Dictionary:

sphere

  (sfĭr)

1. Mathematics. A three-dimensional surface, all points of which are equidistant from a fixed point.
2. A spherical object or figure.
3. A celestial body, such as a planet or star.
4. The sky, appearing as a hemisphere to an observer: the sphere of the heavens.
5. Any of a series of concentric, transparent, revolving globes that together were once thought to contain the moon, sun, planets, and stars.
6. The extent of a person's knowledge, interests, or social position.
7. An area of power, control, or influence; domain. See synonyms at field.

1. To form into a sphere.
2. To put in or within a sphere.
3. To surround or encompass.

[Middle English spere, from Old French espere, from Latin sphaera, from Greek sphaira.]

Both in euclidean solid geometry and in common usage the word sphere denotes a solid of revolution obtained by revolving a semicircle of radius r about its diameter. Its total volume is V = &frac43;πr³.

However, in analytic geometry, and more generally in modern mathematics, the word sphere denotes a spherical surface that bounds a solid sphere. In this sense a sphere is the locus of all points P in three-dimensional space whose distance from a fixed point O (called the center) is equal to a given number. The word radius may refer either to one of the segments OP, or to their common length r. A plane that intersects a sphere in just one point is called a tangent plane and is perpendicular to the radius drawn from the center of the sphere to that point. A plane that intersects a sphere in more than one point intersects it in a circle. The circle is called a great circle or a small circle of the sphere according to whether the plane does or does not pass through the center of the sphere. If two parallel planes intersect a sphere, the spherical surface between them is called a zone.

Any great circle of a sphere divides it into two hemispheres. A second great circle cuts a hemisphere into two lunes. A third great circle cuts each lune into two spherical triangles.

noun

An area within which something or someone exists, acts, or has influence or power: ambit, compass, extension, extent, orbit, purview, range, reach, realm, scope, sweep, swing. See territory.

solid angle The equivalent in three-dimensional space to the revolution for the planar degree in two-dimensional space; 1 sphere = 4π steradians = 129 600π^-1 square degrees, = 160 000π^-1 square grades.

It has been suggested that it, maybe as the spat, be a base unit of the SI.

For more information on sphere, visit Britannica.com.

Divisions of the spirit world, both in spatial and moral-spiritual senses. The doctrine of spheres, in a literal sense, was integral to the ancient world, and much of occult teachings— astrology, magic, Gnosticism—emerged in such a cosmology. It was retained in the occult culture and has passed into modern theosophical and Spiritualist circles, where it has remained, though the spheres are usually thought of as levels of a multidimensional world.

Spiritualists have developed a doctrine of the spheres based upon the communications of spirits in the nineteenth and twentieth centuries. The information conflicts at many points, and there is no authority to declare for one opinion over another, but there is a general agreement as to the number of spheres. They are seven: (1) Hell, (2) Sphere of Desires, (3) Summerland, (4) Mind, (5) Abstract, (6) Meeting of the Sexes, and (7) Union of the Sexes.

There is some contradiction as to whether the Earth should be considered as the first sphere. It is said that the first sphere is the abode of gross and ignorant spirits. It is gloomy and desolate, replete with sadness and misery. After a realization of their state and the circumstances that cast them into it, the desire for progress and betterment will transfer the spirits into the second sphere where, in a scenery as natural as that on Earth, harmony, love, and kindness help to develop the higher qualities of the soul.

The period of the stay in a particular sphere varies individually. The higher spheres cannot be perceived by spirits in the lower ones. Information on the higher spheres is obtained from visitors descending to lower spheres. Owing to a lack of conception, no adequate description can be conveyed to us. It is also said that beyond the spheres are the heavens of boundless extent. These are the ultimate abodes of the glorified and blessed.

Hudson Tuttle, in his book Arcana of Spiritualism (1871), furnishes an interesting exposition of the origin of the spheres. According to Tuttle, the spirit world is built up from atomic emanations. Exhalations from all substances ascend as mist rises from a sheet of water. The spirit world therefore depends on the Earth for its existence and is formed through its refining instrumentality. Without the Earth there could not have been corresponding spirit spheres, actually zones rather than spheres. They are 120 degrees wide; that is, they extend 60 degrees on each side of the equator. If we take the sixtieth parallel of latitude each side of the equator and imagine it projected against the blue dome of the sky, we have the boundaries of these zones.

