# Encyclopedia

A two-dimensional perspective projection of a sphere
.A sphere (from Greek σφαῖραsphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.^ Give them cue words to help them make various shapes, such as geometric figures, letters, and numbers.

^ Convenience, because of all isoperimetric bodies the sphere is the largest and of all shapes the round is most capacious.

^ The sphere looks like a golf ball sitting on a tee, or a round lollipop.

.Like a circle in three dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point.^ A cross section of a sphere is a perfect circle.

^ Hopefully they'll wander around this experiment at some point and read all the comments and maybe read yours too and then consider it...
• Chrome Experiments - Detail - Google Sphere 11 January 2010 4:53 UTC www.chromeexperiments.com [Source type: FILTERED WITH BAYES]

^ In essence, our business today is centered around bringing content to the surface and so, Surphace is a name that, not only defines our business, but one we’ve grown to love.

This distance r is known as the radius of the sphere. .The maximum straight distance through the sphere is known as the diameter of the sphere.^ Transparent sphere straight & level flight into the distance during air force air display in Australia.
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

^ Moreover, a straight line passing through the center of the sphere, with its ends touching the circumference in opposite directions, is called the "axis of the sphere."

.It passes through the center and is thus twice the radius.^ Also, suppose a plane passed through the center of the earth, dividing it and the firmament into equal halves.

^ BETWEEN THE EQUATOR AND TROPIC OF CANCER. -- To those whose zenith is between the equinoctial and the Tropic of Cancer it happens twice a year that the sun passes directly overhead, which is shown thus.

^ Moreover, a straight line passing through the center of the sphere, with its ends touching the circumference in opposite directions, is called the "axis of the sphere."

.In higher mathematics, a careful distinction is made between the sphere (a two-dimensional spherical surface embedded in three-dimensional Euclidean space) and the ball (the three-dimensional shape consisting of a sphere and its interior).^ Those six circles divide the entire surface of the sphere into twelve parts, wide in the middle but narrower toward the poles, and each such part is called a "sign" and has a particular name from the name of that sign which is intercepted between its two lines.

^ The brief is to a) Identify two or three criterion by which to differentiate between Chapters.
• Internet Society (ISOC) : Sphere Project 11 January 2010 4:53 UTC wiki.chapters.isoc.org [Source type: FILTERED WITH BAYES]

^ Explain that they must work with three aspects of space: the other people, the other dancers' "atmospheres" (space bubbles), and the space between the bubbles.

As defined 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.

## Volume of a sphere

In 3 dimensions, the volume inside a sphere (that is, the volume of the ball) is given by the formula
$\!V = \frac{4}{3}\pi r^3$
where .r is the radius of the sphere and π is the constant pi.^ The apparent temperature would be T = (E / (4 pi r^2 eta sigma))^1/4 where E is the energy output of the sun, r the radius of the sphere, eta the emissivity and sigma the constant of Stefan-Boltzman's law.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

This formula was first derived by Archimedes, who showed that the volume of a sphere is 2/3 that of a circumscribed cylinder. (This assertion follows from Cavalieri's principle.) In modern mathematics, this formula can be derived using integral calculus, e.g. disk integration to sum the volumes of an infinite number of circular disks of infinitesimal thickness stacked centered side by side along the x axis from x = 0 where the disk has radius r (i.e. y = r) to x = r where the disk has radius 0 (i.e. y = 0).
At any given x, the incremental volume (δV) is given by the product of the cross-sectional area of the disk at x and its thickness (δx):
$\!\delta V \approx \pi y^2 \cdot \delta x.$
The total volume is the summation of all incremental volumes:
$\!V \approx \sum \pi y^2 \cdot \delta x.$
In the limit as δx approaches zero[1] this becomes:
$\!V = \int_{x=0}^{x=r} \pi y^2 dx.$
At any given x, a right-angled triangle connects x, y and r to the origin, hence it follows from Pythagorean theorem that:
$\!r^2 = x^2 + y^2.$
Thus, substituting y with a function of x gives:
$\!V = \int_{x=0}^{x=r} \pi (r^2 - x^2)dx.$
This can now be evaluated:
$\!V = \pi \left[r^2x - \frac{x^3}{3} \right]_{x=0}^{x=r} = \pi \left(r^3 - \frac{r^3}{3} \right) = \frac{2}{3}\pi r^3.$
This volume as described is for a hemisphere. Doubling it gives the volume of a sphere as:
$\!V = \frac{4}{3}\pi r^3.$
.In higher dimensions, the sphere (or hypersphere) is usually called an n-ball.^ Saturday night, around 7pm or just after, i saw a fiery orange ball, call it a sphere, call it what you will, basically idling in .
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

