A
ring, circular band of gold, silver, or some other precious or
decorative material that is worn on the finger. Rings are worn not only
on the fingers but also on toes, the ears (see earring), and through the
nose. Besides serving to adorn the body, rings have functioned as
symbols of authority, fidelity, or social status.
Basically,
a ring consists of three parts: the circle, or hoop; the shoulders; and
the bezel. The circle can have a circular, semicircular, or square
cross-section, or it can be shaped as a flat band. The shoulders consist
of a thickening or enlargement of the circle wide enough to support the
bezel. The bezel is the top part of a ring; it may simply be a flat
table, or it may be designed to hold a gem or some other ornament.
The
earliest existing rings are those found in the tombs of ancient Egypt.
The Egyptians primarily used signet, or seal, rings, in which a seal
engraved on the bezel can be used to authenticate documents by the
wearer. Egyptian seal rings typically had the name and titles of the
owner deeply sunk in hieroglyphic characters on an oblong gold bezel.
The ancient Greeks were more prone to use rings simply for decoration,
and in the Hellenistic period the bezel began to be used to hold
individual cabochon stones, such as carnelians and garnets, or vitreous
pastes. In Rome rings were an important symbol of social status. In the
early centuries of the Roman Republic, most rings were of iron, and the
wearing of gold rings was restricted to certain classes, such as
patricians who had held high office. But by the 3rd century BC the
privilege of wearing rings had been extended to the class of knights, or
equites, and by the 3rd century AD, during the Roman Empire,
practically any person except a slave was allowed to wear a gold ring.
The Romans are also thought to have originated the custom of betrothal
rings, or engagement rings, symbolizing a promise of marriage to a
member of the opposite sex.
Throughout the
European Middle Ages the signet ring was of great importance in
religious, legal, and commercial transactions. The Roman Catholic church
conferred episcopal rings upon newly appointed bishops, and so-called
papal rings were given by popes to cardinals. An enormous papal ring
called the Fisherman’s Ring—made of gilded bronze and bearing the image
of St. Peter fishing—is traditionally used by the pope as a seal for
pontifical documents. Besides these types, there were memorial rings,
upon which were engraved the name, date of death, or even the effigy of a
deceased person; posy rings, upon which were engraved an inscription or
a few lines of verse; occult rings, which functioned as talismans or
amulets and were supposed to have magical powers; and poison rings,
whose hollow bezels contained a poison for purposes of suicide or
homicide. Rings with bezels that opened may also have held sentimental
keepsakes in miniature.
By the
19th century, the traditional distinctions between ring types had mostly
broken down, giving way to rings of all kinds inspired by past styles.
Fine-quality modern rings, many of which are machine-made, usually
consist of gold or silver and feature standard-sized diamonds or other
precious stones. They are worn either for purposes of simple adornment
or as symbols of betrothal and marital fidelity.
A
sphere (from Ancient Greek σφαῖρα (sphaîra) 'globe, ball')[1] is a
geometrical object that is a three-dimensional analogue to a
two-dimensional circle. A sphere is the set of points that are all at
the same distance r from a given point in three-dimensional space.[2]
That given point is the centre 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 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.
Basic terminology
Two orthogonal radii of a sphere
As mentioned earlier r is the sphere's radius; any line from the center to a point on the sphere is also called a radius.[3]
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 the sphere: their
length is twice the radius, d=2r. Two points on the sphere connected by a
diameter are antipodal points of each other.[3]
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 the sphere has the same center and radius as the sphere, and divides it into two equal hemispheres.
Although
the Earth is not perfectly spherical, terms borrowed from geography are
convenient to apply to the sphere. If a particular point on a sphere is
(arbitrarily) designated as its north pole, its antipodal point is
called the south pole. The great circle equidistant to each is then the
equator. Great circles through the poles are called lines of longitude
or meridians. A line connecting the two poles may be called the axis of
rotation. Small circles on the sphere that are parallel to the equator
are lines of latitude. 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.[3]
Mathematicians
consider a sphere to be a two-dimensional closed surface embedded in
three-dimensional Euclidean space. They draw a distinction 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 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.