The first zone, or the innermost one, is 60 miles from the Earth's surface. The next external one is removed from the first by about the same distance. The third is just outside the moon's orbit, or 265,000 miles from the Earth. From the third sphere rise the most sublimated exhalations, which mingle with the emanations of the other planets and form a vast zone around the entire solar system, including even the unknown planets beyond the vast orbit of Neptune (the spirits had yet to inform him of the existence of Pluto).

The first zone is nearly 30 miles in thickness, the second 20, the third but two miles. While the Earth is slowly diminishing, the spheres are gradually increasing. The surface of the zones is diversified with changing scenery. Matter, when it aggregates there, is prone to assume the forms in which it existed below. Hence there are all the forms of life there as on Earth, except those, such as the lowest plants and animals, that cannot exist surrounded by such superior conditions. The scenery is of mountain and plain; river, lake and ocean; and of forest and prairie. It is like Earth with all its imperfections perfected, and its beauties are multiplied.

The first trance reference to spheres in the lineage of modern Spiritualism seems to have been made by Frederica Hauffe, the seeress of Prevorst. The second is contained in a letter from G. P. Billot to J. P. F. Deleuze in 1831. Billot wrote: "They taught that God was a grand Spiritual Sun—life on earth a probation—the spheres, different degrees of comprehensive happiness or states of retributive suffering—each appropriate to the good or evil deeds done on earth. They described the ascending changes open to every soul in proportion to his own efforts to improve."

The first exact dimensions were claimed by J. A. Gridley in his book Astounding Facts from the Spirit World (1854). According to his data, the first sphere is 5,000 miles, the sixth 30,000 miles from the Earth's surface.

Diagrams of the spheres were first drawn by Hauffe. Nahum Koons in the Koon loghouse was the second to provide detailed sketches; his information was supplemented by accounts given through the trumpet (see also Jonathan Koons).

Robert Hare differed from Gridley and agreed with Hudson Tuttle inasmuch as his communicators put the distance of the nearest sphere as 60 miles from the Earth's surface. But his further distances did not tally with Tuttle's calculations. He placed the sixth sphere within the area of the moon. He was told that the spheres are concentric zones, or circles, of exceedingly refined matter encompassing the Earth like belts or girdles. They have atmospheres of peculiar vital air, soft and balmy. Their surfaces are diversified with an immense variety of picturesque landscapes, with lofty mountain ranges, valleys, rivers, lakes, forests, trees and shrubbery, and flowers of every colour and variety, sending forth grateful emanations.

As flights of unverifiable speculation proceeded, almost every trance description of the spheres asserted something different. Eugene Crowell, in The Identity of Primitive Christianity with Modern Spiritualism (2 vols., 1875-79), states that he had received the following figures: the first sphere is within our atmosphere, the second is about 60 miles from the earth, the third about 160, the fourth 310, the fifth 460, the sixth 635, the seventh 865 miles.

Precise information was tendered in J. Hewat McKenzie's Spirit Intercourse (1916). The supposed spirit of William James was quoted as the authority behind the statements. The disagreement is all too apparent. "The third sphere, the Summer Land, is 1,350 miles from the earth, the fourth 2,850, the fifth 5,050, the sixth 9,450, and the seventh 18,250."

The sustenance of the body in superphysical states is derived from the atmosphere by inhalation in the ordinary act of breathing; the material for clothing and houses is manufactured; there is a union of sexes in a bond of affection, with no offspring; the animals that live there have previously existed on Earth; the spiritual worlds of each planet unite at the seventh sphere; the spheres are built of essences cast off by millions of tons of matter that condense into solid substance and float in space like vast continents, by the operation of centripetal and centrifugal attraction; and the passage from one sphere to the other is effected by gradual refinement of the spiritual body under the effect of the spirit.

