Spheres

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Largest Volume for Smallest Surface

How we chose 'justice'. And is one way more correct than the others? How to use a word that literally drives some people nuts. The awkward case of 'his or her'. Identify the word pairs with a common ancestor. Test your visual vocabulary with our question challenge! Synonyms for sphere Synonyms: Noun ball , globe , orb Visit the Thesaurus for More. Examples of sphere in a Sentence Noun All points on a sphere are the same distance from the center.

Sphere packing in a sphere

Women at that time were confined to the domestic sphere. They recognize that jobs in the public sphere are valuable.

Recent Examples on the Web: Noun India is trying to preserve its own sphere of influence in South Asia through projects such as building roads and bridges in Bangladesh, hydroelectric plants in Nepal and ports and railways in Sri Lanka. Stevenson, Washington Post , "The simple reason intelligence officials keep talking about Russian hacking? As good as the uneven series will ever get," 2 July The dish is served with french fries, kale slaw and little spheres of hot sauce on the house buttermilk-herb dressing.

First Known Use of sphere Noun 14th century, in the meaning defined at sense 1a 1 Verb , in the meaning defined at sense 1. History and Etymology for sphere Noun Middle English spere globe, celestial sphere, from Anglo-French espere , from Latin sphaera , from Greek sphaira , literally, ball; perhaps akin to Greek spairein to quiver — more at spurn Combining form extracted from atmosphere.

Learn More about sphere. Resources for sphere Time Traveler! Explore the year a word first appeared. 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.

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. 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.

If the pencil of spheres does not consist of all planes, then there are three types of pencils: All the tangent lines from a fixed point of the radical plane to the spheres of a pencil have the same length. The radical plane is the locus of the centers of all the spheres that are orthogonal to all the spheres in a pencil. Moreover, a sphere orthogonal to any two spheres of a pencil of spheres is orthogonal to all of them and its center lies in the radical plane of the pencil.

Pairs of points on a sphere that lie on a straight line through the sphere's center are called antipodal points. A great circle is a circle on the sphere that has the same center and radius as the sphere and, consequently, divides it into two equal parts. The plane sections of a sphere are called spheric sections.

They are all circles and those that are not great circles are called small circles. The shortest distance along the surface between two distinct non-antipodal points on the sphere is the length of the smaller of the two arcs on the unique great circle that includes the two points. Equipped with this " 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 such approximately spheroidal astronomical bodies as the planet Earth see geoid. Any plane that includes the center of a sphere divides it into two equal hemispheres. Any two intersecting planes that include the center of a sphere 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.

Largest Volume for Smallest Surface

The hemisphere is conjectured to be the optimal least area isometric filling of the Riemannian circle. Spheres can be generalized to spaces of any number of dimensions. The n -sphere of unit radius centered at the origin is denoted S n 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. General recursive formulas also exist for the volume of an n -ball. 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 Z n with Euclidean metric , a sphere of radius r is nonempty only if r 2 can be written as sum of n squares of integers. The n -sphere is denoted S n. 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. S n is also bounded; therefore it is compact. Remarkably, it is possible to turn an ordinary sphere inside out in a three-dimensional space with possible self-intersections but without creating any crease, in a process called sphere eversion.

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. Also, any two similar spherical triangles are congruent. In their book Geometry and the Imagination [16] David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere.

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