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Heptagon
In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using "sept-" (an elision of ''septua-'', a Latin-derived numerical prefix, rather than '' hepta-'', a Greek-derived numerical prefix; both are cognate) together with the Greek suffix "-agon" meaning angle. Regular heptagon A regular heptagon, in which all sides and all angles are equal, has internal angles of 5π/7 radians (128 degrees). Its Schläfli symbol is . Area The area (''A'') of a regular heptagon of side length ''a'' is given by: :A = \fraca^2 \cot \frac \simeq 3.634 a^2. This can be seen by subdividing the unit-sided heptagon into seven triangular "pie slices" with vertices at the center and at the heptagon's vertices, and then halving each triangle using the apothem as the common side. The apothem is half the cotangent of \pi/7, and the area of each of the 14 small triangles is one-fourth of the apothem. The area of a regular heptago ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schläfli Symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more than three dimensions and discovered all their convex regular polytopes, including the six that occur in four dimensions. Definition The Schläfli symbol is a recursive description, starting with for a ''p''-sided regular polygon that is convex. For example, is an equilateral triangle, is a square, a convex regular pentagon, etc. Regular star polygons are not convex, and their Schläfli symbols contain irreducible fractions ''p''/''q'', where ''p'' is the number of vertices, and ''q'' is their turning number. Equivalently, is created from the vertices of , connected every ''q''. For example, is a pentagram; is a pentagon. A regular polyhedron that has ''q'' regular ''p''-sided polygon faces around each vertex is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Number
In geometry and algebra, a real number r is constructible if and only if, given a line segment of unit length, a line segment of length , r, can be constructed with compass and straightedge in a finite number of steps. Equivalently, r is constructible if and only if there is a closed-form expression for r using only integers and the operations for addition, subtraction, multiplication, division, and square roots. The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it can be produced as one of the points of a compass and straight edge construction (an endpoint of a line segment or crossing point of two lines or circles), starting from a given unit length segment. Alternatively and equivalently, taking the two endpoints of the given segment to be the points (0, 0) and (1, 0) of a Cartesian coordinate system, a point is constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compass And Straightedge
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses. The idealized ruler, known as a straightedge, is assumed to be infinite in length, have only one edge, and no markings on it. The compass is assumed to have no maximum or minimum radius, and is assumed to "collapse" when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass; see compass equivalence theorem. Note however that whilst a non-collapsing compass held against a straightedge might seem to be equivalent to marking it, the neusis construction is still impermissible and this is what unmarked really means: see Markable rulers below.) More formall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two together, may be called a polygon. The segments of a polygonal circuit are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' (singular: vertex) or ''corners''. The interior of a solid polygon is sometimes called its ''body''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. Mathematicians are often concerned only with the bounding polygonal chains of simple polygons and they often define a polygon accordingly. A polygonal boundary may be allowed to cross over itself, creating star polygons and other self-intersecting polygons. A polygon is a 2-dimensional example of the more general polytope in any num ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence of regular polygons with an increasing number of sides approximates a circle, if the perimeter or area is fixed, or a regular apeirogon (effectively a straight line), if the edge length is fixed. General properties ''These properties apply to all regular polygons, whether convex or star.'' A regular ''n''-sided polygon has rotational symmetry of order ''n''. All vertices of a regular polygon lie on a common circle (the circumscribed circle); i.e., they are concyclic points. That is, a regular polygon is a cyclic polygon. Together with the property of equal-length sides, this implies that every regular polygon also has an inscribed circle or incircle that is tangent to every side at the midpoint. Thus a regular polygon is a tangential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constructible Polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not. There are infinitely many constructible polygons, but only 31 with an odd number of sides are known. Conditions for constructibility Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides, and they knew how to construct a regular polygon with double the number of sides of a given regular polygon.Bold, Benjamin. ''Famous Problems of Geometry and How to Solve Them'', Dover Publications, 1982 (orig. 1969). This led to the question being posed: is it possible to construct ''all'' regular polygons with compass and straightedge? If not, which ''n''-gons (that is, polygons with ''n'' edges) are constructible and which are not? Carl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angle Trisection