An impressive conception of after-death states was disclosed in Geraldine Cummins's The Road to Immortality (1932), a book said to be dictated by the spirit of F. W. H. Myers. According to the chapter "The Chart of Existence," the journey of the soul takes place through the following stages:

1. The Plane of Matter.
2. Hades or the Intermediate State.
3. The Plane of Illusion.
4. The Plane of Color.
5. The Plane of Flame.
6. The Plane of Light.
7. Out Yonder, Timelessness.

Between each plane or new chapter in experience, there is existence in Hades or in an intermediate state when the soul reviews his past experiences and makes his choice, deciding whether he will go up or down the ladder of consciousness.

Although there is marked disagreement between different accounts of spirit worlds in the afterlife, it will be recalled that this is also characteristic of the eschatology (considerations of the afterlife) of the different Eastern and Western religions.

It has been claimed that spirits who have not become purified and refined and remain tied to earthly desires have been easier to contact and that their communications would be unreliable. Advance spirits would have moved on to more rarified planes of existence. However, that idea seems to be contradicted by the attempts to identify various spirits with advanced beings from the past.

It is interesting to note that many individuals who have experienced out-of-the-body travel, especially as part of a near-death experience, have reported a remarkable similarity of content in terms both of positive experiences of moving toward a bright light and meeting light beings, as well as negative experiences of a purgatorial realm. These experiences, however, have no relation to the spiritualist doctrine of the spheres.

Sources:

Cummins, Geraldine Dorothy. The Road to Immortality. London: I. Nicholson & Watson, 1933.

Tuttle, Hudson. Arcana of Spiritualism. N.p., 1871. Reprint, Manchester: The Two Worlds Publishing, 1900; Chicago: J. R. Francis, 1904.

A ball or globe.

• attraction s. — centrosome.
• segmentation s. — 1. the morula.
— 2. a blastomere.

To convert from spheres to:

steradians, multiply by 12.57.

A sphere.

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface. 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. Thus, in three dimensions, a mathematical sphere is considered to be a spherical surface, rather than the volume contained within it. 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.

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.

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).

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.

The surface area of a sphere of radius r is

and its enclosed volume is

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. 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.

An image of one of the most accurate spheres ever created by humans, as it refracts the image of Einstein in the background. 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. It is thought that only neutron stars are smoother. It was announced on June 15, 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.^[1]

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. This fact, along with the volume and surface formulas given above, was already known to Archimedes.

A sphere can also be defined as the surface formed by rotating a circle about any diameter. 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.

Terminology

Pairs of points on a sphere that lie on a straight line through its 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 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. Great circles through the two poles are called lines (or meridians) of longitude, and the line connecting the two poles is called the axis of rotation. Circles on the sphere that are parallel to the equator are lines of latitude. This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid).

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

Spheres can be generalized to other dimensions. 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 distance r from a fixed point of that space, where r is, as before, a positive real number. For n> 0, the n-sphere is the simply connected n-dimensional manifold of constant, positive curvature, and can also be thought of embedded in an n+1-dimensional manifold, as the surface or boundary of a ball in the n+1-dimensional manifold.

• a 0-sphere is a pair of points on the line at ( - r,r)
• 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.

Spheres for n > 2 are sometimes called hyperspheres.

The n-sphere of unit radius centred at the origin is denoted Sⁿ and is often referred to as "the" n-sphere. Note that the ordinary sphere is a 2-sphere, because it is a 2-dimensional surface, though it is also a 3-dimensional object because it can be embedded in ordinary 3-space.

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

where Γ(z) is Euler's Gamma function.

Another formula for surface area is

and the volume 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

S(x;r) = { y ∈ E | d(x,y) = r }.

If the center is a distinguished point considered as origin of E, e.g. in a normed space, it is not mentioned in the definition and notation. The same applies for the radius if it is taken equal to one, i.e. in the case of a unit sphere. In contrast to a ball, a sphere may be empty, even for a large radius. For example, in Zⁿ with Euclidean metric, a sphere of radius r is nonempty only if r ² can be written as sum of n squares of integers.

