44-457: A whispering gallery is usually a circular, hemispherical , elliptical or ellipsoidal enclosure, often beneath a dome or a vault , in which whispers can be heard clearly in other parts of the gallery. Such galleries can also be set up using two parabolic dishes. Sometimes the phenomenon is detected in caves. A whispering gallery is most simply constructed in the form of a circular wall, and allows whispered communication from any part of
88-538: A code point in Unicode at U+2300 ⌀ DIAMETER SIGN , in the Miscellaneous Technical set. It should not be confused with several other characters (such as U+00D8 Ø LATIN CAPITAL LETTER O WITH STROKE or U+2205 ∅ EMPTY SET ) that resemble it but have unrelated meanings. It has the compose sequence Compose d i . The diameter of
132-457: A convex shape in the plane , the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the width is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers . For a curve of constant width such as the Reuleaux triangle , the width and diameter are
176-425: A more general definition that is valid for any kind of n {\displaystyle n} -dimensional (convex or non-convex) object, such as a hypercube or a set of scattered points. The diameter or metric diameter of a subset of a metric space is the least upper bound of the set of all distances between pairs of points in the subset. Explicitly, if S {\displaystyle S}
220-467: A right-angled triangle connects x , y and r to the origin; hence, applying the Pythagorean theorem yields: Using this substitution gives which can be evaluated to give the result An alternative formula is found using spherical coordinates , with volume element so For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of
264-479: A sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical plane) in the pencil. 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
308-517: A sphere is the boundary of a (closed or open) ball. The distinction between ball and sphere has not always been maintained and especially older mathematical references talk about a sphere as a solid. The distinction between " circle " and " disk " in the plane is similar. Small spheres or balls are sometimes called spherules (e.g., in Martian spherules ). In analytic geometry , a sphere with center ( x 0 , y 0 , z 0 ) and radius r
352-413: A sphere to be a two-dimensional closed surface embedded in three-dimensional Euclidean space . They draw a distinction between a sphere and a ball , which is a three-dimensional manifold with boundary that includes the volume contained by the sphere. An open ball excludes the sphere itself, while a closed ball includes the sphere: a closed ball is the union of the open ball and the sphere, and
396-407: A unique circle in a plane. Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle. By examining the common solutions of the equations of two spheres , it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. Although the radical plane
440-427: A visitor stands at one focus and whispers, the line of sound emanating from this focus reflects directly to the focus at the other end of the gallery, where the whispers may be heard. In a similar way, two large concave parabolic dishes , serving as acoustic mirrors , may be erected facing each other in a room or outdoors to serve as a whispering gallery, a common feature of science museums. Egg-shaped galleries, such as
484-421: Is a geometrical object that is a three-dimensional analogue to a two-dimensional circle . Formally, a sphere is the set of points that are all at the same distance r from a given point in three-dimensional space . That given point is the center of the sphere, and r is the sphere's radius . The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians . The sphere
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#1732787256225528-586: Is a fundamental object in many fields of mathematics . Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography , and the celestial sphere is an important concept in astronomy . Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings . As mentioned earlier r
572-462: Is a real plane, the circle may be imaginary (the spheres have no real point in common) or consist of a single point (the spheres are tangent at that point). The angle between two spheres at a real point of intersection is the dihedral angle determined by the tangent planes to the spheres at that point. Two spheres intersect at the same angle at all points of their circle of intersection. They intersect at right angles (are orthogonal ) if and only if
616-433: Is an equation of a sphere whose center is P 0 {\displaystyle P_{0}} and whose radius is ρ {\displaystyle {\sqrt {\rho }}} . If a in the above equation is zero then f ( x , y , z ) = 0 is the equation of a plane. Thus, a plane may be thought of as a sphere of infinite radius whose center is a point at infinity . A parametric equation for
660-547: Is called the major axis . The word "diameter" is derived from Ancient Greek : διάμετρος ( diametros ), "diameter of a circle", from διά ( dia ), "across, through" and μέτρον ( metron ), "measure". It is often abbreviated DIA , dia , d , {\displaystyle {\text{DIA}},{\text{dia}},d,} or ∅ . {\displaystyle \varnothing .} The definitions given above are only valid for circles, spheres and convex shapes. However, they are special cases of
