since the term "Cayley graph" is also used when is implicitly understood to be a set of generators for the group, in which case the graph is always connected (but in general, still dependent on the choice of generators). This sort of Cayley graph of a group may be computed in the Wolfram Language using CayleyGraph[G], where the generators used are those returned by GroupGenerators[G].To complicate matters further, undirected versions of proper directed Cayley graphs are also known as Cayley graphs.An undirected Cayley graph of a particular generating set of..

If an -sided Poncelet transverse constructed for two given conic sections is closed for one point of origin, it is closed for any position of the point of origin. Specifically, given one ellipse inside another, if there exists one circuminscribed (simultaneously inscribed in the outer and circumscribed on the inner) -gon, then any point on the boundary of the outer ellipse is the vertex of some circuminscribed -gon. If the conic is taken as a circle (Casey 1888, pp. 124-126) , then a polygon which has both an incenter and a circumcenter (and for which the transversals would therefore close) is called a bicentric polygon.Amazingly, this problem is isomorphic to Gelfand'squestion (King 1994).For an even-sided polygon, the diagonals are concurrent at the limiting point of the two circles, whereas for an odd-sided polygon, the lines connecting the vertices to the opposite points of tangency are concurrent at the limiting point.Inverting..

Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent variables, known as "parameters." For example, while the equation of a circle in Cartesian coordinates can be given by , one set of parametric equations for the circle are given by(1)(2)illustrated above. Note that parametric representations are generally nonunique, so the same quantities may be expressed by a number of different parameterizations. A single parameter is usually represented with the parameter , while the symbols and are commonly used for parametric equations in two parameters.Parametric equations provide a convenient way to represent curves and surfaces, as implemented, for example, in the Wolfram Language commands ParametricPlot[x, y, t, t1, t2] and ParametricPlot3D[x, y, z, u, u1, u2, v, v1, v2]. Unsurprisingly, curves and surfaces obtained by way of parametric equation representations..

The perpendicular bisector of a linesegment is the locus of all points that are equidistant from its endpoints.This theorem can be applied to determine the center of a given circle with straightedge and compass. Pick three points , and on the circle. Since the center is equidistant from all of them, it lies on the bisector of segment and also on the bisector of segment , i.e., it is the intersection point of the two bisectors. This construction is shown on a window pane by tutor Justin McLeod (Mel Gibson) to his pupil Chuck Norstadt (Nick Stahl) in the 1993 film The Man Without a Face.

A curve with polar coordinates,(1)studied by the Greek mathematician Nicomedes in about 200 BC, also known as the cochloid. It is the locus of points a fixed distance away from a line as measured along a line from the focus point (MacTutor Archive). Nicomedes recognized the three distinct forms seen in this family for , , and . (For , it obviously degenerates to a circle.)The conchoid of Nicomedes was a favorite with 17th century mathematicians and could be used to solve the problems of cube duplication, angle trisection, heptagon construction, and other Neusis constructions (Johnson 1975).In Cartesian coordinates, the conchoid ofNicomedes may be written(2)or(3)The conchoid has as an asymptote, and the area between either branch and the asymptote is infinite.A conchoid with has a loop for , where , giving area(4)(5)(6)The curvature and tangentialangle are given by(7)(8)..

Consider three mutually tangent circles, and draw their inner Soddy circle. Then draw the inner Soddy circles of this circle with each pair of the original three, and continue iteratively. The steps in the process are illustrated above (Trott 2004, pp. 34-35).An animation illustrating the construction of the gasket is shown above.The points which are never inside a circle form a set of measure 0 having fractal dimension approximately 1.3058 (Mandelbrot 1983, p. 172). The Apollonian gasket corresponds to a limit set that is invariant under a Kleinian group (Wolfram 2002, p. 986).The Apollonian gasket can also be generalized to three dimensions (Boyd 1973, Andrade et al. 2005), as illustrated above. A graph obtained by connecting the centers of touching spheres in a three-dimensional Apollonian gasket by edges is known as an Apollonian network...

