Perimeter
A perimeter is a path that surrounds a two-dimensional shape. The term may be used either for the path itself or its length—it can be thought of as the length of the outline of a shape; the perimeter of a circle or ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a garden; the perimeter of a wheel describes. The amount of string wound around a spool is related to the spool's perimeter; the perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with ∫ 0 L d s, where L is the length of the path and d s is an infinitesimal line element. Both of these must be replaced with by algebraic forms in order to be calculated. If the perimeter is given as a closed piecewise smooth plane curve γ: → R 2 with γ = its length L can be computed as follows: L = ∫ a b x ′ 2 + y ′ 2 d t A generalized notion of perimeter, which includes hypersurfaces bounding volumes in n -dimensional Euclidean spaces, is described by the theory of Caccioppoli sets.
Polygons are fundamental to determining perimeters, not only because they are the simplest shapes but because the perimeters of many shapes are calculated by approximating them with sequences of polygons tending to these shapes. The first mathematician known to have used this kind of reasoning is Archimedes, who approximated the perimeter of a circle by surrounding it with regular polygons; the perimeter of a polygon equals the sum of the lengths of its sides. In particular, the perimeter of a rectangle of width w and length ℓ equals 2 w + 2 ℓ. An equilateral polygon is a polygon. To calculate the perimeter of an equilateral polygon, one must multiply the common length of the sides by the number of sides. A regular polygon may be characterized by the number of its sides and by its circumradius, to say, the constant distance between its centre and each of its vertices; the length of its sides can be calculated using trigonometry. If R is a regular polygon's radius and n is the number of its sides its perimeter is 2 n R sin .
A splitter of a triangle is a cevian that divides the perimeter into two equal lengths, this common length being called the semiperimeter of the triangle. The three splitters of a triangle all intersect each other at the Nagel point of the triangle. A cleaver of a triangle is a segment from the midpoint of a side of a triangle to the opposite side such that the perimeter is divided into two equal lengths; the three cleavers of a triangle all intersect each other at the triangle's Spieker center. The perimeter of a circle called the circumference, is proportional to its diameter and its radius; that is to say, there exists a constant number pi, π, such that if P is the circle's perimeter and D its diameter P = π ⋅ D. In terms of the radius r of the circle, this formula becomes, P = 2 π ⋅ r. To calculate a circle's perimeter, knowledge of its radius or diameter and the number π suffices; the problem is that π is not rational, nor is it algebraic. So, obtaining an accurate approximation of π is important in the calculation.
The computation of the digits of π is relevant to many fields, such as mathematical analysis and computer science. The perimeter and the area are two main measures of geometric figures. Confusing them is a common error, as well as believing that the greater one of them is, the greater the other must be. Indeed, a commonplace observation is that an enlargement of a shape make its area grow as well as its perimeter. For example, if a field is drawn on a 1/10,000 scale map, the actual field perimeter can be calculated multiplying the drawing perimeter by 10,000; the real area is 10,0002 times the area of the shape on the map. There is no relation between the area and the perimeter of an ordinary shape. F
Dihedral group
In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, they play an important role in group theory and chemistry; the notation for the dihedral group differs in abstract algebra. In geometry, Dn or Dihn refers to the symmetries of a group of order 2n. In abstract algebra, D2n refers to this same dihedral group; the geometric convention is used in this article. A regular polygon with n sides has 2 n different symmetries: n rotational symmetries and n reflection symmetries. We take n ≥ 3 here; the associated rotations and reflections make up the dihedral group D n. If n is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If n is there are n/2 axes of symmetry connecting the midpoints of opposite sides and n / 2 axes of symmetry connecting opposite vertices. In either case, there are 2 n elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.
The following picture shows the effect of the sixteen elements of D 8 on a stop sign: The first row shows the effect of the eight rotations, the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left. As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group; the following Cayley table shows the effect of composition in the group D3. R0 denotes the identity. For example, s2s1 = r1, because the reflection s1 followed by the reflection s2 results in a rotation of 120°; the order of elements denoting the composition is right to left, reflecting the convention that the element acts on the expression to its right. The composition operation is not commutative. In general, the group Dn has elements r0, …, rn−1 and s0, …, sn−1, with composition given by the following formulae: r i r j = r i + j, r i s j = s i + j, s i r j = s i − j, s i s j = r i − j.
