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.
Consider the length of the diagonal of a unit square as approximated by piecewise linear steps that may only be taken in the right and up directions. Obviously, the length so obtained is equal to half the perimeter, or 2. As the number of steps becomes large, the path visually appears to approach a diagonal line. However, no matter how small the steps, if they are constrained to be only to the right and up, their total length is always 2, despite the fact that the length of the diagonal is .This apparent paradox arises in physics in the computation of Feynman diagrams, where it has implications for the types of paths that must be included in order to obtain a good approximation to physical quantities.
A snake is an Eulerian path in the -hypercube that has no chords (i.e., any hypercube edge joining snake vertices is a snake edge). Klee (1970) asked for the maximum length of a -snake. Klee (1970) gave the bounds(1)for (Danzer and Klee 1967, Douglas 1969), as well as numerous references. Abbott and Katchalski (1988) show(2)and Snevily (1994) showed that(3)for , and conjectured(4)for . The first few values for for , 2, ..., are 2, 4, 6, 8, 14, 26, ... (OEIS A000937).
An invertible knot is a knot that can be deformed via an ambient isotopy into itself but with the orientation reversed. A knot that is not invertible is said to be noninvertible.Knots on ten and fewer crossing can be tested in the Wolfram Language to see if they are invertible using the command KnotData[knot, "Invertible"].Fox (1962, Problem 10, p. 169) pointed out several knots belonging to the standard table that seemed to be noninvertible. However, no noninvertible knots were proven to exist until Trotter (1964) discovered an infinite family, the smallest of which had 15 crossings.Three prime knots on 9 or fewer crossings are noninvertible: , , and (Cromwell 2004, pp. 297-299). Some noninvertible knots can be obtained in the Wolfram Language as KnotData["Noninvertible"]. The simplest noninvertible knot is (illustrated above) was first postulated to be noninvertible by Fox (1962; Whitten 1972).The..
A magic tesseract is a four-dimensional generalization of the two-dimensional magic square and the three-dimensional magic cube. A magic tesseract has magic constantso for , 2, ..., the magic tesseract constants are 1, 17, 123, 514, 1565, 3891, ... (OEIS A021003).Berlekamp et al. (1982, p. 783) give a magic tesseract. J. Hendricks has constructed magic tesseracts of orders three, four, five (Hendricks 1999a, pp. 128-129), and six (Heinz). M. Houlton has used Hendricks' techniques to construct magic tesseracts of orders 5, 7, and 9.There are 58 distinct magic tesseracts of order three, modulo rotations and reflections (Heinz, Hendricks 1999), one of which is illustrated above. Each of the 27 rows (e.g., 1-72-50), columns (e.g., 1-80-42), pillars (e.g., 1-54-68), and files (e.g., 1-78-44) sum to the magic constant 123.Hendricks (1968) has constructed a pan-4-agonal magic tesseract of order 4. No pan-4-agonal..
A Pierpont prime is a prime number of the form . The first few Pierpont primes are 2, 3, 5, 7, 13, 17, 19, 37, 73, 97, 109, 163, 193, 257, 433, 487, 577, 769, ... (OEIS A005109).A regular polygon of sides can be constructed by ruler, compass and angle-trisector iffwhere , , ..., are distinct Pierpont primes and (Gleason 1998).The numbers of Pierpont primes less than , , ... are 4, 10, 18, 25, 32, 42, 50, 58, ... (OEIS A113420) and the number less than , , , , ... are 4, 10, 25, 58, 125, 250, 505, 1020, 2075, 4227, ... (OEIS A113412; Caldwell).As of Apr. 2010, the largest known Pierpont prime is , which has decimal digits (https://primes.utm.edu/primes/page.php?id=87449).
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.
Compass and straightedge geometric constructions dating back to Euclid were capable of inscribing regular polygons of 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, ..., sides. In 1796 (when he was 19 years old), Gauss gave a sufficient condition for a regular -gon to be constructible, which he also conjectured (but did not prove) to be necessary, thus showing that regular -gons were constructible for , 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, ... (OEIS A003401).A complete enumeration of "constructible" polygons is given by those with central angles corresponding to so-called trigonometry angles.Gardner (1977) and independently Watkins (Conway and Guy 1996, Kríek et al. 2001) noticed that the number of sides for constructible polygons with odd numbers of sides are given by the first 32 rows of the Sierpiński sieve interpreted as binary numbers, giving 1, 3, 5, 15, 17, 51, 85,..
A Bill picture is a sequence of nested regular polygons in which subsequent polygons are each rotated so that they begin one vertex further. The term was coined by Trott (2004, pp. 88-89) and commemorates Swiss artist Max Bill, who in 1938 created a picture showing a similar arrangement of the equilateral triangle through octagon (Huttingerr 1978, Bill 1987).The figure above shows the Bill picture including regular polygons up through theregular dodecagon.
There are no tilings of the equilateral triangle of side length 7 by all the polyhexes of order . There are nine distinct solutions of all the polyhexes of order which tile a parallelogram of base length 7 and side length 4, one of which is illustrated above (Beeler 1972).
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..
A ruled surface is a normal developable of a curve if can be parameterized by , where is the normal vector (Gray 1993, pp. 352-354; first edition only).
The partial differential equation(Gray 1997, p. 399), whose solutions are called minimal surfaces. This corresponds to the mean curvature equalling 0 over the surface.d'Alembert's equationis sometimes also known as Lagrange's equation (Zwillinger 1997, pp. 120 and 265-268).
The Weierstrass substitution is the trigonometric substitution which transforms an integral of the forminto one of the formAccording to Spivak (2006, pp. 382-383), this is undoubtably the world's sneakiest substitution.The Weierstrass substitution can also be useful in computing a Gröbner basis to eliminate trigonometric functions from a system of equations (Trott 2006, p. 39).