General recursive formulas exist for deriving the volume of an n-ball.
.For most practical uses, the volume of a sphere can be approximated by 52.4% (pi/6 ~ 0.5236) of the volume of a cube that it will just fill.^ SPHERE MACHINES You can cut and polish your own spheres with these machines Two-cup machines will produce excellent spheres and are generally the most used version.
• Jack Slevkoff's Prized Possessions--SPHERE MACHINES 11 January 2010 4:53 UTC www.gemworld.com [Source type: General]

Therefore, since a cube with side length 1m has a volume of 1m^3, a sphere with diameter 1m has a volume of about 0.524m^3.

## Surface area of a sphere

The surface area of a sphere is given by the formula
$\!A = 4\pi r^2$
This formula was first derived by Archimedes, based upon the fact that the projection to the lateral surface of a circumscribing cylinder (i.e. the Gall-Peters map projection) is area-preserving. .It is also the derivative of the formula for the volume with respect to r because the total volume of a sphere of radius r can be thought of as the summation of the volumes of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r.^ If one assumes a 1 AU radius, there will be around 42 kg/m^2 of the sphere.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ For a statite in the solar system, the density would be around 0.78 g/m^2 A rigid dyson sphere is not stable, since there is no net attraction between a spherical shell and a point mass inside.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ And even if you happen to know and like your particular garbage man, at one point or another we all have limits to our sphere of monkey concern.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

.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.^ The living space has a thickness of 2400 km if you assume that the outer surface is at a pressure equal to that at 3000 m (10,000 ft) above sea-level on Earth.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ The ring at angle theta, which subtends an angle dtheta, has a circumference 2 pi R sin theta, width R dtheta an thickness t, which gives it a volume of dV=2 pi R^2 t sin theta d theta and a mass of (M/2) sin theta dtheta where rho is the density of the shell.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ A Dyson sphere in the solar system, with a radius of one AU would have a surface area of at least 2.72e17 km^2, around 600 million times the surface area of the Earth.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

At any given radius r, the incremental volume (δV) is given by the product of the surface area at radius r (A(r)) and the thickness of a shell (δr):
$\delta V \approx A(r) \cdot \delta r.$
The total volume is the summation of all shell volumes:
$V \approx \sum A(r) \cdot \delta r.$
In the limit as δr approaches zero[1] this becomes:
$V = \int_{0}^{r}A(r) dr.$
Since we have already proved what the volume is, we can substitute V:
$\frac{4}{3}\pi r^3 = \int_{0}^{r}A(r) dr.$
Differentiating both sides of this equation with respect to r yields A as a function of r:
$\!4\pi r^2 = A(r).$
Which is generally abbreviated as:
$\!A = 4\pi r^2.$
Alternatively, the area element on the sphere is given in spherical coordinates by:
$dA = r^2 \sin heta\, d heta\, d\phi.$
The total area can thus be obtained by integration:
$A = \int_0^{2\pi}\int_0^\pi r^2 \sin heta \, d heta \, d\phi = 4\pi r^2.$

## 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
$\, (x - x_0 )^2 + (y - y_0 )^2 + ( z - z_0 )^2 = r^2.$
The points on the sphere with radius r can be parametrized via
$\, x = x_0 + r \sin heta \; \cos \varphi$
$\, y = y_0 + r \sin heta \; \sin \varphi \qquad (0 \leq \varphi \leq 2\pi \mbox{ and } 0 \leq heta \leq \pi ) \,$
$\, z = z_0 + r \cos heta \,$
A sphere of any radius centered at zero is an integral surface of the following differential form:
$\, x \, dx + y \, dy + z \, dz = 0.$
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 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.^ A Dyson sphere in the solar system, with a radius of one AU would have a surface area of at least 2.72e17 km^2, around 600 million times the surface area of the Earth.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ Convenience, because of all isoperimetric bodies the sphere is the largest and of all shapes the round is most capacious.