Equations
In analytic geometry, a sphere with center (x0, y0, z0) and radius r is the locus of all points (x, y, z) such that
{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.}{\displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}=r^{2}.}
Since it can be expressed as a quadratic polynomial, a sphere is a quadric surface, a type of algebraic surface.[3]
Let a, b, c, d, e be real numbers with a ≠ 0 and put
{\displaystyle
x_{0}={\frac {-b}{a}},\quad y_{0}={\frac {-c}{a}},\quad z_{0}={\frac
{-d}{a}},\quad \rho ={\frac
{b^{2}+c^{2}+d^{2}-ae}{a^{2}}}.}{\displaystyle x_{0}={\frac
{-b}{a}},\quad y_{0}={\frac {-c}{a}},\quad z_{0}={\frac {-d}{a}},\quad
\rho ={\frac {b^{2}+c^{2}+d^{2}-ae}{a^{2}}}.}
Then the equation
{\displaystyle f(x,y,z)=a(x^{2}+y^{2}+z^{2})+2(bx+cy+dz)+e=0}{\displaystyle f(x,y,z)=a(x^{2}+y^{2}+z^{2})+2(bx+cy+dz)+e=0}
has
no real points as solutions if {\displaystyle \rho <0}\rho < 0
and is called the equation of an imaginary sphere. If {\displaystyle
\rho =0}\rho =0, the only solution of {\displaystyle
f(x,y,z)=0}{\displaystyle f(x,y,z)=0} is the point {\displaystyle
P_{0}=(x_{0},y_{0},z_{0})}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
{\displaystyle \rho >0}\rho >0, {\displaystyle
f(x,y,z)=0}{\displaystyle f(x,y,z)=0} is an equation of a sphere whose
center is {\displaystyle P_{0}}P_{0} and whose radius is {\displaystyle
{\sqrt {\rho }}}{\displaystyle {\sqrt {\rho }}}.[2]
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.[4]
Parametric
A
parametric equation for the sphere with radius {\displaystyle r>0}r
> 0 and center {\displaystyle (x_{0},y_{0},z_{0})}(x_{0},y_{0},z_{0})
can be parameterized using trigonometric functions.
{\displaystyle
{\begin{aligned}x&=x_{0}+r\sin \theta \;\cos \varphi
\\y&=y_{0}+r\sin \theta \;\sin \varphi \\z&=z_{0}+r\cos \theta
\,\end{aligned}}}{\displaystyle {\begin{aligned}x&=x_{0}+r\sin
\theta \;\cos \varphi \\y&=y_{0}+r\sin \theta \;\sin \varphi
\\z&=z_{0}+r\cos \theta \,\end{aligned}}}[5]
The symbols
used here are the same as those used in spherical coordinates. r is
constant, while θ varies from 0 to π and {\displaystyle \varphi
}\varphi varies from 0 to 2π.
Enclosed volume
Sphere and circumscribed cylinder
In
three dimensions, the volume inside a sphere (that is, the volume of a
ball, but classically referred to as the volume of a sphere) is
{\displaystyle
V={\frac {4}{3}}\pi r^{3}={\frac {\pi }{6}}\ d^{3}\approx 0.5236\cdot
d^{3}}{\displaystyle V={\frac {4}{3}}\pi r^{3}={\frac {\pi }{6}}\
d^{3}\approx 0.5236\cdot d^{3}}
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 the sphere).[6] 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.[7] 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 the x-axis from x = −r to x = r, assuming the sphere of radius r
is centered at the origin.
Proof of sphere volume, using calculus
For
most practical purposes, the volume inside a sphere inscribed in a cube
can be approximated as 52.4% of the volume of the cube, since V =
π
/
6
d3, 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 m3.
Surface area
The surface area of a sphere of radius r is:
{\displaystyle A=4\pi r^{2}.}A=4\pi r^{2}.
Archimedes
first derived this formula[9] from the fact that the projection to the
lateral surface of a circumscribed cylinder is area-preserving.[10]
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 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.
Proof of surface area, using calculus
The
sphere has 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.[11] 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 the above stated equations as
{\displaystyle
\mathrm {SSA} ={\frac {A}{V\rho }}={\frac {3}{r\rho }},}{\displaystyle
\mathrm {SSA} ={\frac {A}{V\rho }}={\frac {3}{r\rho }},}
where ρ is the density (the ratio of mass to volume).