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 box integral for dimension with parameters and is defined as the expectation of distance from a fixed point of a point chosen at random over the unit -cube,(1)(Bailey et al. 2006).Two special cases include(2)(3)which, with , correspond to hypercube point picking (to a fixed vertex) and hypercube line picking, respectively.Hypercube point picking to the center isgiven by(4)
The Fibonacci number gives the number of ways for dominoes to cover a checkerboard, as illustrated in the diagrams above (Dickau).The numbers of domino tilings, also known as dimer coverings, of a square for , 2, ... are given by 2, 36, 6728, 12988816, ... (OEIS A004003). The 36 tilings on the square are illustrated above. A formula for these numbers is given by(1)Writing(2)gives the surprising result(3)(John and Sachs 2000). For , 2, ..., the first few terms are 1, 3, 29, 5, 5, 7, 25, 9, 9, 11, 21, ... (OEIS A143234).Writing(4)(5)(6)(7)(OEIS A143233), where is Catalan's constant.
A polyomino tiling is a tiling of the plane by specified types of polyominoes. Tiling by polyominoes has been investigated since at least the late 1950s, particularly by S. Golomb (Wolfram 2002, p. 943).Interestingly, the Fibonacci number gives the number of ways for dominoes to cover a checkerboard.Each monomino, domino, triomino, tetromino, pentomino, and hexomino tiles the plane without requiring flipping. In addition, each heptomino with the exception of the four illustrated above can tile the plane, also without flipping (Schroeppel 1972).Recently, sets of polyominoes that force non-periodic patterns have been found. The set illustrated at left above was announced by Roger Penrose in 1994, and the slightly smaller set illustrated at right below was found by Matthew Cook (Wolfram 2002, p. 943).Both of these sets yield nested patterns, as illustrated above for Cook's tiles (Wolfram2002, p. 943).Consider..
Integrals over the unit square arising in geometricprobability are(1)which give the average distances in square point picking from a point picked at random in a unit square to a corner and to the center, respectively.Unit square integrals involving the absolute valueare given by(2)(3)for and , respectively.Another simple integral is given by(4)(Bailey et al. 2007, p. 67). Squaring the denominator gives(5)(6)(7)(8)(9)(OEIS A093754; M. Trott, pers. comm.), where is Catalan's constant and is a generalized hypergeometric function. A related integral is given by(10)which diverges in the Riemannian sense, as can quickly seen by transforming to polar coordinates. However, taking instead Hadamard integral to discard the divergent portion inside the unit circle gives(11)(12)(13)(14)(OEIS A093753), where is Catalan's constant.A collection of beautiful integrals over the unit squareare given by Guillera and Sondow..
A curve on a surface whose tangents are always in the direction of principalcurvature. The equation of the lines of curvature can be writtenwhere and are the coefficients of the first and second fundamental forms.
Two points and are symmetric with respect to a circle or straight line if all circles and straight lines passing through and are orthogonal to . Möbius transformations preserve symmetry. Let a straight line be given by a point and a unit vector , thenwhere is the complex conjugate. Let a circle be given by center and radius , then
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..
The envelope of the lines connecting corresponding points on the Jacobian curve and Steinerian curve. The Cayleyian curve of a net of curves of order has the same curve genus as the Jacobian curve and Steinerian curve and, in general, the class .
The bifolium is a folium with . The bifolium is a quartic curve and is given by the implicit equation is(1)and the polar equation(2)The bifolium has area(3)(4)(5)Its arc length is(6)(7)(OEIS A118307), where , , , and are elliptic integrals with(8)(9)The curvature is given by(10)(11)The bifolium is the pedal curve of the deltoid where the pedal point is the midpoint of one of the three curved sides.
The necessary and sufficient condition that an algebraic curve has an algebraic involute is that the arc length is a two-valued algebraic function of the coordinates of the extremities. Furthermore, this function is a root of a quadratic equation whose coefficients are rational functions of and .
There are several definitions of a ray.When viewed as a vector, a ray is a vector from a point to a point .In geometry, a ray is usually taken as a half-infinite line (also known as a half-line) with one of the two points and taken to be at infinity.
A vector perpendicular to a given vector is a vector (voiced "-perp") such that and form a right angle.In the plane, there are two vectors perpendicular to any given vector, one rotated counterclockwise and the other rotated clockwise. Hill (1994) defines to be the perpendicular vector obtained from an initial vector(1)by a counterclockwise rotation by , i.e.,(2)In the plane, a vector perpendicular to can therefore be obtained by transposing the Cartesian components and taking the minus sign of one. This operation is implemented in the Wolfram Language as Cross[ax, ay].In three dimensions, there are an infinite number of vectors perpendicular to a given vector, all satisfying the equations(3)
Two lines, vectors, planes, etc., are said to be perpendicular if they meet at a right angle. In , two vectors and are perpendicular if their dot product(1)In , a line with slope is perpendicular to a line with slope . Perpendicular objects are sometimes said to be "orthogonal."In the above figure, the line segment is perpendicular to the line segment . This relationship is commonly denoted with a small square at the vertex where perpendicular objects meet, as shown above, and is denoted .Two trilinear lines(2)(3)are perpendicular if(4)(Kimberling 1998, p. 29).
If is a linear transformation of , then the null space Null(), also called the kernel , is the set of all vectors such thati.e.,The term "null space" is most commonly written as two separate words (e.g., Golub and Van Loan 1989, pp. 49 and 602; Zwillinger 1995, p. 128), although other authors write it as a single word "nullspace" (e.g., Anton 1994, p. 259; Robbin 1995, pp. 123 and 180).The Wolfram Language command NullSpace[v1, v2, ...] returns a list of vectors forming a vector basis for the nullspace of a set of vectors over the rationals (or more generally, over whatever base field contains the input vectors).
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..