^ The sphere would consist of a shell of solar collectors or habitats around the star, so that all (or at least a significant amount) energy will hit a receiving surface where it can be used.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

.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.^ To persons on the earth's surface the stars appear of the same size whether they are in mid-sky or just rising or about to set, and this is because the earth is equally distant from them.

^ A Dyson sphere in the solar system, with a radius of one AU would have a surface area of at least 2.72e17 km^2, around 600 million times the surface area of the Earth.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ Thankfully, the World’s Tallest Water Sphere appears to be safe at this level.

.The surface area in relation to the mass of a sphere is called the specific surface area.^ Knowing these factors, you can combine them to get an equation which relates the mass of the star to the desired temperature and gravity of the sphere: k M^(nu - 1) = 4 pi e sigma G T^4/g.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ A Dyson sphere in the solar system, with a radius of one AU would have a surface area of at least 2.72e17 km^2, around 600 million times the surface area of the Earth.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ If the surface is a sphere surrounding the dyson sphere, there is obviously an inward force on the surface of the sphere since there is a mass inside it.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

From the above stated equations it can be expressed as follows:
$SSA = \frac{A}{V\rho} = \frac{3}{r\rho}.$
.
An image of one of the most accurate man-made spheres, as it refracts the image of Einstein in the background.
^ And even if you happen to know and like your particular garbage man, at one point or another we all have limits to our sphere of monkey concern.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

^ Most water towers made in the last 50 years are water spheroids and not water spheres.

.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 of thickness.^ Therefore, your translation is no more correct than the one in the article.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

^ Large sphere, surface covered with different colored 'domes', so that no smooth surface was visible.
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

^ Bright spheric shaped object observed in the southwest sky moving erratically, it was very bright and observed for more than 30 minut .
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

It is thought that only neutron stars are smoother. .It was announced on 1 July 2008 that Australian scientists had created even more perfect spheres, accurate to 0.3 nanometers, as part of an international hunt to find a new global standard kilogram.^ Getting very tough to find, these are all premium parts, perfect for Tube or Guitar amp repairs or construction.
• Sphere's Used Electronic Test Equipment 11 January 2010 4:53 UTC www.sphere.bc.ca [Source type: FILTERED WITH BAYES]

^ Ted wagers that if they measured it with laser micrometers that theyd find it was a perfect sphere to within 1/1,000 th of an inch.
• Sphere 11 January 2010 4:53 UTC jabootu.com [Source type: Original source]

^ This allows you to find the 1N, 2N or other industry standard part number for house marked HP semiconductors.
• Sphere's Used Electronic Test Equipment 11 January 2010 4:53 UTC www.sphere.bc.ca [Source type: FILTERED WITH BAYES]

[2]
.A sphere can also be defined as the surface formed by rotating a circle about any diameter.^ A rotating dyson sphere would be under immense strains; see the section about the ringworld for a simple calculation.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ The answer is a non-rotating bubbleworld can be as large as 480,000 km in diameter (about 3 times the diameter of Jupiter), if you make certain assumptions.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ White sphere with 4 cm diameter, about 4 meters from eyes.
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

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.^ Also the sun describes these circles with the sweep of the firmament as it descends from the first point of Cancer through Libra to the first point of Capricorn; and those circles are called the "circles of natural days."

^ STATIONARY, DIRECT, AND RETROGRADE. -- If, then, two lines are drawn from the center of the earth to include an epicycle, one on the east and the other on the west, the point of contact on the east is called the "first station," while the point of contact to the west is called the "second station."

^ Those have a right horizon and right sphere whose zenith is on the equinoctial, since their horizon is a circle passing through the poles of the world cutting the equinoctial at right angles, wherefore it is called "right horizon" and "right sphere."

.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.^ And as the zodiac is divided by astronomers, so each circle in the sphere, whether great or small, is divided into similar parts.

^ THE ZODIAC. -- There is another circle in the sphere which intersects the equinoctial and is intersected by it into two equal parts.

^ Any circle is called "eccentric" which, like that of the sun, dividing the earth into equal parts, does not have the same center as the earth but one outside it.

.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.^ DAY AND NIGHT. -- Wherefore it appears that, if two circles are taken equidistant in their various parts from the equinoctial, as great as is the arc of day in the one, so great is the arc of night in the other.