Geometric properties
A
sphere can be constructed as the surface formed by rotating a circle
about any of its diameters; this is essentially 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 the circle with an ellipse rotated about its major
axis, the shape becomes a prolate spheroid; rotated about the minor
axis, an oblate spheroid.[12]
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.[13] This
property is analogous to the property that three non-collinear points
determine 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.[14]
Although the radical plane 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).[15]
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.[16] They intersect at right angles (are
orthogonal) if and only if the square of the distance between their
centers is equal to the sum of the squares of their radii.[4]
Pencil of spheres
Main article: Pencil (mathematics) § Pencil of spheres
If f(x, y, z) = 0 and g(x, y, z) = 0 are the equations of two distinct spheres then
{\displaystyle sf(x,y,z)+tg(x,y,z)=0}{\displaystyle sf(x,y,z)+tg(x,y,z)=0}
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 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.[4]
Spherical geometry
Great circle on a sphere
Main article: Spherical geometry
The
basic elements of Euclidean plane geometry are points and lines. On the
sphere, points are defined in the usual sense. The analogue of the
"line" is the geodesic, which is a great circle; the defining
characteristic of a great circle is that the plane containing all its
points also passes through the center of the sphere. Measuring by arc
length shows that the shortest path between two points lying on the
sphere is the shorter segment of the great circle that includes the
points.
Many theorems from classical geometry
hold true for spherical geometry as well, but not all do because the
sphere fails to satisfy some of classical geometry's postulates,
including the parallel postulate. In spherical trigonometry, angles are
defined between great circles. Spherical trigonometry differs from
ordinary trigonometry in many respects. For example, the sum of the
interior angles of a spherical triangle always exceeds 180 degrees.
Also, any two similar spherical triangles are congruent.
Any
pair of points on a sphere that lie on a straight line through the
sphere's center (i.e. the diameter) are called antipodal points—on the
sphere, the distance between them is exactly half the length of the
circumference.[note 2] Any other (i.e. not antipodal) pair of distinct
points on a sphere
lie on a unique great circle,
segment it into one minor (i.e. shorter) and one major (i.e. longer) arc, and
have the minor arc's length be the shortest distance between them on the sphere.[note 3]
Spherical geometry is a form of elliptic geometry, which together with hyperbolic geometry makes up non-Euclidean geometry.
Differential geometry
The
sphere is a smooth surface with constant Gaussian curvature at each
point equal to 1/r2.[9] As per Gauss's Theorema Egregium, this curvature
is independent of the sphere's embedding in 3-dimensional space. Also
following from Gauss, a sphere cannot be mapped to a plane while
maintaining both areas and angles. Therefore, any map projection
introduces some form of distortion.
A sphere of
radius r has area element {\displaystyle dA=r^{2}\sin \theta \,d\theta
\,d\varphi }{\displaystyle dA=r^{2}\sin \theta \,d\theta \,d\varphi }.
This can be found from the volume element in spherical coordinates with r
held constant.[9]
A sphere of any radius centered at zero is an integral surface of the following differential form:
{\displaystyle x\,dx+y\,dy+z\,dz=0.}{\displaystyle x\,dx+y\,dy+z\,dz=0.}
This
equation reflects that the position vector and tangent plane at a point
are always orthogonal to each other. Furthermore, the outward-facing
normal vector is equal to the position vector scaled by 1/r.
In
Riemannian geometry, the filling area conjecture states that the
hemisphere is the optimal (least area) isometric filling of the
Riemannian circle.
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 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 (an exotic
sphere).
The sphere is the inverse image of a
one-point set under the continuous function ||x||, so it is closed; Sn
is also bounded, so it is compact by the Heine–Borel theorem.
Remarkably,
it is possible to turn an ordinary sphere inside out in a
three-dimensional space with possible self-intersections but without
creating any creases, in a process called sphere eversion.
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.
Curves on a sphere
Plane section of a sphere: 1 circle
Coaxial intersection of a sphere and a cylinder: 2 circles
Circles
Main article: Circle of a sphere
Circles
on the sphere are, like circles in the plane, made up of all points a
certain distance from a fixed point on the sphere. The intersection of a
sphere and a plane is a circle, a point, or empty.[17] Great circles
are the intersection of the sphere with a plane passing through the
center of a sphere: others are called small circles.
More
complicated surfaces may intersect a sphere in circles, too: the
intersection of a sphere with a surface of revolution whose axis
contains the center of the sphere (are coaxial) consists of circles
and/or points if not empty. For example, the diagram to the right shows
the intersection of a sphere and a cylinder, which consists of two
circles. If the cylinder radius were that of the sphere, the
intersection would be a single circle. If the cylinder radius were
larger than that of the sphere, the intersection would be empty.
