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Encyclopedia > Sphere  A sphere.

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface. In mathematics, a sphere is the set of all points in three-dimensional space (R3) which are at distance r from a fixed point of that space, where r is a positive real number called the radius of the sphere. Thus, in three dimensions, a mathematical sphere is considered to be a spherical surface, rather than the volume contained within it. The fixed point is called the center or centre, and is not part of the sphere itself. The special case of r = 1 is called a unit sphere. Look up sphere in Wiktionary, the free dictionary. ... Image File history File links Sphere-wireframe. ... Image File history File links Sphere-wireframe. ... Symmetry is a characteristic of geometrical shapes, equations and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... For other uses, see Ball (disambiguation). ... An open surface with X-, Y-, and Z-contours shown. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space â€” for practical purposes the kind of space we live in. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. ...

This article deals with the mathematical concept of a sphere. In physics, a sphere is an object (usually idealized for the sake of simplicity) capable of colliding or stacking with other objects which occupy space. A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ...

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In analytic geometry, a sphere with center (x0, y0, z0) and radius r is the locus of all points (x, y, z) such that Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ...

(xx0)2 + (yy0)2 + (zz0)2 = r2.

The points on the sphere with radius r can be parametrized via $x = x_0 + r cos theta ; sin phi$ $y = y_0 + r sin theta ; sin phi qquad (0 leq theta leq 2pi mbox{ and } 0 < phi leq pi ) ,$ $z = z_0 + r cos phi ,$

(see also trigonometric functions and spherical coordinates). In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... This article describes some of the common coordinate systems that appear in elementary mathematics. ...

A sphere of any radius centered at the origin is described by the following differential equation: A simulation of airflow into a duct using the Navier-Stokes equations A differential equation is a mathematical equation for an unknown function of one or several variables which relates the values of the function itself and of its derivatives of various orders. ... $x , dx + y , dy + z , dz = 0.$

This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...

The surface area of a sphere of radius r is Area is the measure of how much exposed area any two dimensional object has. ... $A = 4 pi r^2 ,$

and its enclosed volume is For other uses, see Volume (disambiguation). ... $V = frac{4}{3}pi r^3.$

Radius from volume is $r = left(V frac{3}{4pi}right)^frac{1}{3}.$

The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension locally minimizes surface area. Surface tension is an effect within the surface layer of a liquid that causes that layer to behave as an elastic sheet. ...

An image of one of the most accurate spheres ever created by humans, as it refracts the image of Einstein in the background. This sphere was a fused quartz gyroscope for the Gravity Probe B experiment which differs in shape from a perfect sphere by no more than 40 atoms of thickness. It is thought that only neutron stars are smoother. It was announced on June 15, 2007 that Australian scientists are planning on making even more perfect spheres, accurate to 35 millionths of a millimeter, as part of an international hunt to find a new global standard kilogram.

Terminology

Pairs of points on a sphere that lie on a straight line through its center are called antipodal points. A great circle is a circle on the sphere that has the same center and radius as the sphere, and consequently divides it into two equal parts. The shortest distance between two distinct non-antipodal points on the surface and measured along the surface, is on the unique great circle passing through the two points. In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it â€” so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. ... For the Brisbane bus routes known collectively as the Great Circle Line (598 & 599), see the following list of Brisbane Transport routes A great circle on a sphere A great circle is a circle on the surface of a sphere that has the same diameter as the sphere, dividing the...

A sphere is divided into two equal hemispheres by any plane that passes through its center. If two intersecting planes pass through its center, then they will subdivide the sphere into four lunes or biangles, the vertices of which all coincide with the antipodal points lying on the line of intersection of the planes.

Generalization to other dimensions

Main article: n-sphere

Spheres can be generalized to spaces of any dimension. For any natural number n, an n-sphere, often written as Sn, is the set of points in (n+1)-dimensional Euclidean space which are at a fixed distance r from a central point of that space, where r is, as before, a positive real number. In particular: For other uses, see sphere (disambiguation). ... 2-dimensional renderings (ie. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

• a 0-sphere is a pair of endpoints of an interval (-r, r) of the real line
• a 1-sphere is a circle of radius r
• a 2-sphere is an ordinary sphere
• a 3-sphere is a sphere in 4-dimensional Euclidean space.

Spheres for n > 2 are sometimes called hyperspheres. Circle illustration This article is about the shape and mathematical concept of circle. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... 2-sphere wireframe as an orthogonal projection Just as a stereographic projection can project a spheres surface to a plane, it can also project a 3-spheres surface into 3-space. ...

The n-sphere of unit radius centred at the origin is denoted Sn and is often referred to as "the" n-sphere. Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface, though it is also a 3-dimensional object because it can be embedded in ordinary 3-space.

The surface area of the (n − 1)-sphere of radius 1 is $2 frac{pi^{n/2}}{Gamma(n/2)}$

where Γ(z) is Euler's Gamma function. The Gamma function along part of the real axis In mathematics, the Gamma function (represented by the capitalized Greek letter Î“) is an extension of the factorial function to real and complex numbers. ...

Another formula for surface area is $begin{cases} {(2pi)^{n/2}r^{n-1} over 2 cdot 4 cdots n-2} & mbox{if } n mbox{ is even}; {2(2pi)^{(n-1)/2}r^{n-1} over 1 cdot 3 cdots n-2} & mbox{if } n mbox{ is odd}. end{cases}$

and the volume within is the surface area times ${r over n}$ or $begin{cases} {(2pi)^{n/2}r^n over 2 cdot 4 cdots n} & mbox{if } n mbox{ is even}; {2(2pi)^{(n-1)/2}r^n over 1 cdot 3 cdots n} & mbox{if } n mbox{ is odd}. end{cases}$

Generalization to metric spaces

More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

S(x;r) = { yE | d(x,y) = r }.