^ THE MERIDIAN. -- There are yet two other great circles in the sphere, namely, the meridian and the horizon.

^ COLURES. -- There are two other great circles in the sphere which are called "colures," whose function is to distinguish solstices and equinoxes.

Equipped with the 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.^ And even if you happen to know and like your particular garbage man, at one point or another we all have limits to our sphere of monkey concern.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

^ Both spheres and spheroids are special-case ellipsoids: spheres have symmetry in 3 directions, spheroids have symmetry in 2 directions (east-west, north-south, but not top-bottom)4, and scalene ellipsoids have 3 unequal length axes.

.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.^ And the two ends of the axis are called the "poles of the world."

^ The other is the slower movement of the sphere itself on the poles of the axis of the circle of the signs, and it is equal to the movement of the sphere of the fixed stars, namely, 1 degree in a hundred years.

^ Here Lucan calls the equinoctial "the circle of the high solstice," on which two high solstices happen to those living at the equator.

.Circles on the sphere that are parallel to the equator are lines of latitude.^ The length of a clime may be said to be the line drawn from east to west parallel to the equator; wherefore the length of the first clime is greater than the length of the second and so on, which happens because the sphere narrows down.

^ Suppose, then, a line parallel to the equator dividing the parts uninhabitable on account of heat from those habitable parts toward the north.

^ While every circle in the sphere except the zodiac is understood to be a line or circumference, the zodiac alone is understood to be a surface, 12 degrees wide of degrees such as we have just mentioned.

This terminology is also used for astronomical bodies such as the planet Earth, even though it is neither spherical nor even spheroidal (see geoid).

## Hemisphere

.A sphere is divided into two equal hemispheres by any plane that passes through its center.^ Also, suppose a plane passed through the center of the earth, dividing it and the firmament into equal halves.

^ According to Frank Palmer: Any sphere about a gravitating body can be analysed into two hemispheres joined at a seam.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ THE NORTH AND SOUTH POLES. -- 'Tis called the "belt of the first movement" because it divides the primum mobile or ninth sphere into two equal parts and is itself equally distant from the poles of the world.

.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.^ According to Frank Palmer: Any sphere about a gravitating body can be analysed into two hemispheres joined at a seam.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ And if two orbits intersect, they can be adjusted by using solar sails, ion engines, magsails or similar low-energy devices.
• http://www.nada.kth.se/%7Easa/dysonFAQ.html 11 January 2010 4:53 UTC www.nada.kth.se [Source type: FILTERED WITH BAYES]

^ Saw sphere over treetops ,thought it was moon ,change colors, spin and split into two spheres .
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

.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.^ The two fixed points in the firmament are called the "poles of the world" because they terminate the axis of the sphere and the world revolves on them.

^ By Theodosius a sphere is described thus: A sphere is a solid body contained within a single surface, in the middle of which there is a point from which all straight lines drawn to the circumference are equal, and that point is called the "center of the sphere."

## Generalization to other dimensions

Spheres can be generalized to spaces of any dimension. .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.^ The two fixed points in the firmament are called the "poles of the world" because they terminate the axis of the sphere and the world revolves on them.

^ From a fixed point, the object zigzad north and south, covering a 16 miles distance in no more than a second or two.
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

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.
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 (which is embedded in 3-dimensional space).
The surface area of the (n − 1)-sphere of radius 1 is
$2 \frac{\pi^{n/2}}{\Gamma(n/2)}$
where Γ(z) is Euler's Gamma function.
Another formula for surface area is
$\begin{cases} \displaystyle \frac{(2\pi)^{n/2}\,r^{n-1}}{2 \cdot 4 \cdots (n-2)}, & ext{if } n ext{ is even}; \\ \ \displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^{n-1}}{1 \cdot 3 \cdots (n-2)}, & ext{if } n ext{ is odd}. \end{cases}$
and the volume is the surface area times ${r \over n}$ or
$\begin{cases} \displaystyle \frac{(2\pi)^{n/2}\,r^n}{2 \cdot 4 \cdots n}, & ext{if } n ext{ is even}; \\ \ \displaystyle \frac{2(2\pi)^{(n-1)/2}\,r^n}{1 \cdot 3 \cdots n}, & ext{if } n ext{ is odd}. \end{cases}$

## 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.^ The first step is to make a general outline shape of the WTWS. The top is sphere and likely will be the focal point of your drawing.