Topology

In topology, an n-sphere is defined as a space homeomorphic to the boundary of an (n+1)-ball; thus, it is homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric.

• 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.

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

Eleven properties of the sphere

In their book Geometry and the imagination^[2] David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane which can be thought of as a sphere with infinite radius. These properties are:

1. The points on the sphere are all the same distance from a fixed point. Also, the ratio of the distance of its points from two fixed points is constant.

   The first part is the usual definition of the sphere and determines it uniquely. The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle. This second part also holds for the plane.

2. The contours and plane sections of the sphere are circles.

   This property defines the sphere uniquely.

3. The sphere has constant width and constant girth.

   The width of a surface is the distance between pairs of parallel tangent planes. There are numerous other closed convex surfaces which have constant width, for example Meissner's tetrahedron. The girth of a surface is the circumference of the boundary of its orthogonal projection on to a plane. It can be proved that each of these properties implies the other.

   

   

   A normal vector to a sphere, a normal plane and its normal section. The curvature of the curve of intersection is the sectional curvature. For the sphere each normal section through a given point will be a circle of the same radius, the radius of the sphere. This means that every point on the sphere will be an umbilical point.

4. All points of a sphere are umbilics.

   At any point on a surface we can find a normal direction which is at right angles to the surface, for the sphere these on the lines radiating out from the center of the sphere. The intersection of a plane containing the normal with the surface will form a curve called a normal section and the curvature of this curve is the sectional curvature. For most points on a surfaces different sections will have different curvatures, the maximum and minimum values of these are called the principal curvatures. It can be proved that any closed surface will have at least four points called umbilical points. At an umbilic all the sectional curvatures are equal, in particular the principal curvature's are equal. Umbilical points can be thought of as the points where the surface is closely approximated by a sphere.

   For the sphere the curvatures of all normal sections are equal, so every point is an umbilic. The sphere and plane are the only surfaces with this property.

5. The sphere does not have a surface of centers.

   For a given normal section there is a circle whose curvature is the same as the sectional curvature, is tangent to the surface and whose center lines along on the normal line. Take the two 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 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 canal surfaces one sheet forms a curve and the other sheet is a surface; For cones, cylinders, toruses and cyclides both sheets form curves. For the sphere the center of every osculating circle is at the center of the sphere and the focal surface forms a single point. This is a unique property of the sphere.

6. All geodesics of the sphere are closed curves.

   Geodesics are curves on a surface which give the shortest distance between two points. They are generalisation of the concept of a straight line in the plane. For the sphere the geodesics are great circles. There are many other surfaces with this property.

7. Of all the solids having a given volume, the sphere is the one with the smallest surface area; of all solids having a given surface area, the sphere is the one having the greatest volume.

   These properties define the sphere uniquely. These properties can be seen by observing soap bubbles. A soap bubble will enclose a fixed volume and due to surface tension it will try to minimize its surface area. 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.

9. The sphere has constant positive mean curvature.

   The sphere is the only surface without boundary or singularities with constant positive mean curvature. There are other surfaces with constant mean curvature, the minimal surfaces have zero mean curvature.

10. The sphere has constant positive Gaussian curvature.

    Gaussian curvature is the product of the two principle curvatures. It is an intrinsic property which can be determined by measuring length and angles and does not depend on the way the surface is embedded in space. Hence, bending a surface will not alter the Gaussian curvature and other surfaces with constant positive Gaussian curvature can be obtained by cutting a small slit in the sphere and bending it. All these other surfaces would have boundaries and the sphere is the only surface without boundary with constant positive Gaussian curvature. The pseudosphere is an example of a surface with constant negative Gaussian curvature.

11. The sphere is transformed into itself by a three-parameter family of rigid motions.

    Consider a unit sphere 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. Thus there is a three parameter family of rotations which transform the sphere onto itself, this is the rotation group, SO(3). The plane is the only other surface with a three parameter family of transformations (translations along the x and y axis and rotations around the origin). Circular cylinders are the only surfaces with two parameter families of rigid motions and the surfaces of revolution and helicoids are the only surfaces with a one parameter family.

