A sphere (from Greek Greek , an independent branch of the Indo-European family of languages, is the language of the Greeks. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. In its ancient form, it is the language of classical ancient Greek literature and the New Testament of σφαίρα—sphaira, "globe, ball,"[1]) is a perfectly round geometrical Geometry is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment object in three-dimensional space In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry. Stereometry deals with the measurements of volumes of various solid figures: cylinder, circular cone, truncated cone,, such as the shape of a round ball A ball is a round object with various uses. It is usually spherical but can be ovoid. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for simpler activities, such as catch, marbles and juggling. Balls made from hard-wearing metal are used in. Like a circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius in two dimensions, a perfect sphere is completely symmetrical Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection. The second meaning is a precise and well-defined concept of balance or "patterned self-similarity" that can be demonstrated or proved according around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter of the sphere. The maximum straight distance through the sphere is known as the diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle. The word "diameter" derives from Greek διάμετρος , "diagonal of a circle", from δια- (dia-), "across, through& of the sphere. It passes through the center and is thus twice the radius.

In higher mathematics, a careful distinction is made between the surface In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball or bagel. On the other hand, there are surfaces which cannot be embedded in three- of a sphere (referred to as a “sphere”), and the inside of a sphere (referred to as a “ball 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”). Thus, a sphere in three dimensions is considered to be a two-dimensional spherical surface embedded In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup in three-dimensional Euclidean space In mathematics, Euclidean space refers to the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, while a ball is a solid figure bounded by a sphere.

This article deals with the mathematical concept of a sphere. In physics Physics is a natural science; it is the study of matter and its motion through spacetime and all that derives from these, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the world and universe behave, a sphere is an object (usually idealized for the sake of simplicity) capable of colliding or stacking with other objects which occupy space.

Contents

Volume of a sphere

In 3 dimensions, the volume The volume of any solid, liquid, gas, plasma, theoretical object, or vacuum is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space. Volume is commonly presented in units such as cubic meters, cubic inside a sphere is given by the formula

where r is the radius of the sphere, d = 2r is the diameter of the sphere and π is the constant pi. This formula was first derived by Archimedes Archimedes of Syracuse was a Greek mathematician, physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the leading scientists in classical antiquity. Among his advances in physics are the foundations of hydrostatics, statics and the explanation of the principle of the lever. He is, who showed that the volume of a sphere is 2/3 that of a circumscribed cylinder A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder. The surface area and the volume of a cylinder have been known since. (This assertion follows from Cavalieri's principle If one knows that the volume of a cone is 1/3 base × height, then one can use Cavalieri's principle to derive the fact that the volume of a sphere is 4/3 × π × r 3, where r is the radius.) In modern mathematics, this formula can be derived using integral calculus Integration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral, e.g. disk integration Disk integration is a means of calculating the volume of a solid of revolution, when integrating along the axis of revolution. This method models the generated 3 dimensional shape as a "stack" of an infinite number of disks of infinitesimal thickness. It is possible to use "washers" instead of "disks" (the washer to sum the volumes of an infinite number of circular disks of infinitesimal thickness stacked centered side by side along the x axis from x = 0 where the disk has radius r (i.e. y = r) to x = r where the disk has radius 0 (i.e. y = 0).

At any given x, the incremental volume (δV) is given by the product of the cross-sectional area of the disk The area of a disk is πr2 when the circle has radius r. Here the symbol π (Greek letter pi) denotes, as usual, the constant ratio of the circumference of a circle to its diameter at x and its thickness (δx):

The total volume is the summation of all incremental volumes:

In the limit as δx approaches zero[2] this becomes:

At any given x, a right-angled triangle connects x, y and r to the origin, whence it follows from Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem (in British English) is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle in British English). It states: that:

Thus, substituting y with a function of x gives:

This can now be evaluated:

This volume as described is for a hemisphere. Doubling it gives the volume of a sphere as:

In higher dimensions, the sphere (or hypersphere) is usually referred to as an n-ball. General recursive formulas exist for deriving the volume of an n-ball.

Surface area of a sphere

The surface area Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved then the definition of arc length of a curve. For polyhedra the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface of a sphere is given by the formula

This formula was first derived by Archimedes, based upon the fact that the projection to the lateral surface of a circumscribing cylinder (i.e. the Gall-Peters map projection The Gall-Peters projection is one specialization of a configurable equal-area map projection known as the equal-area cylindric or cylindric equal-area projection. The Gall-Peters achieved considerable notoriety in the late 20th century as the centerpiece of a controversy surrounding the political implications of map design. Maps based on the) is area-preserving. It is also the derivative In calculus the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much a quantity is changing at a given point; for example, the derivative of the position of a vehicle with respect to time is the instantaneous velocity at which the vehicle is traveling. Conversely, the of the formula for the volume with respect to r because the total volume of a sphere of radius r can be thought of as the summation of the volumes of an infinite number of spherical shells of infinitesimal thickness concentrically stacked on top of one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness.