The golden gnomon is the obtuse isosceles triangle whose ratio of side to base lengths is given by , where is the golden ratio. Such a triangle has angles of -- and can be constructed from a regular pentagon as illustrated above in red. The corresponding 36-72-72 triangle with side-to-base ratio is a golden triangle.Golden triangles and gnomons can be dissected into smaller triangles that are golden gnomons and golden triangles (Livio 2003, p. 79).
A golden rhombohedron is a rhombohedron whose faces consist of congruent golden rhombi. Golden rhombohedra are therefore special cases of a trigonal trapezohedron as well as zonohedra.There are two distinct golden rhombohedra: the acute golden rhombohedron and obtuse golden rhombohedron. Both are built from six golden rhombi and comprise two of the five golden isozonohedra. These polyhedra are implemented in the Wolfram Language as PolyhedronData["AcuteGoldenRhombohedron"] and PolyhedronData["ObtuseGoldenRhombohedron"], respectively.The acute and obtuse golden rhombohedra with edge length both have surface area(1)and have volumes(2)(3)respectively.
The golden triangle, sometimes also called the sublime triangle, is an isosceles triangle such that the ratio of the hypotenuse to base is equal to the golden ratio, . From the above figure, this means that the triangle has vertex angle equal to(1)or , and that the height is related to the base through(2)(3)(4)The inradius of a golden triangle is(5)The triangles at the tips of a pentagram (left figure) and obtained by dividing a decagon by connecting opposite vertices (right figure) are golden triangles. This follows from the fact that(6)for a pentagram and that the circumradius of a decagon of side length is(7)Golden triangles and gnomons can be dissected into smaller triangles that are golden gnomons and golden triangles (Livio 2002, p. 79).Successive points dividing a golden triangle into golden gnomons and triangles lieon a logarithmic spiral (Livio 2002, p. 119).Kimberling (1991) defines a second type of golden triangle..
A nonuniform rational B-spline surface of degree is defined bywhere and are the B-spline basis functions, are control points, and the weight of is the last ordinate of the homogeneous point .NURBS surfaces are implemented in the WolframLanguage as BSplineSurface[array].
The integral of over the unit disk is given by(1)(2)(3)(4)In general,(5)provided .Additional integrals include(6)(7)(8)
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...
Given a rectangle having sides in the ratio , the golden ratio is defined such that partitioning the original rectangle into a square and new rectangle results in a new rectangle having sides with a ratio . Such a rectangle is called a golden rectangle. Euclid used the following construction to construct them. Draw the square , call the midpoint of , so that . Now draw the segment , which has length(1)and construct with this length. Now complete the rectangle , which is golden since(2)Successive points dividing a golden rectangle into squares lie on a logarithmic spiral (Wells 1991, p. 39; Livio 2002, p. 119) which is sometimes known as the golden spiral.The spiral is not actually tangent at these points, however, but passes through them and intersects the adjacent side, as illustrated above.If the top left corner of the original square is positioned at (0, 0), the center of the spiral occurs at the position(3)(4)(5)(6)(7)(8)(9)(10)(11)and..
A golden rhombus is a rhombus whose diagonals are in the ratio , where is the golden ratio.The faces of the acute golden rhombohedron, Bilinski dodecahedron, obtuse golden rhombohedron, rhombic hexecontahedron, and rhombic triacontahedron are golden rhombi.The half-angle is given by(1)(2)(3)(4)(OEIS A195693).Labeling the smaller interior angle as and the larger as , then(5)and(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(OEIS A105199 and A137218).The diagonal lengths of a golden rhombus with edge length are given by(18)(19)(20)(21)(22)(23)(24)(25)(OEIS A121570 and A179290),the inradius by(26)and the area by(27)
An unfolding is the cutting along edges and flattening out of a polyhedron to form a net. Determining how to unfold a polyhedron into a net is tricky. For example, cuts cannot be made along all edges that surround a face or the face will completely separate. Furthermore, for a polyhedron with no coplanar faces, at least one edge cut must be made from each vertex or else the polyhedron will not flatten. In fact, the edges that must be cut corresponds to a special kind of graph called a spanning tree of the skeleton of the polyhedron (Malkevitch).In 1987, K. Fukuda conjectured that no convex polyhedra admit a self-overlapping unfolding. The top figure above shows a counterexample to the conjecture found by M. Namiki. An unfoldable tetrahedron was also subsequently found (bottom figure above). Another nonregular convex polyhedra admitting an overlapping unfolding was found by G. Valette (shown in Buekenhout and Parker 1998).Examples..
Given rods of length 1, 2, ..., , how many distinct triangles can be made? Lengths for which(1)obviously do not give triangles, but all other combinations of three rods do. The answer is(2)The values for , 2, ... are 0, 0, 0, 1, 3, 7, 13, 22, 34, 50, ... (OEIS A002623). Somewhat surprisingly, this sequence is also given by the generating function(3)
Given a Pythagorean triple , the fractions and are called Pythagorean fractions. Diophantus showed that the Pythagorean fractions consist precisely of fractions of the form .
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.)
A fast boat is overtaking a slower one when fog suddenly sets in. At this point, the boat being pursued changes course, but not speed, and proceeds straight in a new direction which is not known to the fast boat. How should the pursuing vessel proceed in order to be sure of catching the other boat?The amazing answer is that the pursuing boat should continue to the point where the slow boat would be if it had set its course directly for the pursuing boat when the fog set in. If the boat is not there, it should proceed in a spiral whose origin is the point where the slow boat was when the fog set in. The spiral must be constructed in such a way that, while circling the origin, the fast boat's distance from it increases at the same rate as the boat being pursued. The two courses must therefore intersect before the fast boat has completed one circuit. In order to make the problem reasonably practical, the fast boat should be capable of maintaining a speed four or five times..