In all cases and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin elements of the dihedral group act as linear transformations of the plane; this lets us represent elements of Dn with composition being matrix multiplication. This is an example of a group representation. For example, the elements of the group D4 can be represented by the following eight matrices: r 0 =, r 1 =, r 2 =, r 3 =, s 0 =, s 1 =, s 2 =
Cylinder
A cylinder has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. It is the idealized version of a solid physical tin can having lids on bottom; this traditional view is still used in elementary treatments of geometry, but the advanced mathematical viewpoint has shifted to the infinite curvilinear surface and this is how a cylinder is now defined in various modern branches of geometry and topology. The shift in the basic meaning has created some ambiguity with terminology, it is hoped that context makes the meaning clear. In this article both points of view are presented and distinguished by referring to solid cylinders and cylindrical surfaces, but keep in mind that in the literature the unadorned term cylinder could refer to either of these or to an more specialized object, the right circular cylinder; the definitions and results in this section are taken from the 1913 text and Solid Geometry by George Wentworth and David Eugene Smith. A cylindrical surface is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to the given line.
Any line in this family of parallel lines is called an element of the cylindrical surface. From a kinematics point of view, given a plane curve, called the directrix, a cylindrical surface is that surface traced out by a line, called the generatrix, not in the plane of the directrix, moving parallel to itself and always passing through the directrix. Any particular position of the generatrix is an element of the cylindrical surface. A solid bounded by a cylindrical surface and two parallel planes is called a cylinder; the line segments determined by an element of the cylindrical surface between the two parallel planes is called an element of the cylinder. All the elements of a cylinder have equal lengths; the region bounded by the cylindrical surface in either of the parallel planes is called a base of the cylinder. The two bases of a cylinder are congruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a right cylinder, otherwise it is called an oblique cylinder.
If the bases are disks the cylinder is called a circular cylinder. In some elementary treatments, a cylinder always means a circular cylinder; the height of a cylinder is the perpendicular distance between its bases. The cylinder obtained by rotating a line segment about a fixed line that it is parallel to is a cylinder of revolution. A cylinder of revolution is a right circular cylinder; the height of a cylinder of revolution is the length of the generating line segment. The line that the segment is revolved about is called the axis of the cylinder and it passes through the centers of the two bases; the bare term cylinder refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an open cylinder; the formulae for the surface area and the volume of a right circular cylinder have been known from early antiquity. A right circular cylinder can be thought of as the solid of revolution generated by rotating a rectangle about one of its sides.
These cylinders are used in an integration technique for obtaining volumes of solids of revolution. A cylindric section is the intersection of a cylinder's surface with a plane, they are, in general and are special types of plane sections. The cylindric section by a plane that contains two elements of a cylinder is a parallelogram; such a cylindric section of a right cylinder is a rectangle. A cylindric section in which the intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section. If a right section of a cylinder is a circle the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a conic section the solid cylinder is said to be parabolic, elliptic or hyperbolic respectively. For a right circular cylinder, there are several ways. First, consider planes that intersect a base in at most one point. A plane is tangent to the cylinder; the right sections are circles and all other planes intersect the cylindrical surface in an ellipse.
If a plane intersects a base of the cylinder in two points the line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. If a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle. In the case of a right circular cylinder with a cylindric section, an ellipse, the eccentricity e of the cylindric section and semi-major axis a of the cylindric section depend on the radius of the cylinder r and the angle α between the secant plane and cylinder axis, in the following way: e = cos α, a = r sin α. If the base of a circular cylinder has a radius r and the cylinder has height h its volume is given by V = πr2h; this formula holds. This formula may be established by using Cavalieri's principle. In more generality, by the same principle, the volume of an
Conway polyhedron notation
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations. Conway and Hart extended the idea of using operators, like truncation as defined by Kepler, to build related polyhedra of the same symmetry. For example, tC represents a truncated cube, taC, parsed as t, is a truncated cuboctahedron; the simplest operator dual swaps vertex and face elements. Applied in a series, these operators allow many higher order polyhedra to be generated. Conway defined the operators abdegjkmost, while Hart added r and p. Conway's basic operations are sufficient to generate the Archimedean and Catalan solids from the Platonic solids; some basic operations can be made as composites of others. Implementations named further operators, sometimes referred to as "extended" operators. In general, it is difficult to predict the resulting appearance of the composite of two or more operations from a given seed polyhedron.