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. .The same applies for the radius if it is taken equal to one, as in the case of a unit sphere.^ From this it seems to follow that if two natural days in the year are taken equally remote from either equinoctial point in opposite directions, as long as is the artificial day in one case, so long is the night in the other, and conversely.

^ Since the zenith is the pole of the horizon, those two quarters, since they are quarters of one and the same circle, are equal.

^ From this it is evident that two equal and opposite arcs in the slanting sphere have their combined ascensions equal to the ascensions of the same arcs taken together in the right sphere, because as much as is the diminution on the one hand, so much is the addition on the other.

In contrast to a ball, a sphere may be an empty set, even for a large radius. 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.

## 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.^ SPHERE DEFINED. -- A sphere is thus described by Euclid: A sphere is the transit of the circumference of a half-circle upon a fixed diameter until it revolves back to its original position.

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 implies that a Euclidean n-sphere is compact. .The sphere is the inverse image of a one-point set under the continuous function ||x||.^ And even if you happen to know and like your particular garbage man, at one point or another we all have limits to our sphere of monkey concern.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

^ In the second we give information concerning the circles of which this material sphere is composed and that supercelestial one, of which this is the image, is understood to be composed.

^ Just how he was able to deduce this from an utterly unfamiliar setting with which he has no prior reference points is one of the true mysteries of the film.
• Sphere 11 January 2010 4:53 UTC jabootu.com [Source type: Original source]

Therefore, the sphere is closed. Sn is also bounded; therefore it is compact.

## Spherical geometry

Great circle on a sphere
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.^ The rule, indeed, is that any two arcs which are equal and equally distant from either of the equinoctial points have unequal (?

^ DAY AND NIGHT. -- Wherefore it appears that, if two circles are taken equidistant in their various parts from the equinoctial, as great as is the arc of day in the one, so great is the arc of night in the other.

^ Be it noted, then, that the sun, when in the first point of Cancer or the summer solstice, as it is carried by the firmament describes a circle, which is the one last described by the sun in the direction of the Arctic pole.

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.^ And it is called "right" because neither pole is elevated more for them than the other, or because their horizon intersects the equinoctial circle and is intersected by it at spherical right angles.

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[3] 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.^ And even if you happen to know and like your particular garbage man, at one point or another we all have limits to our sphere of monkey concern.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

^ Since all quarters of one and the same circle are equal, the quarter of this colure between equator and pole is equal to the quarter of the same colure from the first point of Cancer to the pole of the zodiac.

^ The two fixed points in the firmament are called the "poles of the world" because they terminate the axis of the sphere and the world revolves on them.

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 the Meissner body. 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.^ Also the sun describes these circles with the sweep of the firmament as it descends from the first point of Cancer through Libra to the first point of Capricorn; and those circles are called the "circles of natural days."

^ In the right sphere the horizon, since it passes through the poles of the world, divides all those circles into equal parts, whence the arcs of days are the same as those of nights for persons living at the equator.

^ Those have a right horizon and right sphere whose zenith is on the equinoctial, since their horizon is a circle passing through the poles of the world cutting the equinoctial at right angles, wherefore it is called "right horizon" and "right 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 are 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.^ Also the sun describes these circles with the sweep of the firmament as it descends from the first point of Cancer through Libra to the first point of Capricorn; and those circles are called the "circles of natural days."

^ The thing that I think most of those who have posted negatively to this article fail to understand that these are ideas are free from value judgements.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

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 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.
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 centers 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 channel 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.^ While every circle in the sphere except the zodiac is understood to be a line or circumference, the zodiac alone is understood to be a surface, 12 degrees wide of degrees such as we have just mentioned.

^ Major point: Ted says this, even though the entire surface of this perfect sphere is roiling and undulating, with valleys forming that look deep enough to hold a cantaloupe.
• Sphere 11 January 2010 4:53 UTC jabootu.com [Source type: Original source]

^ The equinoctial is a circle dividing the sphere into two equal parts and equidistant at its every point from either pole.

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.^ BETWEEN THE ARCTIC CIRCLE AND THE NORTH POLE. -- To those whose zenith is between the Arctic circle and the North Pole, it happens that their horizon will intersect the zodiac in two points equidistant from the beginning of Cancer.