The tractrix arises in the following problem posed to Leibniz: What is the path of an object starting off with a vertical offset when it is dragged along by a string of constant length being pulled along a straight horizontal line (Steinhaus 1999, pp. 250-251)? By associating the object with a dog, the string with a leash, and the pull along a horizontal line with the dog's master, the curve has the descriptive name "hundkurve" (dog curve) in German. Leibniz found the curve using the fact that the axis is an asymptote to the tractrix (MacTutor Archive).From its definition, the tractrix is precisely the catenary involute described by a point initially on the vertex (so the catenary is the tractrix evolute). The tractrix is sometimes called the tractory or equitangential curve. The tractrix was first studied by Huygens in 1692, who gave it the name "tractrix." Later, Leibniz, Johann Bernoulli, and others studied the curve.In..
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 first Brocard point is the interior point (also denoted or ) of a triangle with points labeled in counterclockwise order for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(1)(Kimberling 1998, p. 47).Note that extreme care is needed when consulting the literature, since reversing the order in which the points of the triangle are labeled results in exchanging the Brocard points.The second Brocard point is the interior point (also denoted or ) for which the angles , , and are equal, with the unique such angle denoted . It is not a triangle center, but has trilinear coordinates(2)(Kimberling 1998, p. 47).Moreover, the two angles are equal, and this angle is called the Brocard angle,(3)(4)The first two Brocard points are isogonal conjugates (Johnson 1929, p. 266). They were described by French army officer Henri Brocard in 1875, although they..
It is conjectured that any convex body in -dimensional Euclidean space has an interior point lying on normals through distinct boundary points (Croft et al. 1991). This has been proved for and 3 by Heil (1979ab, 1985). It is known that higher dimensions always contain at least a 6-normal point, but the general conjecture remains open.
A hypersphere is parallelizable if there are vector fields that are linearly independent at each point. There exist only three parallelizable spheres: , , and (Adams 1958, 1960, Le Lionnais 1983).More generally, an -dimensional manifold is parallelizable if its tangent bundle is a trivial bundle (i.e., if is globally of the form ).
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 generalization of the Schönflies theorem to dimensions. A smoothly embedded -hypersphere in an -hypersphere separates the -hypersphere into two components, each homeomorphic to -balls. It can be proved using Morse theory.
The Klein bottle is a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside, originally described by Felix Klein (Hilbert and Cohn-Vossen 1999, p. 308). It can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, but can be physically realized only in four dimensions, since it must pass through itself without the presence of a hole. Its topology is equivalent to a pair of cross-caps with coinciding boundaries (Francis and Weeks 1999). It can be represented by connecting the side of a square in the orientations illustrated in the right figure above (Gardner 1984, pp. 15-17; Gray 1997, pp. 323-324).It can be cut in half along its length to make two Möbius strips (Dodson and Parker 1997, p. 88), but can also be cut into a single Möbius strip (Gardner 1984, pp. 14 and 17).The above picture..
In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors and of the real plane or the real space are orthogonal iff their dot product . This condition has been exploited to define orthogonality in the more abstract context of the -dimensional real space .More generally, two elements and of an inner product space are called orthogonal if the inner product of and is 0. Two subspaces and of are called orthogonal if every element of is orthogonal to every element of . The same definitions can be applied to any symmetric or differential k-form and to any Hermitian form.
A homeomorphism, also called a continuous transformation, is an equivalence relation and one-to-one correspondence between points in two geometric figures or topological spaces that is continuous in both directions. A homeomorphism which also preserves distances is called an isometry. Affine transformations are another type of common geometric homeomorphism.The similarity in meaning and form of the words "homomorphism"and "homeomorphism" is unfortunate and a common source of confusion.
If is a simple closed curve in , then the Jordan curve theorem, also called the Jordan-Brouwer theorem (Spanier 1966) states that has two components (an "inside" and "outside"), with the boundary of each.The Jordan curve theorem is a standard result in algebraic topology with a rich history. A complete proof can be found in Hatcher (2002, p. 169), or in classic texts such as Spanier (1966). Recently, a proof checker was used by a Japanese-Polish team to create a "computer-checked" proof of the theorem (Grabowski 2005).
Home plate in the game of baseball is an irregular pentagon with two parallel sides, each perpendicular to a base. It seems reasonable to dub such a figure (i.e., a rectangle with a coincident isosceles triangle placed on one side) a "isosceles right pentagon."However, specification of the shape of home plate, illustrated above, as specified by both the Major League Baseball Official Rules and the Little League rulebook (Kreutzer and Kerley 1990) is not physically realizable, since it requires the existence of a (12, 12, 17) right triangle, whereas(Bradley 1996). More specifically, the standards require the existence of an isosceles right triangle with side lengths 8.5 inches and a hypotenuse of length 12 inches, which does not satisfy the Pythagorean theorem.
The goat problem (or bull-tethering problem) considers a fenced circular field of radius with a goat (or bull, or other animal) tied to a point on the interior or exterior of the fence by means of a tether of length , and asks for the solution to various problems concerning how much of the field can be grazed.Tieing a goat to a point on the interior of the fence with radius 1 using a chain of length , consider the length of chain that must be used in order to allow the goat to graze exactly one half the area of the field. The answer is obtained by using the equation for a circle-circle intersection(1)Taking gives(2)plotted above. Setting (i.e., half of ) leads to the equation(3)which cannot be solved exactly, but which has approximate solution(4)(OEIS A133731).Now instead consider tieing the goat to the exterior of the fence (or equivalently, to the exterior of a silo whose horizontal cross section is a circle) with radius . Assume that , so that the goat is not..