Loxodrome
Loxodrome
Main article: Rhumb line
In
navigation, a rhumb line or loxodrome is an arc crossing all meridians
of longitude at the same angle. Loxodromes are the same as straight
lines in the Mercator projection. A rhumb line is not a spherical
spiral. Except for some simple cases, the formula of a rhumb line is
complicated.
Clelia curves
Main article: Clélie
spherical spiral with {\displaystyle c=8}{\displaystyle c=8}
A
Clelia curve is a curve on a sphere for which the longitude
{\displaystyle \varphi }\varphi and the colatitude {\displaystyle
\theta }\theta satisfy the equation
{\displaystyle \varphi =c\;\theta ,\quad c>0}{\displaystyle \varphi =c\;\theta ,\quad c>0}.
Special
cases are: Viviani's curve ({\displaystyle c=1}{\displaystyle c=1}) and
spherical spirals ({\displaystyle c>2}c>2) such as Seiffert's
spiral. Clelia curves approximate the path of satellites in polar orbit.
Spherical conics
Main article: Spherical conic
The
analog of a conic section on the sphere is a spherical conic, a quartic
curve which can be defined in several equivalent ways, including:
as the intersection of a sphere with a quadratic cone whose vertex is the sphere center;
as the intersection of a sphere with an elliptic or hyperbolic cylinder whose axis passes through the sphere center;
as the locus of points whose sum or difference of great-circle distances from a pair of foci is a constant.
Many theorems relating to planar conic sections also extend to spherical conics.
Intersection of a sphere with a more general surface
General intersection sphere-cylinder
If a sphere is intersected by another surface, there may be more complicated spherical curves.
Example
sphere – cylinder
Main article: Sphere–cylinder intersection
The
intersection of the sphere with equation {\displaystyle
\;x^{2}+y^{2}+z^{2}=r^{2}\;}{\displaystyle \;x^{2}+y^{2}+z^{2}=r^{2}\;}
and the cylinder with equation {\displaystyle
\;(y-y_{0})^{2}+z^{2}=a^{2},\;y_{0}\neq 0\;}{\displaystyle
\;(y-y_{0})^{2}+z^{2}=a^{2},\;y_{0}\neq 0\;} is not just one or two
circles. It is the solution of the non-linear system of equations
{\displaystyle x^{2}+y^{2}+z^{2}-r^{2}=0}{\displaystyle x^{2}+y^{2}+z^{2}-r^{2}=0}
{\displaystyle (y-y_{0})^{2}+z^{2}-a^{2}=0\ .}{\displaystyle (y-y_{0})^{2}+z^{2}-a^{2}=0\ .}
(see implicit curve and the diagram)
Eleven properties of the sphere
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.
In their book
Geometry and the Imagination,[18] 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:
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.
The contours and plane sections of the sphere are circles.
This property defines the sphere uniquely.
The sphere has constant width and constant girth.
The
width of a surface is the distance between pairs of parallel tangent
planes. Numerous other closed convex surfaces 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. Each of
these properties implies the other.
All points of a sphere are umbilics.
At
any point on a surface a normal direction is at right angles to the
surface because the sphere these are the lines radiating out from the
center of the sphere. The intersection of a plane that contains the
normal with the surface will form a curve that is called a normal
section, and the curvature of this curve is the normal curvature. For
most points on most surfaces, different sections will have different
curvatures; the maximum and minimum values of these are called the
principal curvatures. 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 curvatures 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.
The sphere does not have a surface of centers.
For
a given normal section exists a circle of curvature that equals the
sectional curvature, is tangent to the surface, and the center lines of
which lie along on the normal line. For example, the two centers
corresponding to the maximum and minimum sectional curvatures 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
that are each a surface and meet at umbilical points. Several cases are
special:
* For channel surfaces one sheet forms a curve and the other sheet is a surface
* For cones, cylinders, tori 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 property
is unique to the sphere.
All geodesics of the sphere are closed curves.
Geodesics
are curves on a surface that give the shortest distance between two
points. They are a generalization of the concept of a straight line in
the plane. For the sphere the geodesics are great circles. Many other
surfaces share this property.
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.
It follows from isoperimetric inequality.
These properties define the sphere uniquely and can be seen in soap
bubbles: a soap bubble will enclose a fixed volume, and surface tension
minimizes its surface area for that volume. A freely floating soap
bubble therefore approximates a sphere (though such external forces as
gravity will slightly distort the bubble's shape). It can also be seen
in planets and stars where gravity minimizes surface area for large
celestial bodies.