At any given radius r, the incremental volume (δV) is given by the product of the surface area at radius r (A(r)) and the thickness of a shell (δr):

The total volume is the summation of all shell volumes:

In the limit as δr approaches zero[2] this becomes:

Since we have already proved what the volume is, we can substitute V:

Differentiating both sides of this equation with respect to r yields A as a function of r:

Which is generally abbreviated as:

Alternatively, the area element In mathematics, a volume form on a differentiable manifold is a nowhere vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn(T∗M), that is nowhere equal to zero. A manifold has a volume form if and only if it is orientable, and orientable manifolds have on the sphere is given in spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance from a fixed origin, the elevation angle of that point from a fixed plane, and the azimuth angle of its orthogonal projection on that plane, from a fixed direction on the by:

The total area can thus be obtained by integration Integration is an important concept in mathematics which, together with differentiation, forms one of the main operations in calculus. Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral:

Equations in R3

In analytic geometry Analytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on, a sphere with center (x0, y0, z0) and radius r is the locus In mathematics, a locus is a collection of points which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve. For example, in two-dimensional space a line is the locus of points equidistant from two fixed points or from two parallel lines of all points (x, y, z) such that

The points on the sphere with radius r can be parametrized via

(see also trigonometric functions In mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications and spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance from a fixed origin, the elevation angle of that point from a fixed plane, and the azimuth angle of its orthogonal projection on that plane, from a fixed direction on the).

A sphere of any radius centered at zero is an integral surface of the following differential form In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. A differential form of degree k, or k-form, on a smooth manifold M is a smooth section of the kth exterior power of the cotangent bundle of M. The set of all k-forms on M is a:

This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal In mathematics, two vectors are orthogonal if they are perpendicular, i.e., they form a right angle. The word comes from the Greek ὀρθός , meaning "straight", and γωνία (gonia), meaning "angle" to each other.

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 Surface tension is a property of the surface of a liquid. It is what causes the surface portion of liquid to be attracted to another surface, such as that of another portion of liquid locally minimizes surface area. The surface area in relation to the mass of a sphere is called the specific surface area It is a derived scientific value that can be used to determine the type and properties of a material . It is defined either by surface area divided by mass (with units of m²/kg), or surface area divided by the volume (units of m²/m³ or m-1). From the above stated equations it can be expressed as follows:

An image of one of the most accurate man-made spheres, as it refracts Refraction is the change in direction of a wave due to a change in its velocity. This is most commonly observed when a wave passes from one medium to another. Refraction of light is the most commonly observed phenomenon, but any type of wave can refract when it interacts with a medium, for example when sound waves pass from one medium into another the image of Einstein in the background. This sphere was a fused quartz Fused quartz and fused silica are types of glass containing primarily silica in amorphous form. They are manufactured using several different processes. Note that glasses formed by the traditional 'melt-quench' methods (heating the material to melting temperatures, then rapidly cooling to the solid glass phase), are often referred to as 'vitreous', gyroscope A gyroscope is a device for measuring or maintaining orientation, based on the principles of angular momentum. The device is a spinning wheel or disk whose axle is free to take any orientation. This orientation changes much less in response to a given external torque than it would without the large angular momentum associated with the gyroscope's for the Gravity Probe B Gravity Probe B is a satellite-based mission which launched on 20 April 2004. The spaceflight phase lasted until 2005, and data analysis is expected to continue through 2010. Its aim is to measure spacetime curvature near Earth, and thereby the stress-energy tensor (which is related to the distribution and the motion of matter in space) in and experiment, and differs in shape from a perfect sphere by no more than 40 atoms of thickness. It is thought that only neutron stars A neutron star is a type of remnant that can result from the gravitational collapse of a massive star during a Type II, Type Ib or Type Ic supernova event. Such stars are composed almost entirely of neutrons, which are subatomic particles without electrical charge and roughly the same mass as protons. Neutron stars are very hot and are supported are smoother. It was announced on 1 July July 1 is the 182nd day of the year in the Gregorian calendar. There are 183 days remaining until the end of the year. The end of this day marks the halfway point of a leap year. It also falls on the same day of the week as New Year's Day in a leap year 2008 that Australian Australia , officially the Commonwealth of Australia, is a country in the Southern Hemisphere comprising the continental mainland (the world's smallest), the island of Tasmania, and numerous smaller islands in the Indian and Pacific Oceans.N4 Neighbouring countries include Indonesia, East Timor, and Papua New Guinea to the north, the Solomon scientists had created even more perfect spheres, accurate to 0.3 nanometers A nanometre (Greek: νάνος, nanos, "dwarf"; μέτρον, metrοn, "unit of measurement") is a unit of length in the metric system, equal to one billionth of a metre (i.e., 10-9 m or one millionth of a millimetre), as part of an international hunt to find a new global standard kilogram The kilogram is the base unit of mass in the International System of Units (SI, from the French Le Système International d’Unités).[Note 2] The kilogram is defined as being equal to the mass of the International Prototype Kilogram (IPK),[Note 3] which is almost exactly equal to the mass of one liter of water. It is the only SI base unit with.[3]