Given a straight segment of track of length , add a small segment so that the track bows into a circular arc. Find the maximum displacement of the bowed track. The Pythagorean theorem gives(1)But is simply , so(2)Solving (1) and (2) for gives(3)Expressing the length of the arc in terms of the centralangle,(4)(5)(6)(7)But is given by(8)so plugging in gives(9)(10)This is a transcendental equation that cannot be solved exactly with a closed-form solution for , but for ,(11)Therefore,(12)(13)Keeping only terms to order ,(14)(15)so(16)and(17)If we take and 1 foot, then feet. Solving equation (◇) numerically, we find that the true answer is feet.
Pythagoras's theorem states that the diagonal of a square with sides of integral length cannot be rational. Assume is rational and equal to where and are integers with no common factors. Thensoand , so is even. But if is even, then is even. Since is defined to be expressed in lowest terms, must be odd; otherwise and would have the common factor 2. Since is even, we can let , then . Therefore, , and , so must be even. But cannot be both even and odd, so there are no and such that is rational, and must be irrational.In particular, Pythagoras's constant is irrational. Conway and Guy (1996) give a proof of this fact using paper folding, as well as similar proofs for (the golden ratio) and using a pentagon and hexagon. A collection of 17 computer proofs of the irrationality of is given by Wiedijk (2006)...
Pick two real numbers and at random in with a uniform distribution. What is the probability that , where denotes the nearest integer function, is even?The answer may be found as follows.(1)(2)so(3)(4)(5)(6)(7)(8)(9)(10)(OEIS A091651).
The coordinates representing any point of an -dimensional affine space by an -tuple of real numbers, thus establishing a one-to-one correspondence between and .If is the underlying vector space, and is the origin, every point of is identified with the -tuple of the components of vector with respect to a given basis of .If is a three-dimensional space, each basis can be depicted by choosing its elements as the unit vectors of the -axis, the -axis, and the -axis, respectively. In general, this will produce three axes which are not necessarily perpendicular, and where the units are set differently. Hence, Cartesian coordinates are a very special kind of affine coordinates that correspond to the case where , , .
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)...
The plane spanned by the three points , , and on a curve as . Let be a point on the osculating plane, thenwhere denotes the scalar triple product. The osculating plane passes through the tangent. The intersection of the osculating plane with the normal plane is known as the (principal) normal vector. The vectors and (tangent vector and normal vector) span the osculating plane.
For a plane curve, the tangential angle is defined by(1)where is the arc length and is the radius of curvature. The tangential angle is therefore given by(2)where is the curvature. For a plane curve , the tangential angle can also be defined by(3)Gray (1997) calls the turning angle instead of the tangential angle.
If two single-valued continuous functions (curvature) and (torsion) are given for , then there exists exactly one space curve, determined except for orientation and position in space (i.e., up to a Euclidean motion), where is the arc length, is the curvature, and is the torsion.
The term rectification is sometimes used to refer to the determination of the length of a curve.Rectification also refers to the operation which converts the midpoints of the edges of a regular polyhedron to the vertices of the related "rectified" polyhedron. Rectified forms are bounded by a combination of rectified cells and vertex figures. Therefore, a rectified polychoron is bounded by s and s. For example, is bounded by 600 truncated tetrahedra (truncated cells) and 120 icosahedra (vertex figures). A rectified polyhedron is indicated by prepending an "r" to the Schläfli symbol.polyhedronSchläfli symbolrectified polyhedronSchläfli symboltetrahedronoctahedronoctahedroncuboctahedroncubecuboctahedronicosahedronicosidodecahedrondodecahedronicosidodecahedron16-cell24-cellRectification of the six regular polychora gives five (not six) new polychora since the rectified..
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 reciprocal of a real or complex number is its multiplicative inverse , i.e., to the power . The reciprocal of zero is undefined. A plot of the reciprocal of a real number is plotted above for .Two numbers are reciprocals if and only if their product is 1. To put it another way, a number and its reciprocal are inversely related. Therefore, the larger a (positive) number, the smaller its reciprocal.The reciprocal of a complex number is given byPlots of the reciprocal in the complex plane are given above.Given a geometric figure consisting of an assemblage of points, the polars with respect to an inversion circle constitute another figure. These figures are said to be reciprocal with respect to each other. Then there exists a duality principle which states that theorems for the original figure can be immediately applied to the reciprocal figure after suitable modification (Lachlan 1893)...
The maximum number of pieces into which a cylinder can be divided by oblique cuts is given by(1)(2)(3)where is a binomial coefficient.This problem is sometimes also called cake cutting or pie cutting, and has the same solution as space division by planes. For , 2, ... cuts, the maximum number of pieces is 2, 4, 8, 15, 26, 42, ... (OEIS A000125). Unsurprisingly, the numbers of this sequence are called cake numbers.
The average number of regions into which randomly chosen planes divide a cube is(Finch 2003, p. 482).The maximum number of regions is presumably the same as for spacedivision by planes, namely(Yaglom and Yaglom 1987, pp. 102-106). For , 2, ... planes, this gives the values 2, 4, 8, 15, 26, 42, ... (OEIS A000125), a sequence whose values are sometimes called the "cake numbers" due to their relation to the cake cutting problem.
The number of regions into which space can be divided by mutually intersecting spheres isgiving 2, 4, 8, 16, 30, 52, 84, ... (OEIS A046127) for , 2, ....
Determining the maximum number of pieces in which it is possible to divide a circle for a given number of cuts is called the circle cutting or pancake cutting problem. The minimum number is always , where is the number of cuts, and it is always possible to obtain any number of pieces between the minimum and maximum. The first cut creates 2 regions, and the th cut creates new regions, so(1)(2)(3)Therefore,(4)(5)(6)(7)(8)Evaluating for , 2, ... gives 2, 4, 7, 11, 16, 22, ... (OEIS A000124). This is equivalent to the maximal number of regions into which a plane can be cut by lines.