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, which is
constant because the two principal curvatures are constant at all
points of the sphere.
The sphere has constant mean curvature.
The
sphere is the only imbedded surface that lacks boundary or
singularities with constant positive mean curvature. Other such immersed
surfaces as minimal surfaces have constant mean curvature.
The sphere has constant positive Gaussian curvature.
Gaussian
curvature is the product of the two principal curvatures. It is an
intrinsic property that can be determined by measuring length and angles
and is independent of how 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 that
lacks a boundary with constant, positive Gaussian curvature. The
pseudosphere is an example of a surface with constant negative Gaussian
curvature.
The sphere is transformed into itself by a three-parameter family of rigid motions.
Rotating
around any axis a unit sphere at the origin will map the sphere onto
itself. Any rotation about a line through the origin can be expressed as
a combination of rotations around the three-coordinate axis (see Euler
angles). Therefore, a three-parameter family of rotations exists such
that each rotation transforms the sphere onto itself; this family 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-axes 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.
Generalizations
Ellipsoids
An
ellipsoid is a sphere that has been stretched or compressed in one or
more directions. More exactly, it is the image of a sphere under an
affine transformation. An ellipsoid bears the same relationship to the
sphere that an ellipse does to a circle.
Dimensionality
Main article: n-sphere
Spheres
can be generalized to spaces of any number of dimensions. For any
natural number n, an "n-sphere," often written as Sn, is the set of
points in (n + 1)-dimensional Euclidean space that are at a fixed
distance r from a central point of that space, where r is, as before, a
positive real number. In particular:
S0: a 0-sphere consists of two discrete points, −r and r
S1: a 1-sphere is a circle of radius r
S2: a 2-sphere is an ordinary sphere
S3: a 3-sphere is a sphere in 4-dimensional Euclidean space.
Spheres for n > 2 are sometimes called hyperspheres.
The
n-sphere of unit radius centered at the origin is denoted Sn and is
often referred to as "the" n-sphere. The ordinary sphere is a 2-sphere,
because it is a 2-dimensional surface which is embedded in 3-dimensional
space.
Metric spaces
Main article: Metric space
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 that is considered to be the 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 to equal one,
as in the case of a unit sphere.
Unlike a ball,
even a large sphere may be an empty set. 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.
An octahedron is a sphere in taxicab geometry, and a cube is a sphere in geometry using the Chebyshev distance.
History
The
geometry of the sphere was studied by the Greeks. Euclid's Elements
defines the sphere in book XI, discusses various properties of the
sphere in book XII, and shows how to inscribe the five regular polyhedra
within a sphere in book XIII. Euclid does not include the area and
volume of a sphere, only a theorem that the volume of a sphere varies as
the third power of its diameter, probably due to Eudoxus of Cnidus. The
volume and area formulas were first determined in Archimedes's On the
Sphere and Cylinder by the method of exhaustion. Zenodorus was the first
to state that, for a given surface area, the sphere is the solid of
maximum volume.[3]
Archimedes wrote about the
problem of dividing a sphere into segments whose volumes are in a given
ratio, but did not solve it. A solution by means of the parabola and
hyperbola was given by Dionysodorus of Amisus (c. 1st century B.C.), and
a similar problem — to construct a segment equal in volume to a given
segment, and in surface to another segment — was solved later by
al-Quhi.[3]
Gallery
An
image of one of the most accurate human-made spheres, as it refracts
the image of Einstein in the background. This sphere was a fused quartz
gyroscope for the Gravity Probe B experiment, and differs in shape from a
perfect sphere by no more than 40 atoms (less than 10 nm) of thickness.
It was announced on 1 July 2008 that Australian scientists had created
even more nearly perfect spheres, accurate to 0.3 nm, as part of an
international hunt to find a new global standard kilogram.[19]
Deck of playing cards illustrating engineering instruments, England, 1702. King of spades: Spheres
Regions
See also: Ball (mathematics) § Regions
Hemisphere
Spherical cap
Spherical lune
Spherical polygon
Spherical sector
Spherical segment
Spherical wedge
Spherical zone
See also
3-sphere
Affine sphere
Alexander horned sphere
Celestial spheres
Curvature
Directional statistics
Dyson sphere
Gauss map
Hand
with Reflecting Sphere, M.C. Escher self-portrait drawing illustrating
reflection and the optical properties of a mirror sphere
Hoberman sphere
Homology sphere
Homotopy groups of spheres
Homotopy sphere
Lenart Sphere
Napkin ring problem
Orb (optics)
Pseudosphere
Riemann sphere
Solid angle
Sphere packing
Spherical coordinates
Spherical cow
Spherical helix, tangent indicatrix of a curve of constant precession
Spherical polyhedron
Sphericity
Tennis ball theorem
Zoll sphere
Notes and references
Notes
r is being considered as a variable in this computation.