^ It likewise happens that the portion intercepted between two points equidistant from the beginning of Capricorn is always left below the horizon.

They are generalisation of the concept of a straight line in the plane. .For the sphere the geodesics are great circles.^ THE MERIDIAN. -- There are yet two other great circles in the sphere, namely, the meridian and the horizon.

^ But, since the firmament is in continual motion, the circle of the horizon will intersect the zodiac instantaneously, and, since they are great circles in the sphere, they will intersect in equal parts.

^ COLURES. -- There are two other great circles in the sphere which are called "colures," whose function is to distinguish solstices and equinoxes.

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.^ And even if you happen to know and like your particular garbage man, at one point or another we all have limits to our sphere of monkey concern.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

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. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).
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.^ Small mettalic spheres "chasing" each other .
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

^ Given the other action seen in the cut-aways, even if time was compressed with editing, Norman has gotten to the sphere in an impossibly short time.
• Sphere 11 January 2010 4:53 UTC jabootu.com [Source type: Original source]

^ Phosphorescent green sphere, with no other surface characteristics.
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

.All these other surfaces would have boundaries and the sphere is the only surface without boundary with constant positive Gaussian curvature.^ The shots of Harry and Normans reflection showing up in the surface of the sphere where they were previously the only things not reflected would have engendered a combination of surprise and dread in the viewer, had they not (you guessed it) been shown in the trailer.
• Sphere 11 January 2010 4:53 UTC jabootu.com [Source type: Original source]

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 placed 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 other is the slower movement of the sphere itself on the poles of the axis of the circle of the signs, and it is equal to the movement of the sphere of the fixed stars, namely, 1 degree in a hundred years.

^ I through out the trush I look it the sky in the Northwest it fly and stop in about three then .
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

^ UFO spotted over Gwinnett county; metallic sphere surrounded by three or four smaller spheres which were rotating around the center.
• ndxLocOut 11 January 2010 4:53 UTC www.nuforc.org [Source type: General]

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.

## Cubes in relation to spheres

For every sphere there are multiple cuboids that may be inscribed within the sphere. When briefly considered it becomes apparent that the largest of the multiple cuboids which may be inscribed is a cube.

## Notes

1. ^ a b Pages 141, 149. E.J. Borowski, J.M. Borwein. Collins Dictionary of Mathematics. ISBN 0-00-434347-6.
2. ^ New Scientist | Technology | Roundest objects in the world created
3. ^ Hilbert, David; Cohn-Vossen, Stephan (1952). Geometry and the Imagination (2nd ed.). Chelsea. ISBN 0-8284-1087-9.

## References

• William Dunham. "Pages 28, 226", The Mathematical Universe: An Alphabetical Journey Through the Great Proofs, Problems and Personalities, ISBN 0-471-17661-3.
• Surface area of sphere proof.

# Study guide

Up to date as of January 14, 2010

### From Wikiversity

.A sphere is the set of all points in three-dimensional space.^ And even if you happen to know and like your particular garbage man, at one point or another we all have limits to our sphere of monkey concern.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

^ If set to true , normals will be pointed inside the sphere.

^ Sphere extends TriMesh Sphere represents a 3D object with all points equidistant from a center point.
• Sphere (jME API) 11 January 2010 4:53 UTC jmonkeyengine.com [Source type: Reference]

 Run a search on Sphere at Wikipedia.
 Search Wikimedia Commons for images, sounds and other media related to: Sphere
 Search for Sphere on the following projects:
 Lost on Wikiversity? Please help by choosing project boxes to classify this resource by: subject educational level resource type

# 1911 encyclopedia

Up to date as of January 14, 2010

### From LoveToKnow 1911

SPHERE (Gr. .acf aZpa, a ball or globe), in geometry, the solid or surface traced out by the revolution of a semicircle about its diameter; this is essentially Euclid's definition; 1 in the modern geometry of surfaces it is defined as the quadric surface passing through the circle at infinity.^ The surface and solid traced by the revolution of the lesser segment of a circle is termed a "spindle."

^ Zpa, a ball or globe), in geometry , the solid or surface traced out by the revolution of a semicircle about its diameter; this is essentially Euclid's definition; 1 in the modern geometry of surfaces it is defined as the quadric surface passing through the circle at infinity.

^ The size of the circle is maximized when the plane defining the cross section passes through a diameter .