A sphere can also be defined as the surface formed by rotating a circle A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are the same distance from a given point called the centre. The common distance of the points of a circle from its center is called its radius about any diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints are on the circle. The diameters are the longest chords of the circle. The word "diameter" derives from Greek διάμετρος , "diagonal of a circle", from δια- (dia-), "across, through&. If the circle is replaced by an ellipse In mathematics, an ellipse is the finite or bounded case of a conic section, the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane (the other two cases being the parabola and the hyperbola). It is also the locus of all points of the plane whose distances to two fixed points add to the same, and rotated about the major axis, the shape becomes a prolate spheroid A spheroid is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters, rotated about the minor axis, an oblate spheroid.

Terminology

Pairs of points on a sphere that lie on a straight line through its center are called antipodal points In mathematics, the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite it — so situated that a line drawn from the one to the other passes through the centre of the sphere and forms a true diameter. A great circle A great circle of a sphere is a circle that runs along the surface of that sphere so as to cut it into two equal halves. The great circle therefore has both the same circumference and the same center as the sphere. It is the largest circle that can be drawn on a given sphere 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. Equipped with the great-circle distance, a great circle becomes the Riemannian circle.

If a particular point on a sphere is (arbitrarily) designated as its north pole, then the corresponding antipodal point is called the south pole and the equator is the great circle that is equidistant to them. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the two poles is called the axis of rotation. Circles on the sphere that are parallel to the equator are lines of latitude. This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid).

Hemisphere

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.

The antipodal quotient of the sphere is the surface called the real projective plane, which can also be thought of as the northern hemisphere with antipodal points of the equator identified.

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:

Spheres for n > 2 are sometimes called hyperspheres.

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 (which is embedded in 3-dimensional space).

The surface area of the (n − 1)-sphere of radius 1 is

where Γ(z) is Euler's Gamma function.

Another formula for surface area is

and the volume is the surface area times or

Generalization to metric spaces

More generally, in a metric space (E,d), the sphere of center x and radius r > 0 is the set of points y such that d(x,y) = r.

If the center is a distinguished point considered as origin of E, as 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, as in the case of a unit sphere.

In contrast to a ball, a sphere may be an empty set, even for a large radius. For example, in Zn with Euclidean metric, a sphere of radius r is nonempty only if r2 can be written as sum of n squares of integers.

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.

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.

The Heine-Borel theorem implies 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 closed. Sn is also bounded, therefore it is compact.

Spherical geometry

Great circle on a sphere Main article: Spherical geometry

The basic elements of plane geometry are points and lines. On the sphere, points are defined in the usual sense, but the analogue of "line" may not be immediately apparent. If one measures by arc length one finds that the shortest path connecting two points lying entirely in the sphere is a segment of the great circle containing the points; see geodesic. Many theorems from classical geometry hold true for this spherical geometry as well, but many do not (see parallel postulate). In spherical trigonometry, angles are defined between great circles. Thus spherical trigonometry is different from ordinary trigonometry in many respects. For example, the sum of the interior angles of a spherical triangle exceeds 180 degrees. Also, any two similar spherical triangles are congruent.

Eleven properties of the sphere

In their book Geometry and the imagination[4] 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:

  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 the Meissner body. 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 centers 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 channel 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. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).
  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 placed 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.

Cubes in relation to spheres

For every sphere there are multiple cuboids that may be inscribed within the sphere. When briefly considered it becomes apparent that the largest of the multiple cuboids which may be inscribed is a cube.

See also

Notes

  1. ^ Sphaira, Henry George Liddell, Robert Scott, A Greek-English Lexicon, at Perseus
  2. ^ a b Pages 141, 149. E.J. Borowski, J.M. Borwein. Collins Dictionary of Mathematics. ISBN 0-00-434347-6.
  3. ^ New Scientist | Technology | Roundest objects in the world created
  4. ^ Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. ISBN 0-8284-1087-9.

References

External links

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