It does not matter which direction is chosen, the distance is the sphere's radius × π.
The distance between two non-distinct points (i.e. a point and itself) on the sphere is zero.
References
σφαῖρα, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus.
Albert 2016, p. 54.
Chisholm, Hugh, ed. (1911). "Sphere" . Encyclopædia Britannica. Vol. 25 (11th ed.). Cambridge University Press. pp. 647–648.
Woods 1961, p. 266.
Kreyszig (1972, p. 342).
Steinhaus 1969, p. 223.
"The volume of a sphere - Math Central". mathcentral.uregina.ca. Retrieved 10 June 2019.
E.J. Borowski; J.M. Borwein (1989). Collins Dictionary of Mathematics. pp. 141, 149. ISBN 978-0-00-434347-1.
Weisstein, Eric W. "Sphere". MathWorld.
Steinhaus 1969, p. 221.
Osserman,
Robert (1978). "The isoperimetric inequality". Bulletin of the American
Mathematical Society. 84 (6): 1187.
doi:10.1090/S0002-9904-1978-14553-4. Retrieved 14 December 2019.
Albert 2016, p. 60.
Albert 2016, p. 55.
Albert 2016, p. 57.
Woods 1961, p. 267.
Albert 2016, p. 58.
Weisstein, Eric W. "Spheric section". MathWorld.
Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. ISBN 978-0-8284-1087-8.
New Scientist | Technology | Roundest objects in the world created.
Further reading
Wikisource has the text of the 1911 Encyclopædia Britannica article "Sphere".
Albert, Abraham Adrian (2016) [1949], Solid Analytic Geometry, Dover, ISBN 978-0-486-81026-3.
Dunham,
William (1997). The Mathematical Universe: An Alphabetical Journey
Through the Great Proofs, Problems and Personalities. Wiley. New York.
pp. 28, 226. Bibcode:1994muaa.book.....D. ISBN 978-0-471-17661-9.
Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York: Wiley, ISBN 978-0-471-50728-4.
Steinhaus, H. (1969), Mathematical Snapshots (Third American ed.), Oxford University Press.
Woods, Frederick S. (1961) [1922], Higher Geometry / An Introduction to Advanced Methods in Analytic Geometry, Dover.
John C. Polking (15 April 1999). "The Geometry of the Sphere". www.math.csi.cuny.edu. Retrieved 21 January 2022.
External links
Sphere
at Wikipedia's sister projects
Definitions from Wiktionary
Media from Commons
News from Wikinews
Quotations from Wikiquote
Texts from Wikisource
Textbooks from Wikibooks
Resources from Wikiversity
Mathematica/Uniform Spherical Distribution
Surface area of sphere proof
vte
Compact topological surfaces and their immersions in 3D
Without boundary
Orientable
Sphere (genus 0)Torus (genus 1)Number 8 (genus 2)Pretzel (genus 3) ...
Non-orientable
Real projective plane genus 1; Boy's surfaceRoman surfaceKlein bottle (genus 2)Dyck's surface (genus 3) ...
With boundary
Disk SemisphereRibbon AnnulusCylinderMöbius strip Cross-capSphere with three holes ...
Related
notions
Properties
ConnectednessCompactnessTriangulatedness or smoothnessOrientability
Characteristics
Number of boundary componentsGenusEuler characteristic
Operations
Connected sumMaking a holeGluing a handleGluing a cross-capImmersion
Uniqueness
is a state or condition wherein someone or something is unlike anything
else in comparison.[1] When used in relation to humans, it is often in
relation to a person's personality, or some specific characteristics of
it, signalling that it is unlike the personality traits that are
prevalent in that individual's culture.[2] When the term uniqueness is
used in relation to an object, it is often within the realm of product,
with the term being a factor used to publicize or market the product in
order to make it stand out from other products within the same
category.[3]
The notion of American exceptionalism is premised on the uniqueness of the West, particularly its well-defined secularism