"Globose" redirects here. See also Globose nucleus. The globose nucleus is one of the deep cerebellar nuclei It is located medial to the Emboliform nucleus and lateral to the Fastigial nucleus.

A sphere.

A sphere (from Greek σφαίρα - sphaira, "globe, ball,"^[1]) is a symmetrical geometrical object. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly 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 Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface. BALL ( Biochemical Algorithms Library) is a C++ library containing common algorithms used in Biochemistry and Bioinformatics. In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. 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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, solid geometry was the traditional name for the Geometry of three-dimensional Euclidean space &mdash for practical purposes the kind of In Mathematics, the real numbers may be described informally in several different ways Thus, in three dimensions, a mathematical sphere is considered to be a two-dimensional spherical surface embedded in three-dimensional space, rather than the volume contained within it (which mathematicians would instead describe 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 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. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed

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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion.

Equations in R³

In analytic geometry, a sphere with center (x₀, y₀, z₀) and radius r is the locus of all points (x, y, z) such that

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

(see also trigonometric functions and spherical coordinates). Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial

A sphere of any radius centered at the origin is described by the following differential equation:

This equation reflects the fact that the position and velocity vectors of a point travelling on the sphere are always orthogonal to each other. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, two Vectors are orthogonal if they are Perpendicular, i

The surface area of a sphere of radius r is

so the radius from surface area is

Its volume is

so the radius from volume is

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. Surface area is the measure of how much exposed Area an object has Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers Surface area is the measure of how much exposed Area an object has The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically 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. For the work of fiction see Surface Tension (short story. Surface tension is a property of the surface of a Liquid that causes it to The surface area in relation to the mass of a sphere is called the specific surface area. Specific surface area is a material property of Solids which measures the total Surface area per unit of Mass, solid or bulk Volume, or cross-sectional 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 the image of Einstein in the background. Refraction is the change in direction of a Wave due to a change in its Speed. 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. Fused quartz and fused silica are types of Glass containing primarily Silica in amorphous (non- Crystalline form A gyroscope is a device for measuring or maintaining orientation, based on the principles of Angular momentum. Gravity Probe B ( GP-B) is a Satellite -based mission which launched in 2004 It is thought that only neutron stars are smoother. 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 It was announced on 15 June, 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. Events 763 BC - Assyrians record a Solar eclipse that will be used to fix the Chronology of Mesopotamian history For a topic outline on this subject see List of basic Australia topics. The Millimetre ( American spelling: millimeter, symbol mm) is a unit of Length in the Metric system, equal to ^[2]

The circumscribed cylinder for a given sphere has a volume which is 3/2 times the volume of the sphere, and also the curved portion has a surface area which is equal to the surface area of the sphere. 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 This fact, along with the volume and surface formulas given above, was already known to Archimedes. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer

A sphere can also be defined as the surface formed by rotating a circle about any diameter. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the 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 If the circle is replaced by an ellipse, and rotated about the major axis, the shape becomes a prolate spheroid, rotated about the minor axis, an oblate spheroid. In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a Equation A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation \left(\frac{x}{a}\right^2+\left(\frac{y}{a}\right^2+\left(\frac{z}{b}\right^2

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 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. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. 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.

If a particular point on a sphere is 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. The equator (sometimes referred to colloquially as "the Line") is the intersection of the Earth 's surface with the plane perpendicular to the 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. This article is about the geographical concept For other uses of the word see Meridian. Longitude (ˈlɒndʒɪˌtjuːd or ˈlɒŋgɪˌtjuːd symbolized by the Greek character Lambda (λ is the east-west Geographic coordinate measurement A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation Circles on the sphere that are parallel to the equator are lines of latitude. Latitude, usually denoted symbolically by the Greek letter phi ( Φ) gives the location of a place on Earth (or other planetary body north or south of the This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid). EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 Equation A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation \left(\frac{x}{a}\right^2+\left(\frac{y}{a}\right^2+\left(\frac{z}{b}\right^2 The geoid is that Equipotential surface which would coincide exactly with the mean ocean surface of the Earth if the oceans were in equilibrium at rest and extended through

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. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it 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 Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In particular:

• 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. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere.

Spheres for n > 2 are sometimes called hyperspheres. In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension.

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.

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

where Γ(z) is Euler's Gamma function. In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function

Another formula for surface area is

and the volume within 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. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined

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. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length The same applies for the radius if it is taken equal to one, as in the case of a unit sphere. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed

In contrast to a ball, a sphere may be an empty set, even for a large radius. 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 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. In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler

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. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Topological equivalence redirects here see also Topological equivalence (dynamical systems. 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 Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, a metric space is a set where a notion of Distance (called a metric) 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. In Topology, a discrete space is a particularly simple example of a Topological space or similar structure one in which the points are " isolated " 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 Topological equivalence redirects here see also Topological equivalence (dynamical systems. In Mathematics, a knot is an Embedding of a Circle in 3-dimensional Euclidean space, R 3 considered up to continuous deformations 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 Topological equivalence redirects here see also Topological equivalence (dynamical systems. Equation A spheroid centered at the origin and rotated about the z axis is defined by the implicit equation \left(\frac{x}{a}\right^2+\left(\frac{y}{a}\right^2+\left(\frac{z}{b}\right^2 It is an example of a compact topological manifold without boundary. In Mathematics, a topological manifold is a Hausdorff Topological space which looks locally like Euclidean space in a sense defined below For a different notion of boundary related to Manifolds see that article A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable

The Heine-Borel theorem implies that a Euclidean n-sphere is compact. In the Topology of Metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states For a Subset 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.