Let be a finite-dimensional split semisimple Lie algebra over a field of field characteristic 0, a splitting Cartan subalgebra, and a weight of in a representation of . Thenis also a weight. Furthermore, the reflections with a root, generate a group of linear transformations in called the Weyl group of relative to , where is the algebraic conjugate space of and is the Q-space spanned by the roots (Jacobson 1979, pp. 112, 117, and 119).The Weyl group acts on the roots of a semisimple Lie algebra, and it is a finite group. The animations above illustrate this action for Weyl Group acting on the roots of a homotopy from one Weyl matrix to the next one (i.e., it slides the arrows from to ) in the first two figures, while the third figure shows the Weyl Group acting on the roots of the Cartan matrix of the infinite family of semisimple lie algebras (cf. Dynkin diagram), which is the special linear Lie algebra, ...

Turmites, also called turning machines, are 2-dimensional Turing machines in which the "tape" consists of a grid of spaces that can be written and erased by an active ("head") element that turns at each iteration on the basis of the state of its current grid square. The "head" of the system is usually called a "vant," "ant," or "turmite" on square grids, and a "bee," "worm," or "turtle" on hexagonal grids. (The term "turtle" is named after Seymour Papert's turtle geometry). The turmite tracks its position, direction, and current state.Amazingly, the turmite with rule illustrated above mimics binary counting. In this turmite, the pattern of bands above the red line corresponds to incrementing binary digits. After each cycle constructing the upper pattern, the same pattern is produced (mirrored left-to-right) below the red..

A 4-state two-dimensional Turing machine invented in the 1980s. The ant starts out on a grid containing black and white cells, and then follows the following set of rules. 1. If the ant is on a black square, it turns right and moves forward one unit. 2. If the ant is on a white square, it turns left and moves forward one unit. 3. When the ant leaves a square, it inverts the color. When the ant is started on an empty grid, it eventually builds a "highway" that is a series of 104 steps that repeat indefinitely, each time displacing the ant two pixels vertically and horizontally. The plots above show the ant starting from a completely white grid after 386 (left figure) and (right figure) steps. In the right figure, the highway is being constructed towards the lower right-hand corner. The fact that the ant's path is unbounded is guaranteed by the Cohen-Kung theorem. It is believed that no matter what initial pattern the ant is started on, it will eventually..

The th roots of unity are roots of the cyclotomic equationwhich are known as the de Moivre numbers. The notations , , and , where the value of is understood by context, are variously used to denote the th th root of unity. is always an th root of unity, but is such a root only if is even. In general, the roots of unity form a regular polygon with sides, and each vertex lies on the unit circle.

A polynomial given by(1)where are the roots of unity in given by(2)and runs over integers relatively prime to . The prime may be dropped if the product is instead taken over primitive roots of unity, so that(3)The notation is also frequently encountered. Dickson et al. (1923) and Apostol (1975) give extensive bibliographies for cyclotomic polynomials.The cyclotomic polynomial for can also be defined as(4)where is the Möbius function and the product is taken over the divisors of (Vardi 1991, p. 225). is an integer polynomial and an irreducible polynomial with polynomial degree , where is the totient function. Cyclotomic polynomials are returned by the Wolfram Language command Cyclotomic[n, x]. The roots of cyclotomic polynomials lie on the unit circle in the complex plane, as illustrated above for the first few cyclotomic polynomials.The first few cyclotomic polynomials are(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)The cyclotomic..

Deck transformations, also called covering transformations, are defined for any cover . They act on by homeomorphisms which preserve the projection . Deck transformations can be defined by lifting paths from a space to its universal cover , which is a simply connected space and is a cover of . Every loop in , say a function on the unit interval with , lifts to a path , which only depends on the choice of , i.e., the starting point in the preimage of . Moreover, the endpoint depends only on the homotopy class of and . Given a point , and , a member of the fundamental group of , a point is defined to be the endpoint of a lift of a path which represents .The deck transformations of a universal cover form a group , which is the fundamental group of the quotient spaceFor example, when is the square torus then is the plane and the preimage is a translation of the integer lattice . Any loop in the torus lifts to a path in the plane, with the endpoints lying in the integer lattice...