If a circular pizza is divided into 8, 12, 16, ... slices by making cuts at equal angles from an arbitrary point, then the sums of the areas of alternate slices are equal.There is also a second pizza theorem. This one gives the volume of a pizza of thickness and radius :
In homogeneous coordinates, the first positive quadrant joins with by "points" , and is mapped onto the hyperbolic line by the correspondence . Now define(1)Let be any bounded linear transformation of a Banach space that maps a closed convex cone of onto itself. Then the -norm of is defined by(2)for pairs with finite . Birkhoff's inequality then states that if the transform of under has finite diameter under , then(3)(Birkhoff 1957).
A geodesic on a paraboloid(1)(2)(3)has differential parameters defined by(4)(5)(6)(7)(8)(9)The geodesic is then given by solving the Euler-Lagrangedifferential equation(10)As given by Weinstock (1974), the solution simplifies to(11)
The word "median" has several different meanings in mathematics all related to the "middle" of mathematical objects.The statistical median is an order statistic that gives the "middle" value of a sample. More specifically, it is the value such that an equal number of samples are less than and greater than the value (for an odd sample size), or the average of the two central values (for an even sample size). The Wolfram Language function Median[list] can be used to find the statistical median of the elements in a list.A triangle median is the Cevian from one of its vertices to the midpoint of the opposite side. The medians intersect in a point known as the triangle centroid that is sometimes also called the median point.Similarly, a tetrahedron median is a line joining a vertex of a tetrahedron to the geometric centroid of the opposite face.The median of a trapezoid is the line segment determinedby the midpoints of..
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..
The intersection of two sets and is the set of elements common to and . This is written , and is pronounced " intersection " or " cap ." The intersection of sets through is written .The intersection of two lines and is written . The intersection of two or more geometric objects is the point (points, lines, etc.) at which they concur.
The incidence matrix of a graph gives the (0,1)-matrix which has a row for each vertex and column for each edge, and iff vertex is incident upon edge (Skiena 1990, p. 135). However, some authors define the incidence matrix to be the transpose of this, with a column for each vertex and a row for each edge. The physicist Kirchhoff (1847) was the first to define the incidence matrix.The incidence matrix of a graph (using the first definition) can be computed in the Wolfram Language using IncidenceMatrix[g]. Precomputed incidence matrices for a many named graphs are given in the Wolfram Language by GraphData[graph, "IncidenceMatrix"].The incidence matrix of a graph and adjacency matrix of its line graph are related by(1)where is the identity matrix (Skiena 1990, p. 136).For a -D polytope , the incidence matrix is defined by(2)The th row shows which s surround , and the th column shows which s bound . Incidence matrices are also..
The golden angle is the angle that divides a full angle in a golden ratio (but measured in the opposite direction so that it measures less than ), i.e.,(1)(2)(3)(4)(5)(6)(7)(OEIS A131988 and A096627;Livio 2002, p. 112).It is implemented in the Wolfram Languageas GoldenAngle.van Iterson showed in 1907 that points separated by on a tightly bound spiral tends to produce interlocked spirals winding in opposite directions, and that the number of spirals in these two families tend to be consecutive Fibonacci numbers (Livio 2002, p. 112).Another angle related to the golden ratio is theangle(8)or twice this angle(9)the later of which is the smaller interior angle in the goldenrhombus.
There are at least two meanings on the word congruent in mathematics. Two geometric figures are said to be congruent if one can be transformed into the other by an isometry (Coxeter and Greitzer 1967, p. 80). This relationship, called geometric congruence, is written . (Unfortunately, the symbol is also used to denote an isomorphism.)A number is said to be congruent to modulo if ( divides ).
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 approximation for pi given by(1)(2)(3)In the above figure, let , and construct the circle centered at of radius 1. This intersects at point . Now construct the circle about with radius 1. The circles and intersect in , and the line intersects the perpendicular to through in the point . Now construct the point to be a distance 3 along . The line segment is then of length(4)This construction was given by the Polish Jesuit priest Kochansky (Steinhaus 1999).
"The" Sylvester graph is a quintic graph on 36 nodes and 90 edges that is the unique distance-regular graph with intersection array (Brouwer et al. 1989, §13.1.2; Brouwer and Haemers 1993). It is a subgraph of the Hoffman-Singleton graph obtainable by choosing any edge, then deleting the 14 vertices within distance 2 of that edge.It has graph diameter 3, girth 5, graph radius 3, is Hamiltonian, and nonplanar. It has chromatic number 4, edge connectivity 5, vertex connectivity 5, and edge chromatic number 5.It is an integral graph and has graph spectrum (Brouwer and Haemers 1993).The Sylvester graph of a configuration is the set of ordinarypoints and ordinary lines.
It is possible to find six points in the plane, no three on a line and no four on a circle (i.e., none of which are collinear or concyclic), such that all the mutual distances are rational. An example is illustrated by Guy (1994, p. 185).It is not known if a triangle with integer sides, triangle medians, and area exists (although there are incorrect proofs of the impossibility in the literature). However, R. L. Rathbun, A. Kemnitz, and R. H. Buchholz have showed that there are infinitely many triangles with rational sides (Heronian triangles) with two rational triangle medians (Guy 1994, p. 188).
The rational distance problem asks to find a geometric configuration satisfying given properties such that all distances along specific edges are rational numbers. (This is equivalent to having all edge lengths be integers, since the denominators of rational numbers can be cleared by multiplication.)A cuboid whose edges and face diagonals are integers is called an Euler brick. It is not known if there exists a point in a unit square all of whose distances from the corners are rational, although J. H. Conway and M. Guy found an infinite numbers of solutions to the problem of three such distances being integers, which involves solvingwhere , , and are the three distances and is the side length of the square (Guy 1994, p. 181). There are infinitely many solutions of the corresponding problem of integer distances from the corners of an equilateral triangle (Guy 1994, p. 183).In 2001, E. Pegg found a small scalene..