For instance, ambo applied twice is the expand operation: aa = e, while a truncation after ambo produces bevel: ta = b. Many basic questions about Conway operators remain open, for instance, how many operators of a given "size" exist. In Conway's notation, operations on polyhedra are applied from right to left. For example, a cuboctahedron is an ambo cube, i.e. a = a C, a truncated cuboctahedron is t = t = t a C. Repeated application of an operator can be denoted with an exponent: j2. In general, Conway operators are not commutative; the resulting polyhedron has a fixed topology, while exact geometry is not specified: it can be thought of as one of many embeddings of a polyhedral graph on the sphere. The polyhedron is put into canonical form. Individual operators can be visualized in terms of "chambers", as below; each white chamber is a rotated version of the others. For achiral operators, the red chambers are a reflection of the white chambers. Achiral and chiral operators are called local symmetry-preserving operations and local operations that preserve orientation-preserving symmetries although the exact definition is a little more restrictive.
The relationship between the number of vertices and faces of the seed and the polyhedron created by the operations listed in this article can be expressed as a matrix M x. When x is the operator, v, e, f are the vertices and faces of the seed, v ′, e ′, f ′ are the vertices and faces of the result M x =; the matrix for the composition of two operators is just the product of the matrixes for the two operators. Distinct operators may have the same matrix, for p and l; the edge count of the result is an integer multiple d of that of the seed: this is called the inflation rate, or the edge factor. The simplest operators, the identity operator S and the dual operator d, have simple matrix forms: M S = = I 3, M d = Two dual operators cancel out; when applied to other operators, the dual operator corresponds to horizontal and vertical reflections of the matrix. Operators can be grouped into groups of four by identifying the operators x, xd, dx, dxd. In this article, only the matrix for x is given. Hart introduced the reflection operator r.
This is not a LOPSP, since it does not preserve orientation. R has no effect on achiral seeds, rr returns the original seed. An overline can be used to indicate the other chiral form of an operator. R does not affect the matrix. An operation is irreducible if it cannot be expressed as a composition of operators aside from d and r; the majority of Conway's original operators are irreducible: the exceptions are e, b, o, m. Some open questions about Conway
Geometry
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp
Rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal, it can be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square; the term oblong is used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD; the word rectangle comes from the Latin rectangulus, a combination of rectus and angulus. A crossed rectangle is a crossed quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals, it is a special case of an antiparallelogram, its angles are not right angles. Other geometries, such as spherical and hyperbolic, have so-called rectangles with opposite sides equal in length and equal angles that are not right angles. Rectangles are involved in many tiling problems, such as tiling the plane by rectangles or tiling a rectangle by polygons. A convex quadrilateral is a rectangle if and only if it is any one of the following: a parallelogram with at least one right angle a parallelogram with diagonals of equal length a parallelogram ABCD where triangles ABD and DCA are congruent an equiangular quadrilateral a quadrilateral with four right angles a quadrilateral where the two diagonals are equal in length and bisect each other a convex quadrilateral with successive sides a, b, c, d whose area is 1 4.
A convex quadrilateral with successive sides a, b, c, d whose area is 1 2. A rectangle is a special case of a parallelogram in which each pair of adjacent sides is perpendicular. A parallelogram is a special case of a trapezium in which both pairs of opposite sides are parallel and equal in length. A trapezium is a convex quadrilateral. A convex quadrilateral is Simple: The boundary does not cross itself. Star-shaped: The whole interior is visible from a single point, without crossing any edge. De Villiers defines a rectangle more as any quadrilateral with axes of symmetry through each pair of opposite sides; this definition crossed rectangles. Each has an axis of symmetry parallel to and equidistant from a pair of opposite sides, another, the perpendicular bisector of those sides, but, in the case of the crossed rectangle, the first axis is not an axis of symmetry for either side that it bisects. Quadrilaterals with two axes of symmetry, each through a pair of opposite sides, belong to the larger class of quadrilaterals with at least one axis of symmetry through a pair of opposite sides.
These quadrilaterals crossed isosceles trapezia. A rectangle is cyclic: all corners lie on a single circle, it is equiangular: all its corner angles are equal. It is isogonal or vertex-transitive: all corners lie within the same symmetry orbit, it has two lines of reflectional symmetry and rotational symmetry of order 2. The dual polygon of a rectangle is a rhombus; the figure formed by joining, in order, the midpoints of the sides of a rectangle is a rhombus and vice versa. A rectangle is rectilinear: its sides meet at right angles. A rectangle in the plane can be defined by five independent degrees of freedom consisting, for example, of three for position, one for shape, one for overall size. Two rectangles, neither of which will fit inside the other, are said to be incomparable. If a rectangle has length ℓ and width w it has area A = ℓ w, it has perimeter P = 2 ℓ + 2 w = 2, each diagonal has length d = ℓ 2 + w 2, when ℓ = w, the rectangle is a square; the isoperimetric theorem for rectangles states that among all rectangles of a given perimeter, the square has the largest area.
