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Inscribed Square Problem
The inscribed square problem, also known as the square peg problem or the Toeplitz conjecture, is an unsolved question in geometry: ''Does every plane simple closed curve contain all four vertices of some square?'' This is true if the curve is convex or piecewise smooth and in other special cases. The problem was proposed by Otto Toeplitz in 1911. Some early positive results were obtained by Arnold Emch and Lev Schnirelmann. The general case remains open. Problem statement Let C be a Jordan curve. A polygon P is inscribed in C if all vertices of P belong to C. The inscribed square problem asks: : ''Does every Jordan curve admit an inscribed square?'' It is ''not'' required that the vertices of the square appear along the curve in any particular order. Examples Some figures, such as circles and squares, admit infinitely many inscribed squares. There is one inscribed square in a triangle for any obtuse triangle, two squares for any right triangle, and three squares ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is ��the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which ��will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background. Paid circulation in 2008 was 9,500 and total circulation was 10,000. ''Mathematics Magazine'' is a continuation of ''Mathematics News Letter'' (1926–1934) and ''National Mathematics Magazine'' (1934–1945). Doris Schattschneider became the first female editor of ''Mathematics Magazine'' in 1981. .. The MAA gives the Carl B. Allendoerfer Awards annually "for articles of expository excellence" published in ''Mathematics Magazine''. See also *''American Mathematical Mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous (mathematics)
In mathematics, the terms continuity, continuous, and continuum are used in a variety of related ways. Continuity of functions and measures * Continuous function * Absolutely continuous function * Absolute continuity of a measure with respect to another measure * Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous. Sometimes it has a less inclusive meaning: a distribution whose c.d.f. is absolutely continuous with respect to Lebesgue measure. This less inclusive sense is equivalent to the condition that every set whose Lebesgue measure is 0 has probability 0. * Geometric continuity * Parametric continuity Continuum * Continuum (set theory), the real line or the corresponding cardinal number * Linear continuum, any ordered set that shares certain properties of the real line * Continuum (topology), a nonempty compact connected metric space (sometimes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rhombus
In plane Euclidean geometry, a rhombus (: rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The rhombus is often called a "diamond", after the Diamonds (suit), diamonds suit in playing cards which resembles the projection of an Octahedron#Orthogonal projections, octahedral diamond, or a lozenge (shape), lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle (which some authors call a calisson after calisson, the French sweet—also see Polyiamond), and the latter sometimes refers specifically to a rhombus with a 45° angle. Every rhombus is simple polygon, simple (non-self-intersecting), and is a special case of a parallelogram and a Kite (geometry), kite. A rhombus with right angles is a square. Etymology The word "rhombus" comes from , meaning something that spins, which derives from the verb , roman ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwise it is even—as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. Perpendicular intersections can happen between two lines (or two line segments), between a line and a plane, and between two planes. ''Perpendicular'' is also used as a noun: a perpendicular is a line which is perpendicular to a given line or plane. Perpendicularity is one particular instance of the more general mathematical concept of '' orthogonality''; perpendicularity is the orthogonality of classical geometric objects. Thus, in advanced mathematics, the word "perpendicular" is sometimes used to describe much more complicated geometric orthogonality conditions, such as that between a surface and its '' normal vector''. A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel (geometry)
In geometry, parallel lines are coplanar infinite straight line (geometry), lines that do not intersecting lines, intersect at any point. Parallel planes are plane (geometry), planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not tangent, touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called ''skew lines''. Line segments and Euclidean vectors are parallel if they have the same direction (geometry), direction or opposite direction (geometry), opposite direction (not necessarily the same length). Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometry, affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Segments
In geometry, a line segment is a part of a straight line that is bounded by two distinct endpoints (its extreme points), and contains every point on the line that is between its endpoints. It is a special case of an '' arc'', with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using an overline ( vinculum) above the symbols for the two endpoints, such as in . Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. When the end points both lie on a curve (such as a circle), a line segment is called a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Line
In geometry, a secant is a line (geometry), line that intersects a curve at a minimum of two distinct Point (geometry), points.. The word ''secant'' comes from the Latin word ''secare'', meaning ''to cut''. In the case of a circle, a secant intersects the circle at exactly two points. A Chord (geometry), chord is the line segment determined by the two points, that is, the interval (mathematics), interval on the secant whose ends are the two points. Circles A straight line can intersect a circle at zero, one, or two points. A line with intersections at two points is called a ''secant line'', at one point a ''tangent line'' and at no points an ''exterior line''. A ''chord'' is the line segment that joins two distinct points of a circle. A chord is therefore contained in a unique secant line and each secant line determines a unique chord. In rigorous modern treatments of plane geometry, results that seem obvious and were assumed (without statement) by Euclid in Euclid's Elements, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Midpoint
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dimensional space whose endpoints are A = (a_1, a_2, \dots , a_n) and B = (b_1, b_2, \dots , b_n) is given by :\frac. That is, the ''i''th coordinate of the midpoint (''i'' = 1, 2, ..., ''n'') is :\frac 2. Construction Given two points of interest, finding the midpoint of the line segment they determine can be accomplished by a compass and straightedge construction. The midpoint of a line segment, embedded in a plane, can be located by first constructing a lens using circular arcs of equal (and large enough) radii centered at the two endpoints, then connecting the cusps of the lens (the two points where the arcs intersect). The point where the line connecting the cusps intersects the segment is then the midpoint of the segment. It i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Curve
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function is analytic if and only if for every x_0 in its domain, its Taylor series about x_0 converges to the function in some neighborhood of x_0 . This is stronger than merely being infinitely differentiable at x_0 , and therefore having a well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. Definitions Formally, a function f is ''real analytic'' on an open set D in the real line if for any x_0\in D one can write f(x) = \sum_^\infty a_ \left( x-x_0 \right)^ = a_0 + a_1 (x-x_0) + a_2 (x-x_0)^2 + \cdots in which the coefficients a_0, a_1, \dots are rea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise
In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be defined differently. Piecewise definition is actually a way of specifying the function, rather than a characteristic of the resulting function itself, as every function whose domain contains at least two points can be rewritten as a piecewise function. The first three paragraphs of this article only deal with this first meaning of "piecewise". Terms like piecewise linear, piecewise smooth, piecewise continuous, and others are also very common. The meaning of a function being piecewise P, for a property P is roughly that the domain of the function can be partitioned into pieces on which the property P holds, but is used slightly differently by different authors. Unlike the first meaning, this is a property of the function itself and not on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