Spherical geometry

Great circle on a sphere

Main article: Spherical geometry

The basic elements of plane geometry are points and lines. A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere. In Mathematics, plane geometry may mean geometry of a plane, geometry of the Euclidean plane, or sometimes In Geometry, Topology and related branches of mathematics a spatial point describes a specific point within a given space that consists of neither Volume 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. Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult A great circle is a Circle on the surface of a Sphere that has the same circumference as the sphere dividing the sphere into two equal Hemispheres. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces Many theorems from classical geometry hold true for this spherical geometry as well, but many do not (see parallel postulate). In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive In spherical trigonometry, angles are defined between great circles. Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Thus spherical trigonometry is different from ordinary trigonometry in many respects. Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. For example, the sum of the interior angles of a spherical triangle exceeds 180 degrees. Also, any two similar spherical triangles are congruent. Geometry Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking of the other

Eleven properties of the sphere

In their book Geometry and the imagination^[3] David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Stefan or Stephan Cohn-Vossen ( 28 May[[ 902]] Breslau, Germany – 25 June[[ 936]] Moscow, USSR) was a 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. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the 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. A Reuleaux polygon is a Curve of constant width - that is a curve in which all diameters are the same length 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. In Differential geometry, the two principal curvatures at a given point of a Surface measure how the surface bends by different amounts in different directions It can be proved that any closed surface will have at least four points called umbilical points. In the Differential geometry of surfaces in three dimensions umbilics or umbilical points are points which are locally spherical At an umbilic all the sectional curvatures are equal, in particular the principal curvature's are equal. In Differential geometry, the two principal curvatures at a given point of a Surface measure how the surface bends by different amounts in different directions 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 a Surface in three dimension the focal surface, surface of centers or evolute is formed by taking the centers of the Curvature spheres which

   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. A channel or canal surface is a Surface formed as the envelope of a family of Spheres whose centers lie on a Space curve. A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar In Mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of any Standard torus. 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. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces 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 is a very thin film of Soap water that forms a Sphere with an iridescent Surface. A soap bubble will enclose a fixed volume and due to surface tension it will try to minimize its surface area. For the work of fiction see Surface Tension (short story. Surface tension is a property of the surface of a Liquid that causes it to 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. In Mathematics, the mean curvature H of a Surface S is an extrinsic measure of Curvature that comes from Differential

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. In Mathematics, a Minimal surface is a surface with a Mean curvature of zero

10. The sphere has constant positive Gaussian curvature.

    Gaussian curvature is the product of the two principle curvatures. In Differential geometry, the Gaussian curvature or Gauss curvature of a point on a Surface is the product of the Principal 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. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group 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. In Geometry, a pseudosphere of radius R is a surface of curvature &minus1/ R 2 (precisely a complete, Simply connected

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. The Euler angles were developed by Leonhard Euler to describe the orientation of a Rigid body (a body in which the relative position of all its points is constant Thus there is a three parameter family of rotations which transform the sphere onto itself, this is the rotation group, SO(3). This article is about rotations in three-dimensional Euclidean space 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. A surface of revolution is a Surface created by rotating a Curve lying on some plane (the Generatrix) around a Straight line (the Axis The helicoid, after the plane and the Catenoid, is the third Minimal surface to be known

References

See also

External links

• Surface Area MATHguide
• Volume MATHguide
• Mathworld website
• calculate area and volume with your own radius-values to understand the equations
• MathAce » Spheres MathAce's basic article on spheres - good step by step explanation of equation transformation. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. The Alexander horned sphere is one of the most famous pathological examples in Mathematics discovered in 1924 by J 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 The Banach–Tarski paradox is a Theorem in set theoretic Geometry which states that a solid ball in 3-dimensional space can be split into several Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry Directional statistics is the subdiscipline of Statistics that deals with directions ( Unit vectors in R n) axes (lines through In Mathematics, a dome is a closed geometrical Surface which can be obtained by sectioning off a portion of a Sphere with an intersecting A Dyson sphere (or shell as it appeared in the original paper is a hypothetical Megastructure originally described by Freeman Dyson. In Algebraic topology, a homology sphere is an n -manifold X having the Homology groups of an n - Sphere, for some integer In the mathematical field of Algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other In Algebraic topology, a branch of Mathematics, a homotopy sphere is an n - Manifold homotopy equivalent to the n - sphere In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. In Mathematics, a metric space is a set where a notion of Distance (called a metric) between elements of the set is defined In Geometry, a pseudosphere of radius R is a surface of curvature &minus1/ R 2 (precisely a complete, Simply connected In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that In Differential topology, Smale's Paradox states that it is possible to turn a sphere inside out in 3-space with possible self-intersections but without creating The solid angle, Ω, is the angle in three-dimensional space that an object Subtends at a point In Geometry, a spherical cap is a portion of a Sphere cut off by a plane. In Mathematics, the spherical coordinate system is a Coordinate system for representing geometric figures in three dimensions using three coordinates the radial The concept of a spherical Earth dates back to around the 6th century BC in ancient Greek philosophy and possibly ancient Indian philosophy.
• Outside In. Retrieved on 2007-11-24. Year 2007 ( MMVII) was a Common year starting on Monday of the Gregorian calendar in the 21st century. Events 380 - Theodosius I makes his adventus, or formal (computer animation showing how the inside of a sphere can turn outside. )