704-497: Is the locus of all points ( x , y , z ) such that Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface , a type of algebraic surface . Let a, b, c, d, e be real numbers with a ≠ 0 and put Then the equation has no real points as solutions if ρ < 0 {\displaystyle \rho <0} and is called the equation of an imaginary sphere . If ρ = 0 {\displaystyle \rho =0} ,
748-446: Is the same as the diameter of its convex hull . In medical terminology concerning a lesion or in geology concerning a rock, the diameter of an object is the least upper bound of the set of all distances between pairs of points in the object. In differential geometry , the diameter is an important global Riemannian invariant . In planar geometry , a diameter of a conic section is typically defined as any chord which passes through
792-449: Is the sphere's radius; any line from the center to a point on the sphere is also called a radius. 'Radius' is used in two senses: as a line segment and also as its length. If a radius is extended through the center to the opposite side of the sphere, it creates a diameter . Like the radius, the length of a diameter is also called the diameter, and denoted d . Diameters are the longest line segments that can be drawn between two points on
836-453: Is the subset and if ρ {\displaystyle \rho } is the metric , the diameter is diam ( S ) = sup x , y ∈ S ρ ( x , y ) . {\displaystyle \operatorname {diam} (S)=\sup _{x,y\in S}\rho (x,y).} If the metric ρ {\displaystyle \rho }
880-511: Is the summation of all shell volumes: In the limit as δr approaches zero this equation becomes: Substitute V : Differentiating both sides of this equation with respect to r yields A as a function of r : This is generally abbreviated as: where r is now considered to be the fixed radius of the sphere. Alternatively, the area element on the sphere is given in spherical coordinates by dA = r sin θ dθ dφ . The total area can thus be obtained by integration : The sphere has
924-400: Is viewed here as having codomain R {\displaystyle \mathbb {R} } (the set of all real numbers ), this implies that the diameter of the empty set (the case S = ∅ {\displaystyle S=\varnothing } ) equals − ∞ {\displaystyle -\infty } ( negative infinity ). Some authors prefer to treat
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#1732787256225968-684: The Gol Gumbaz mausoleum in Bijapur, India and the Echo Wall of the Temple of Heaven in Beijing. A hemispherical enclosure will also guide whispering gallery waves. The waves carry the words so that others will be able to hear them from the opposite side of the gallery. The gallery may also be in the form of an ellipse or ellipsoid , with an accessible point at each focus . In this case, when
1012-572: The Golghar Granary at Bankipore, and irregularly shaped smooth-walled galleries in the form of caves, such as the Ear of Dionysius in Syracuse, also exist. The term whispering gallery has been borrowed in the physical sciences to describe other forms of whispering-gallery waves such as light or matter waves . Sphere#Hemisphere A sphere (from Greek σφαῖρα , sphaîra )
1056-478: The conic's centre ; such diameters are not necessarily of uniform length, except in the case of the circle, which has eccentricity e = 0. {\displaystyle e=0.} The symbol or variable for diameter, ⌀ , is sometimes used in technical drawings or specifications as a prefix or suffix for a number (e.g. "⌀ 55 mm"), indicating that it represents diameter. Photographic filter thread sizes are often denoted in this way. The symbol has
1100-414: The volume inside a sphere (that is, the volume of a ball , but classically referred to as the volume of a sphere) is where r is the radius and d is the diameter of the sphere. Archimedes first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere (having the height and diameter equal to the diameter of
1144-399: The x -axis from x = − r to x = r , assuming the sphere of radius r is centered at the origin. At any given x , the incremental volume ( δV ) equals the product of the cross-sectional area of the disk at x and its thickness ( δx ): The total volume is the summation of all incremental volumes: In the limit as δx approaches zero, this equation becomes: At any given x ,
1188-476: The above stated equations as where ρ is the density (the ratio of mass to volume). A sphere can be constructed as the surface formed by rotating a circle one half revolution about any of its diameters ; this is very similar to the traditional definition of a sphere as given in Euclid's Elements . Since a circle is a special type of ellipse , a sphere is a special type of ellipsoid of revolution . Replacing
1232-442: The circle with an ellipse rotated about its major axis , the shape becomes a prolate spheroid ; rotated about the minor axis, an oblate spheroid. A sphere is uniquely determined by four points that are not coplanar . More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. This property is analogous to the property that three non-collinear points determine