If the center is a distinguished point considered as origin of E, e.g. in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken equal to one, i.e. in the case of a unit sphere. In contrast to a ball, a sphere may be empty, even for a large radius. For example, in Zn with Euclidean metric, a sphere of radius r is nonempty only if r 2 can be written as sum of n squares of integers. In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ... In mathematics, unit ball and unit sphere refer to a ball with radius equal to 1. ... In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ... The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ...

Topology

In topology, an n-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric. A MÃ¶bius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... This word should not be confused with homomorphism. ... In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ... This word should not be confused with homomorphism. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

• a 0-sphere is a pair of points with the discrete topology
• a 1-sphere is a circle (up to homeomorphism); thus, for example, (the image of) any knot is a 1-sphere
• a 2-sphere is an ordinary sphere (up to homeomorphism); thus, for example, any spheroid is a 2-sphere

The n-sphere is denoted Sn. It is an example of a compact topological manifold without boundary. A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... A trefoil knot. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ... In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In topology, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More formally, it is the set of points in the closure of S, not belonging to the interior of... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

The Heine-Borel theorem is used in a short proof that a Euclidean n-sphere is compact. The sphere is the inverse image of a one-point set under the continuous function ||x||. Therefore the sphere is a closed. Sn is also bounded. Therefore it is compact. In mathematical analysis, the Heine-Borel theorem, named after Eduard Heine and Ã‰mile Borel, states: A subset of the real numbers R is compact iff it is closed and bounded. ...

Spherical geometry

Main article: Spherical geometry

Eleven properties of the sphere

In their book Geometry and the imagination David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane which can be thought of as a sphere with infinite radius. These properties are: David Hilbert (January 23, 1862, KÃ¶nigsberg, East Prussia â€“ February 14, 1943, GÃ¶ttingen, Germany) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. ... Stefan or Stephan Cohn-Vossen (28 May 1902-25 June 1936) was a German-Jewish mathematician, now best known for his collaboration with David Hilbert on the 1932 book Anschauliche Geometrie. ... This article is about the mathematical construct. ...

1. The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.
The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle. This second part also holds for the plane.
2. The contours and plane sections of the sphere are circles.
This property defines the sphere uniquely.
3. The sphere has constant width and constant girth.
The width of a surface is the distance between pairs of parallel tangent planes. There are numerous other closed convex surfaces which have constant width, for example Meissner's tetrahedron. The girth of a surface is the circumference of the boundary of its orthogonal projection on to a plane. It can be proved that each of these properties implies the other. A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius, the radius of the sphere. This means that every point on the sphere will be an umbilical point.
4. All points of a sphere are umbilics.
At any point on a surface we can find a normal direction which is at right angles to the surface, for the sphere these on the lines radiating out from the center of the sphere. The intersection of a plane containing the normal with the surface will form a curve called a normal section and the curvature of this curve is the sectional curvature. For most points on a surfaces different sections will have different curvatures, the maximum and minimum values of these are called the principal curvatures. It can be proved that any closed surface will have at least four points called umbilical points. At an umbilic all the sectional curvatures are equal, in particular the principal curvature's are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.
For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.
5. The sphere does not have a surface of centers.
For a given normal section there is a circle whose curvature is the same as the sectional curvature, is tangent to the surface and whose center lines along on the normal line. Take the two center corresponding to the maximum and minimum sectional curvatures these are called the focal points, and the set of all such centers forms the focal surface.
For most surfaces the focal surface forms two sheets each of which is a surface and which come together at umbilical points. There are a number of special cases. For canal surfaces one sheet forms a curve and the other sheet is a surface; For cones, cylinders, toruses and cyclides both sheets form curves. For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This is a unique property of the sphere.
6. All geodesics of the sphere are closed curves.
Geodesics are curves on a surface which give the shortest distance between two points. They are generalisation of the concept of a straight line in the plane. For the sphere the geodesics are great circles. There are many other surfaces with this property.
7. Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.
These properties define the sphere uniquely. These properties can be seen by observing soap bubbles. A soap bubble will enclose a fixed volume and due to surface tension it will try to minimize its surface area. Therefore a free floating soap bubble will be approximately a sphere, factors like gravity will cause a slight distortion.
8. The sphere has the smallest total mean curvature among all convex solids with a given surface area.
The mean curvature is the average of the two principal curvatures and as these are constant at all points of the sphere then so is the mean curvature.
9. The sphere has constant positive mean curvature.
The sphere is the only surface without boundary or singularities with constant positive mean curvature. There are other surfaces with constant mean curvature, the minimal surfaces have zero mean curvature.
10. The sphere has constant positive Gaussian curvature.
Gaussian curvature is the product of the two principle curvatures. It is an intrinsic property which can be determined by measuring length and angles and does not depend on the way the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries and the sphere is the only surface without boundary with constant positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature.
11. The sphere is transformed into itself by a three-parameter family of rigid motions.
Consider a unit sphere place at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three coordinate axis, see Euler angles. Thus there is a three parameter family of rotations which transform the sphere onto itself, this is the rotation group, SO(3). The plane is the only other surface with a three parameter family of transformations (translations along the x and y axis and rotations around the origin). Circular cylinders are the only surfaces with two parameter families of rigid motions and the surfaces of revolution and helicoids are the only surfaces with a one parameter family. Results from FactBites:

 Sphere - definition of Sphere in Encyclopedia (609 words) The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension minimizes surface area. The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also a surface area which is 3/2 times the surface area of the sphere.
More results at FactBites »

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