The problem of finding the curve down which a bead placed anywhere will fall to the bottom in the same amount of time. The solution is a cycloid, a fact first discovered and published by Huygens in Horologium oscillatorium (1673). This property was also alluded to in the following passage from Moby Dick: "[The try-pot] is also a place for profound mathematical meditation. It was in the left-hand try-pot of the Pequod, with the soapstone diligently circling round me, that I was first indirectly struck by the remarkable fact, that in geometry all bodies gliding along a cycloid, my soapstone, for example, will descend from any point in precisely the same time" (Melville 1851).Huygens also constructed the first pendulum clock with a device to ensure that the pendulum was isochronous by forcing the pendulum to swing in an arc of a cycloid. This is accomplished by placing two evolutes of inverted cycloid arcs on each side of the pendulum's point..

Replacing the logistic equation(1)with the quadratic recurrence equation(2)where (sometimes also denoted ) is a positive constant sometimes known as the "biotic potential" gives the so-called logistic map. This quadratic map is capable of very complicated behavior. While John von Neumann had suggested using the logistic map as a random number generator in the late 1940s, it was not until work by W. Ricker in 1954 and detailed analytic studies of logistic maps beginning in the 1950s with Paul Stein and Stanislaw Ulam that the complicated properties of this type of map beyond simple oscillatory behavior were widely noted (Wolfram 2002, pp. 918-919).The first few iterations of the logistic map (2) give(3)(4)(5)where is the initial value, plotted above through five iterations (with increasing iteration number indicated by colors; 1 is red, 2 is yellow, 3 is green, 4 is blue, and 5 is violet) for various values of .The..

On a Riemannian manifold , tangent vectors can be moved along a path by parallel transport, which preserves vector addition and scalar multiplication. So a closed loop at a base point , gives rise to a invertible linear map of , the tangent vectors at . It is possible to compose closed loops by following one after the other, and to invert them by going backwards. Hence, the set of linear transformations arising from parallel transport along closed loops is a group, called the holonomy group.Since parallel transport preserves the Riemannian metric, the holonomy group is contained in the orthogonal group . Moreover, if the manifold is orientable, then it is contained in the special orthogonal group. A generic Riemannian metric on an orientable manifold has holonomy group , but for some special metrics it can be a subgroup, in which case the manifold is said to have special holonomy.A Kähler manifold is a -dimensional manifold whose holonomy lies..

Analytic continuation (sometimes called simply "continuation") provides a way of extending the domain over which a complex function is defined. The most common application is to a complex analytic function determined near a point by a power series(1)Such a power series expansion is in general valid only within its radius of convergence. However, under fortunate circumstances (that are very fortunately also rather common!), the function will have a power series expansion that is valid within a larger-than-expected radius of convergence, and this power series can be used to define the function outside its original domain of definition. This allows, for example, the natural extension of the definition trigonometric, exponential, logarithmic, power, and hyperbolic functions from the real line to the entire complex plane . Similarly, analytic continuation can be used to extend the values of an analytic function across a branch..

A group is said to act on a set when there is a map such that the following conditions hold for all elements . 1. where is the identity element of . 2. for all . In this case, is called a transformation group, is a called a -set, and is called the group action.In a group action, a group permutes the elements of . The identity does nothing, while a composition of actions corresponds to the action of the composition. For example, as illustrated above, the symmetric group acts on the digits 0 to 9 by permutations.For a given , the set , where the group action moves , is called the group orbit of . The subgroup which fixes is the isotropy group of .For example, the group acts on the real numbers by multiplication by . The identity leaves everything fixed, while sends to . Note that , which corresponds to . For , the orbit of is , and the isotropy subgroup is trivial, . The only group fixed point of this action is .In a group representation, a group acts by invertible linear transformations..

A circle bundle is a fiber bundle whose fibers are circles. It may also have the structure of a principal bundle if there is an action of that preserves the fibers, and is locally trivial. That is, if every point has a trivialization such that the action of on is the usual one.

The fundamental group of an arcwise-connected set is the group formed by the sets of equivalence classes of the set of all loops, i.e., paths with initial and final points at a given basepoint , under the equivalence relation of homotopy. The identity element of this group is the set of all paths homotopic to the degenerate path consisting of the point . The fundamental groups of homeomorphic spaces are isomorphic. In fact, the fundamental group only depends on the homotopy type of . The fundamental group of a topological space was introduced by Poincaré (Munkres 1993, p. 1).The following is a table of the fundamental group for some common spaces , where denotes the fundamental group, is the first integral homology group, denotes the group direct product, denotes the free product, denotes the ring of integers, and is the cyclic group of order .space ()symbolcirclecomplex projective space00figure eightKlein bottle-torusreal projective..