1276-423: The cube, since V = π / 6 d , where d is the diameter of the sphere and also the length of a side of the cube and π / 6 ≈ 0.5236. For example, a sphere with diameter 1 m has 52.4% the volume of a cube with edge length 1 m, or about 0.524 m . The surface area of a sphere of radius r is: Archimedes first derived this formula from
1320-420: The diameter of a sphere . In more modern usage, the length d {\displaystyle d} of a diameter is also called the diameter. In this sense one speaks of the diameter rather than a diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r . {\displaystyle r.} For
1364-401: 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 ) equals the product of the surface area at radius r ( A ( r ) ) and the thickness of a shell ( δr ): The total volume
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1408-414: The empty set as a special case, assigning it a diameter of 0 , {\displaystyle 0,} which corresponds to taking the codomain of ρ {\displaystyle \rho } to be the set of nonnegative reals. For any solid object or set of scattered points in n {\displaystyle n} -dimensional Euclidean space , the diameter of the object or set
1452-523: The fact that the projection to the lateral surface of a circumscribed cylinder is area-preserving. Another approach to obtaining the formula comes from the fact that it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r . At infinitesimal thickness
1496-486: The internal side of the circumference to any other part. The sound is carried by waves, known as whispering-gallery waves , that travel around the circumference clinging to the walls, an effect that was discovered in the whispering gallery of St Paul's Cathedral in London. The extent to which the sound travels at St Paul's can also be judged by clapping in the gallery, which produces four echoes. Other historical examples are
1540-550: The only solution of f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0} is the point P 0 = ( x 0 , y 0 , z 0 ) {\displaystyle P_{0}=(x_{0},y_{0},z_{0})} and the equation is said to be the equation of a point sphere . Finally, in the case ρ > 0 {\displaystyle \rho >0} , f ( x , y , z ) = 0 {\displaystyle f(x,y,z)=0}
1584-422: The poles is called the equator . Great circles through the poles are called lines of longitude or meridians . Small circles on the sphere that are parallel to the equator are circles of latitude (or parallels ). In geometry unrelated to astronomical bodies, geocentric terminology should be used only for illustration and noted as such, unless there is no chance of misunderstanding. Mathematicians consider
1628-407: The same because all such pairs of parallel tangent lines have the same distance. For an ellipse , the standard terminology is different. A diameter of an ellipse is any chord passing through the centre of the ellipse. For example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one diameter is parallel to the conjugate diameter. The longest diameter
1672-440: The smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area. The surface area relative to the mass of a ball is called the specific surface area and can be expressed from
1716-510: The sphere has the same center and radius as the sphere, and divides it into two equal hemispheres . Although the figure of Earth is not perfectly spherical, terms borrowed from geography are convenient to apply to the sphere. A particular line passing through its center defines an axis (as in Earth's axis of rotation ). The sphere-axis intersection defines two antipodal poles ( north pole and south pole ). The great circle equidistant to
1760-522: The sphere with radius r > 0 {\displaystyle r>0} and center ( x 0 , y 0 , z 0 ) {\displaystyle (x_{0},y_{0},z_{0})} can be parameterized using trigonometric functions . The symbols used here are the same as those used in spherical coordinates . r is constant, while θ varies from 0 to π and φ {\displaystyle \varphi } varies from 0 to 2 π . In three dimensions,
1804-522: The sphere). This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle . This formula can also be derived using integral calculus (i.e., disk integration ) to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked side by side and centered along
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1848-407: The sphere. Several properties hold for the plane , which can be thought of as a sphere with infinite radius. These properties are: Diameter In geometry , a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for
1892-426: The sphere: their length is twice the radius, d = 2 r . Two points on the sphere connected by a diameter are antipodal points of each other. A unit sphere is a sphere with unit radius ( r = 1 ). For convenience, spheres are often taken to have their center at the origin of the coordinate system , and spheres in this article have their center at the origin unless a center is mentioned. A great circle on
1936-432: The square of the distance between their centers is equal to the sum of the squares of their radii. If f ( x , y , z ) = 0 and g ( x , y , z ) = 0 are the equations of two distinct spheres then is also the equation of a sphere for arbitrary values of the parameters s and t . The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition
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