The midpoints of the sides of any quadrilateral with perpendicular diagonals form a rectangle. A parallelogram with equal diagonals is a rectangle; the Japanese theorem for cyclic quadrilaterals states that the incentres of the four triangles determined by the vertices of a cyclic quadrilateral taken three at a time form a rectangle. The British flag theorem states that with vertices denoted A, B, C, D, for any point P on the same plane of a rectangle: 2 + 2 = 2 + 2
Cross section (geometry)
In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross sections; the boundary of a cross section in three-dimensional space, parallel to two of the axes, that is, parallel to the plane determined by these axes, is sometimes referred to as a contour line. In technical drawing a cross-section, being a projection of an object onto a plane that intersects it, is a common tool used to depict the internal arrangement of a 3-dimensional object in two dimensions, it is traditionally crosshatched with the style of crosshatching indicating the types of materials being used. With computed axial tomography, computers construct cross-sections from x-ray data. If a plane intersects a solid the region common to the plane and the solid is called a cross-section of the solid. A plane containing a cross-section of the solid may be referred to as a cutting plane.
The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid. For instance, while all the cross-sections of a ball are disks, the cross-sections of a cube depend on how the cutting plane is related to the cube. If the cutting plane is perpendicular to a line joining the centers of two opposite faces of the cube, the cross-section will be a square, however, if the cutting plane is perpendicular to a diagonal of the cube joining opposite vertices, the cross-section can be either a point, a triangle or a hexagon. A related concept is that of a plane section, the curve of intersection of a plane with a surface. Thus, a plane section is the boundary of a cross-section of a solid in a cutting plane. If a surface in a three-dimensional space is defined by a function of two variables, i.e. z = f, the plane sections by cutting planes that are parallel to a coordinate plane are called level curves or isolines. More cutting planes with equations of the form z = k produce plane sections that are called contour lines in application areas.
A cross section of a polyhedron is a polygon. The conic sections – circles, ellipses and hyperbolas – are plane sections of a cone with the cutting planes at various different angles, as seen in the diagram at left. Any cross-section passing through the center of an ellipsoid forms an elliptic region, while the corresponding plane sections are ellipses on its surface; these degenerate to disks and circles when the cutting planes are perpendicular to a symmetry axis. In more generality, the plane sections of a quadric are conic sections. A cross-section of a solid right circular cylinder extending between two bases is a disk if the cross-section is parallel to the cylinder's base, or an elliptic region if it is neither parallel nor perpendicular to the base. If the cutting plane is perpendicular to the base it consists of a rectangle unless it is just tangent to the cylinder, in which case it is a single line segment; the term cylinder can mean the lateral surface of a solid cylinder. If cylinder is used in this sense, the above paragraph would read as follows: A plane section of a right circular cylinder of finite length is a circle if the cutting plane is perpendicular to the cylinder's axis of symmetry, or an ellipse if it is neither parallel nor perpendicular to that axis.
If the cutting plane is parallel to the axis the plane section consists of a pair of parallel line segments unless the cutting plane is tangent to the cylinder, in which case, the plane section is a single line segment. A plane section can be used to visualize the partial derivative of a function with respect to one of its arguments, as shown. Suppose z = f. In taking the partial derivative of f with respect to x, one can take a plane section of the function f at a fixed value of y to plot the level curve of z against x. A plane section of a probability density function of two random variables in which the cutting plane is at a fixed value of one of the variables is a conditional density function of the other variable. If instead the plane section is taken for a fixed value of the density, the result is an iso-density contour. For the normal distribution, these contours are ellipses. In economics, a production function f specifies the output that can be produced by various quantities x and y of inputs labor and physical capital.
The production function of a firm or a society can be plotted in three-dimensional space. If a plane section is taken parallel to the xy-plane, the result is an isoquant showing the various combinations of labor and capital usage that would result in the level of output given by the height of the plane section. Alternatively, if a plane section of the production function is taken at a fixed level of y—that is, parallel to the xz-plane—then the result is a two-dimensional graph showing how much output can be produced at each of various values of usage x of one input combined with the fixed value of the other input y. In economics, a cardinal or ordinal utility function u gives the degree of satisfaction of a consumer obtained by consuming quantities w and v of two goods. If a plane section of the utility function is taken at a given height (level of u