References

1. ^ http://www.cnn.com/2007/TECH/science/06/15/australia.spheres.reut/index.html
2. ^ Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination, 2nd ed., Chelsea. ISBN 0-8284-1087-9. 

See also

• 3-sphere
• Alexander horned sphere
• Ball (mathematics)
• Banach-Tarski Paradox
• Circle
• Curvature
• Directional statistics
• Dome (mathematics)
• Dyson sphere
• Homology sphere
• Homotopy groups of spheres
• Homotopy sphere
• Hypersphere
• Metric space
• Pseudosphere
• Riemann sphere
• Smale's paradox
• Solid angle
• Spherical cap
• Spherical coordinates
• Spherical Earth

External links

• Surface Area MATHguide
• Volume MATHguide
• Mathworld website

• calculate area and volume with your own radius-values to understand the equations

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

Dansk (Danish)
n. - kugle, klode, himmel, sfære, fag, fagkreds
v. tr. - omkredse, gøre rund, anbringe i en sfære, sende rundt, ophøje, fylde

idioms:

• sphere of action    virkefelt
• sphere of influence    indflydelsesområde

Nederlands (Dutch)
bol, gebied, sfeer, globe

Français (French)
n. - (Astron) sphère, (Astron) sphère céleste, domaine, milieu
v. tr. - former une sphère, enfermer dans une sphère, entourer, envelopper

idioms:

• sphere of influence    sphère d'influence

Deutsch (German)
n. - Bereich, Kugel, Sphäre
v. - umkreisen, in den Himmel heben

idioms:

• sphere of influence    Einflußbereich

Ελληνική (Greek)
n. - (αστρον., μαθημ., μτφ.) σφαίρα, κοινωνικό στρώμα, κοινωνικός κύκλος
v. - κλείνω σε σφαίρα

idioms:

• sphere of action    περιοχή δράσεως
• sphere of influence    σφαίρα επιρροής

Italiano (Italian)
campo, sfera, globo

idioms:

• sphere of influence    sfera d'influenza

Português (Portuguese)
n. - globo (m), esfera (f), ambiente (m)

idioms:

• sphere of influence    esfera de ação

Русский (Russian)
сфера, шар, глобус, земной шар, планета, небеса, окружать, придавать форму шара

idioms:

• sphere of influence    сфера влияния

Español (Spanish)
n. - terreno, campo, dominio, esfera, bola, globo, orbe
v. tr. - colocar en una esfera o entre las esferas, redondear, completar, rodear, abarcar

idioms:

• sphere of influence    esfera de influencia

Svenska (Swedish)
n. - sfär, klot, himlakropp, område, fack, klass
v. - innesluta i en sfär, omsluta, göra sfärisk

中文（简体） (Chinese (Simplified))
球, 球形, 球体, 球面, 包围, 放入球内, 使成球体

idioms:

• sphere of action    行动范围
• sphere of influence    势力范围

中文（繁體） (Chinese (Traditional))
n. - 球, 球形, 球體, 球面
v. tr. - 包圍, 放入球內, 使成球體

idioms:

• sphere of action    行動範圍
• sphere of influence    勢力範圍

한국어 (Korean)
n. - (기하학상의) 구체, 지구의, (활동) 범위
v. tr. - 구상으로 하다, 둘러싸다, 천체 사이에 놓다

日本語 (Japanese)
n. - 球, 天体, 地球儀, 地位, 範囲

idioms:

• sphere of action    活動範囲
• sphere of influence    勢力範囲
• sphere of power    権力圏

العربيه (Arabic)
‏(الاسم) جسم كروي, ميدان, نطاق, حقل (فعل) يجعله كروي الشكل‏

עברית (Hebrew)
n. - ‮כדור, גלובוס, כוכב, שמיים, גלגל, חוג, סביבה, היקף, ספירה, תחום (השפעה)‬
v. tr. - ‮סגר בתוך כדור (ספרותי), נעשה לכדור (ספרותי)‬

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