Angle trisection is a classical problem of straightedge and compass construction of ancient Greek mathematics. It concerns construction of an angle equal to one third of a given arbitrary angle, using only two tools: an unmarked straightedge and a compass. Pierre Wantzel proved in 1837 that the problem, as stated, is impossible to solve for arbitrary angles. However, although there is no way to trisect an angle ''in general'' with just a compass and a straightedge, some special angles can be trisected. For example, it is relatively straightforward to trisect a right angle (that is, to construct an angle of measure 30 degrees). It is possible to trisect an arbitrary angle by using tools other than straightedge and compass. For example, neusis construction, also known to ancient Greeks, involves simultaneous sliding and rotation of a marked straightedge, which cannot be achieved with the original tools. Other techniques were developed by mathematicians over the centuries. Bec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Septua-
Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cycle) * dyad, triad (2 parts, 3 parts) * biped, quadruped (2 legs, 4 legs) * September, October, November, December (month 7, month 8, month 9, month 10) * decimal, hexadecimal (base-10, base-16) * septuagenarian, octogenarian (70-79 years old, 80-89 years old) * centipede, millipede (around 100 legs, around 1000 legs) In many European languages there are two principal systems, taken from Latin and Greek, each with several subsystems; in addition, Sanskrit occupies a marginal position. There is also an international set of metric prefixes, which are used in the metric system and which for the most part are either distorted from the forms below or not based on actual number words. Table of number prefixes in English In the following p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hepta-
Numeral or number prefixes are prefixes derived from Numeral (linguistics), numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cycle) * dyad, triad (2 parts, 3 parts) * biped, quadruped (2 legs, 4 legs) * September, October, November, December (month Julius Caesar, 7, month Augustus, 8, month Septimius Severus, 9, month Octavia E. Butler, 10) * decimal, hexadecimal (base-10, base-16) * septuagenarian, octogenarian (70-79 years old, 80-89 years old) * centipede, millipede (around 100 legs, around 1000 legs) In many European languages there are two principal systems, taken from Latin and Greek language, Greek, each with several subsystems; in addition, Sanskrit occupies a marginal position. There is also an international set of metric prefixes, which are used in the metric system and which for the most part are either distorted from the forms belo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Prefix
Numeral or number prefixes are prefixes derived from numerals or occasionally other numbers. In English and many other languages, they are used to coin numerous series of words. For example: * unicycle, bicycle, tricycle (1-cycle, 2-cycle, 3-cycle) * dyad, triad (2 parts, 3 parts) * biped, quadruped (2 legs, 4 legs) * September, October, November, December (month 7, month 8, month 9, month 10) * decimal, hexadecimal (base-10, base-16) * septuagenarian, octogenarian (70-79 years old, 80-89 years old) * centipede, millipede (around 100 legs, around 1000 legs) In many European languages there are two principal systems, taken from Latin and Greek, each with several subsystems; in addition, Sanskrit occupies a marginal position. There is also an international set of metric prefixes, which are used in the metric system and which for the most part are either distorted from the forms below or not based on actual number words. Table of number prefixes in English In the following pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierpont Prime
In number theory, a Pierpont prime is a prime number of the form 2^u\cdot 3^v + 1\, for some nonnegative integers and . That is, they are the prime numbers for which is 3-smooth. They are named after the mathematician James Pierpont, who used them to characterize the regular polygons that can be constructed using conic sections. The same characterization applies to polygons that can be constructed using ruler, compass, and angle trisector, or using paper folding. Except for 2 and the Fermat primes, every Pierpont prime must be 1 modulo 6. The first few Pierpont primes are: It has been conjectured that there are infinitely many Pierpont primes, but this remains unproven. Distribution A Pierpont prime with is of the form 2^u+1, and is therefore a Fermat prime (unless ). If is positive then must also be positive (because 3^v+1 would be an even number greater than 2 and therefore not prime), and therefore the non-Fermat Piermont primes all have the form , when is a posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