The universal cover of a connected topological space is a simply connected space with a map that is a covering map. If is simply connected, i.e., has a trivial fundamental group, then it is its own universal cover. For instance, the sphere is its own universal cover. The universal cover is always unique and, under very mild assumptions, always exists. In fact, the universal cover of a topological space exists iff the space is connected, locally pathwise-connected, and semilocally simply connected.Any property of can be lifted to its universal cover, as long as it is defined locally. Sometimes, the universal covers with special structures can be classified. For example, a Riemannian metric on defines a metric on its universal cover. If the metric is flat, then its universal cover is Euclidean space. Another example is the complex structure of a Riemann surface , which also lifts to its universal cover. By the uniformization theorem, the only possible..

When discussing a rotation, there are two possible conventions: rotation of the axes, and rotation of the object relative to fixed axes.In , consider the matrix that rotates a given vector by a counterclockwise angle in a fixed coordinate system. Then(1)so(2)This is the convention used by the WolframLanguage command RotationMatrix[theta].On the other hand, consider the matrix that rotates the coordinate system through a counterclockwise angle . The coordinates of the fixed vector in the rotated coordinate system are now given by a rotation matrix which is the transpose of the fixed-axis matrix and, as can be seen in the above diagram, is equivalent to rotating the vector by a counterclockwise angle of relative to a fixed set of axes, giving(3)This is the convention commonly used in textbooks such as Arfken (1985, p. 195).In , coordinate system rotations of the x-, y-, and z-axes in a counterclockwise direction when looking towards..

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..

Given any open set in with compact closure , there exist smooth functions which are identically one on and vanish arbitrarily close to . One way to express this more precisely is that for any open set containing , there is a smooth function such that 1. for all and 2. for all . A function that satisfies (1) and (2) is called a bump function. If then by rescaling , namely , one gets a sequence of smooth functions which converges to the delta function, providing that is a neighborhood of 0.

A web diagram, also called a cobweb plot, is a graph that can be used to visualize successive iterations of a function . In particular, the segments of the diagram connect the points , , , .... The diagram is so-named because its straight line segments "anchored" to the functions and can resemble a spider web. The animation above shows a web diagram for the logistic map with .

An ellipse is a curve that is the locus of all points in the plane the sum of whose distances and from two fixed points and (the foci) separated by a distance of is a given positive constant (Hilbert and Cohn-Vossen 1999, p. 2). This results in the two-center bipolar coordinate equation(1)where is the semimajor axis and the origin of the coordinate system is at one of the foci. The corresponding parameter is known as the semiminor axis.The ellipse is a conic section and a Lissajouscurve.An ellipse can be specified in the Wolfram Language using Circle[x, y, a, b].If the endpoints of a segment are moved along two intersecting lines, a fixed point on the segment (or on the line that prolongs it) describes an arc of an ellipse. This is known as the trammel construction of an ellipse (Eves 1965, p. 177).It is possible to construct elliptical gears that rotate smoothly against one another (Brown 1871, pp. 14-15; Reuleaux and Kennedy 1876,..

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 4-cusped hypocycloid which is sometimes also called a tetracuspid, cubocycloid, or paracycle. The parametric equations of the astroid can be obtained by plugging in or into the equations for a general hypocycloid, giving parametric equations(1)(2)(3)(4)(5)(6)for .The polar equation can be obtained by computing(7)and plugging in to to obtain(8)for .In Cartesian coordinates,(9)A generalization of the curve to(10)gives "squashed" astroids, which are a special case of the superellipse corresponding to parameter .In pedal coordinates with the pedalpoint at the center, the equation is(11)and the Cesàro equation is(12)A further generalization to an equation of the form(13)is known as a superellipse.The arc length, curvature,and tangential angle are(14)(15)(16)where the formula for holds for .The perimeter of the entire astroid can be computedfrom the general hypocycloid formula(17)with ,(18)For a squashed..

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