Young's geometry is a finite geometry which satisfiesthe following five axioms: 1. There exists at least one line. 2. Every line of the geometryhas exactly three points on it. 3. Not all points of the geometryare on the same line. 4. For two distinct points, there exists exactly one line on both of them. 5. If a point does not lie on a given line, then there exists exactly one line on that point that does not intersect the given line. Cherowitzo (2006) notes that the last axiom bears a strong resemblance to the parallel postulate of Euclidean geometry.
Three point geometry is a finite geometry subjectto the following four axioms: 1. There exist exactly three points. 2. Two distinct points are on exactly one line.3. Not all the three points are collinear.4. Two distinct lines are on at least one point.Three point geometry is categorical.Like many finite geometries, the number of provable theorems in three point geometry is small. One can prove from this collection of axioms that two distinct lines are on exactly one point and that three point geometry contains exactly three lines. In this sense, three point geometry is extremely simple. On the other hand, note that the axioms say nothing about whether the lines are straight or curved, whereby it is possible that a number of different (but equivalent) visualizations of three point geometry may exist...
Four line geometry is a finite geometry subjectto the following three axioms: 1. there exist exactly four lines, 2. any two distinct lines have exactly one pointof on both of them, and 3. each point is on exactly two lines.Four line geometry is categorical.Like many finite geometries, the number of provable theorems in three point geometry is small. Of those, one can prove that there exist exactly six points and that each line has exactly three points on it. In that regard, four line geometry is among the simplest finite geometries.Note that by forming the plane dual of the four line geometry axioms (that is, by interchanging the terms "point" and "line" throughout the above discussion), one obtains axioms for a four point geometry. In this new (but equivalent) geometry, the plane duals of the above results still hold...
Five point geometry is a finite geometry subjectto the following three axioms: 1. there exist exactly five points, 2. each two distinct points have exactly one lineon both of them, and 3. each line has exactly two points.Five point geometry is categorical.Like many finite geometries, the number of provable theorems in five point geometry is small. One can show that in this scheme, there are exactly 10 lines and that each point has exactly four lines on it.
Fano's geometry is a finite geometry attributed to Fano from around the year 1892. This geometry comes with five axioms, namely: 1. There exists at least one line. 2. Every line has exactly three pointson it. 3. Not all the points are on the same line.4. For two distinct points, there exists exactly one line on both of them. 5. Each two lines have at least one pointon both of them. Fano's geometry is categorical. Even so, there are several different though equivalent visual representations of Fano's geometry. Perhaps the most common is the so-called Fano plane which shows, among other things, that lines in Fano's geometry need not be straight.Like many finite geometries, the number of provable theorems in Fano's geometry is small. One can show that in Fano's geometry, each two lines have exactly one point in common and that the geometry itself consists of exactly seven points and seven lines...
An interval is a connected portion of the real line. If the endpoints and are finite and are included, the interval is called closed and is denoted . If the endpoints are not included, the interval is called open and denoted . If one endpoint is included but not the other, the interval is denoted or and is called a half-closed (or half-open interval).An interval is called a degenerate interval.If one of the endpoints is , then the interval still contains all of its limit points, so and are also closed intervals. Intervals involving infinity are also called rays or half-lines. If the finite point is included, it is a closed half-line or closed ray. If the finite point is not included, it is an open half-line or open ray.The non-standard notation for an open interval and or for a half-closed interval is sometimes also used.A non-empty subset of is an interval iff, for all and , implies . If the empty set is considered to be an interval, then the following are equivalent:..
Count the number of lattice points inside the boundary of a circle of radius with center at the origin. The exact solution is given by the sum(1)(2)(3)(Hilbert and Cohn-Vossen 1999, p. 39). The first few values for , 1, ... are 1, 5, 13, 29, 49, 81, 113, 149, ... (OEIS A000328).The series for is intimately connected with the sum of squares function (i.e., the number of representations of by two squares), since(4)(Hardy 1999, p. 67). is also closely connected with the Leibniz series since(5)where is a Lerch transcendent and is a digamma function, so taking the limit gives(6)(Hilbert and Cohn-Vossen 1999, p. 39).Gauss showed that(7)where(8)(Hardy 1999, p. 67).The first few values of are 5, 13/4, 29/9, 49/16, 81/25, 113/36, 149/49, 197/64, 253/81, 317/100, 377/121, 49/16, ... (OEIS A000328 and A093837). As can be seen in the plot above, the values of such that are , 3, 4, 6, 11, 16, 21, 36, 52, 53, 86, 101, ... (OEIS A093832).Writing..
There are at least two distinct notions of linear space throughout mathematics.The term linear space is most commonly used within functionalanalysis as a synonym of the term vector space.The term is also used to describe a fundamental notion in the field of incidence geometry. In particular, a linear space is a space consisting of a collection of points and a set of lines subject to the following axioms: 1. Any two distinct points of belong to exactly one line of . 2. Any line of has at least two points of . 3. There are at least three points of not on a common line. Using this terminology, lines are considered to be "distinguished subsets" of the collection of points. Moreover, in this context, one can view a linear space as a generalization of the notions of projective space and affine space (Batten and Beutelspracher 2009)...
A projection matrix is an square matrix that gives a vector space projection from to a subspace . The columns of are the projections of the standard basis vectors, and is the image of . A square matrix is a projection matrix iff .A projection matrix is orthogonal iff(1)where denotes the adjoint matrix of . A projection matrix is a symmetric matrix iff the vector space projection is orthogonal. In an orthogonal projection, any vector can be written , so(2)An example of a nonsymmetric projection matrix is(3)which projects onto the line .The case of a complex vector space is analogous. A projection matrix is a Hermitian matrix iff the vector space projection satisfies(4)where the inner product is the Hermitian inner product. Projection operators play a role in quantum mechanics and quantum computing.Any vector in is fixed by the projection matrix for any in . Consequently, a projection matrix has norm equal to one, unless ,(5)Let be a -algebra. An element..
The conjecture that, for any triangle,(1)where , , and are the vertex angles of the triangle and is the Brocard angle. The Abi-Khuzam inequality states that(2)(Yff 1963, Le Lionnais 1983, Abi-Khuzam and Boghossian 1989), which can be used to prove the conjecture (Abi-Khuzam 1974).The maximum value of occurs when two angles are equal, so taking , and using , the maximum occurs at the maximum of(3)which occurs when(4)Solving numerically gives (OEIS A133844), corresponding to a maximum value of approximately 0.440053 (OEIS A133845).
There are a number of algebraic equations known as the icosahedral equation, all of which derive from the projective geometry of the icosahedron. Consider an icosahedron centered , oriented with -axis along a fivefold () rotational symmetry axis, and with one of the top five edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest icosahedral equation is defined by projecting the vertices of the icosahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not the height above..
The tetrahedral equation, by way of analogy with the icosahedral equation, is a set of related equations derived from the projective geometry of the octahedron. Consider a tetrahedron centered , oriented with -axis along a fourfold () rotational symmetry axis, and with one of the top three edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest tetrahedral equation is defined by projecting the vertices of the tetrahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not..
The octahedral equation, by way of analogy with the icosahedral equation, is a set of related equations derived from the projective geometry of the octahedron. Consider an octahedron centered , oriented with -axis along a fourfold () rotational symmetry axis, and with one of the top four edges lying in the -plane (left figure). In this figure, vertices are shown in black, face centers in red, and edge midpoints in blue.The simplest octahedral equation is defined by projecting the vertices of the octahedron with unit circumradius using a stereographic projection from the south pole of its circumsphere onto the plane , and expressing these vertex locations (interpreted as complex quantities in the complex -plane) as roots of an algebraic equation. The resulting projection is shown as the left figure above, with black dots being the vertex positions. The resulting equation is(1)where here refers to the coordinate in the complex plane (not the..
There are several meanings of the word content in mathematics.The content of a polytope or other -dimensional object is its generalized volume (i.e., its "hypervolume"). Just as a three-dimensional object has volume, surface area, and generalized diameter, an -dimensional object has "measures" of order 1, 2, ..., . The content of a region can be computed in the Wolfram Language using RegionMeasure[reg].The content of an integer polynomial , denoted , is the largest integer such that also has integer coefficients. Gauss's lemma for contents states that if and are two polynomials with integer coefficients, then (Séroul 2000, p. 287).For a general univariate polynomial , the Wolfram Language command FactorTermsList[poly, x] returns a list of three elements, the first being the integer content , the second being the polynomial content, i.e., a primitive (with respect to all variables) polynomial that..
The Ochoa curve is the elliptic curvegiven in Weierstrass form asThe complete set of 23 integer solutions (where solutions of the form are counted as a single solution) to this equation consists of , (, 4520), (, 13356), (, 14616), (, 10656), (91, 8172), (227, 4228), (247, 3528), (271, 2592), (455, 200), (499, 3276), (523, 4356), (530, 4660), (599, 7576), (751, 14112), (1003, 25956), (1862, 75778), (3511, 204552), (5287, 381528), (23527, 3607272), (64507, 16382772), (100102, 31670478), and (1657891, 2134685628) (OEIS A141144 and A141145; Stroeker and de Weger 1994).
An elliptic curve of the form for an integer. This equation has a finite number of solutions in integers for all nonzero . If is a solution, it therefore follows that is as well.Uspensky and Heaslet (1939) give elementary solutions for , , and 2, and then give , , , and 1 as exercises. Euler found that the only integer solutions to the particular case (a special case of Catalan's conjecture) are , , and . This can be proved using Skolem's method, using the Thue equation , using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky and Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński and Schinzel (1988, pp. 75-80), and Metsaenkylae (2003).Solutions of the Mordell curve with are summarized in the table below for small .solutions123456none7none8910Values of such that the Mordell curve has no integer solutions are given by 6, 7, 11, 13,..
The only known classically known algebraic curve of curve genus that has an explicit parametrization in terms of standard special functions (Burnside 1893, Brezhnev 2001). This equation is given by(1)The closed portion of the curve has area(2)(3)where is a gamma function.The closed portion of this curve has a parametrization in terms of the Weierstrasselliptic function given by(4)(5)where(6)the half-periods are given by and ranges over complex values (Brezhnev 2001).
Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections.In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers. The geometry of such a ring is determined by its algebraic structure, in particular its prime ideals. Grothendieck defined schemes as the basic geometric objects, which have the same relationship to the geometry of a ring as a manifold to a coordinate chart. The language of category theory evolved at around the same time, largely in response to the needs of the increasing abstraction in algebraic geometry.As a consequence,..
An algebraic curve over a field is an equation , where is a polynomial in and with coefficients in . A nonsingular algebraic curve is an algebraic curve over which has no singular points over . A point on an algebraic curve is simply a solution of the equation of the curve. A -rational point is a point on the curve, where and are in the field .The following table lists the names of algebraic curves of a given degree.ordercurveexamples2quadratic curvecircle, ellipse, hyperbola, parabola3cubic curvecissoid of Diocles, conchoid of de Sluze, folium of Descartes, Maclaurin trisectrix, Maltese cross curve, Mordell curve, Ochoa curve, right strophoid, semicubical parabola, serpentine curve, Tschirnhausen cubic, witch of Agnesi4quartic curveampersand curve, bean curve, bicorn, bicuspid curve, bifoliate, bifolium, bitangent-rich curve, bow, bullet nose, butterfly curve, capricornoid, cardioid, Cartesian ovals, Cassini ovals, conchoid..