Every point is equidistant from a fixed point within the surface; this point is the "centre," the constant distance the "radius," and any line through the centre and intersecting the sphere is a "diameter." All sections of the 1 The surfaces formed by revolving a circle about any chord also received attention at the hands' of the Greeks. .According to Heron and Geminus they were discussed under the name spire by Perseus (c. 200-100 B.C.), their sections were termed spiral sections, and are probably the same as the hippopede of Eudoxus.^ According to Heron and Geminus they were discussed under the name spire by Perseus ( c.

^ B.C.), their sections were termed spiral sections, and are probably the same as the hippopede of Eudoxus.

.The surface and solid traced by the revolution of the lesser segment of a circle is termed a "spindle."^ The surface and solid traced by the revolution of the lesser segment of a circle is termed a "spindle."

^ Zpa, a ball or globe), in geometry , the solid or surface traced out by the revolution of a semicircle about its diameter; this is essentially Euclid's definition; 1 in the modern geometry of surfaces it is defined as the quadric surface passing through the circle at infinity.

^ If the plane does not contain the centre, the curve of intersection is a "small circle," and the solid cut off is a "segment."

.An "anchor ring" or "tore" results when a circle revolves about an axis in its plane.^ An " anchor ring" or "tore" results when a circle revolves about an axis in its plane.

^ Two spheres intersect in a plane, and the equation to a system of spheres which intersect in a common circle is x 2 + y 2 + z 2 +2Ax -fD = o, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of intersection, and the axis of x the line of centres.

^ All sections of the 1 The surfaces formed by revolving a circle about any chord also received attention at the hands' of the Greeks.

sphere are necessarily circles; if the cutting plane contains the centre, the section is said to be "meridional," the curve of intersection is a "great circle," and the solid cut off a "hemisphere." If the plane does not contain the centre, the curve of intersection is a "small circle," and the solid cut off is a "segment." "Great" circles may also be defined as circles on a sphere which pass through the extremities of a diameter; they are familiar as the meridians or lines of longitude of geographers; lines of latitude are "small circles." The shortest distance between two points on a sphere is the arc of the great circle containing the points. .This proposition is the basis of the "great circle sailing" of navigators, and the arc of the great circle is called the "rhumbline" or "loxodromic curve."^ Such a circle is called a great circle .
• The Geometry of the Sphere 1. 11 January 2010 4:53 UTC math.rice.edu [Source type: Original source]

^ This proposition is the basis of the "great circle sailing" of navigators, and the arc of the great circle is called the "rhumbline" or "loxodromic curve."

^ Given two points on a sphere, the shortest path on the surface of the sphere which connects them (the sphere geodesic ) is an arc of a circle known as a great circle .

.The determination of the shortest distance between two small circles on a sphere is given in the article Variations, Calculus Of.^ A geodesic , the shortest distance between any two points on a sphere, is an arc of the great circle through the two points.
• sphere (mathematics) -- Britannica Online Encyclopedia 11 January 2010 4:53 UTC www.britannica.com [Source type: FILTERED WITH BAYES]

^ The determination of the shortest distance between two small circles on a sphere is given in the article Variations, Calculus Of .

^ The shortest distance between two points on a sphere is the arc of the great circle containing the points.

.The extremities of the diameter perpendicular to a small circle are called the "poles" of that circle, and the distance from the pole to the circle, measured by the arc of the great circle through the pole, is the "polar distance" of the small circle.^ The extremities of the diameter perpendicular to a small circle are called the "poles" of that circle, and the distance from the pole to the circle, measured by the arc of the great circle through the pole, is the "polar distance" of the small circle.

^ Such a circle is called a great circle .
• The Geometry of the Sphere 1. 11 January 2010 4:53 UTC math.rice.edu [Source type: Original source]

^ A meridian is any great circle passing through a point designated a pole.
• sphere (mathematics) -- Britannica Online Encyclopedia 11 January 2010 4:53 UTC www.britannica.com [Source type: FILTERED WITH BAYES]

The solid enclosed by a small circle and the radii vectores from the centre of the sphere is a "spherical sector"; and the solid contained between two spherical sectors standing on copolar small circles is a "spherical cone." A "spherical sector" and "spherical cone" may be also regarded as the solids of revolution of a circular sector about one of its bounding radii, and about any other line through the vertex respectively. .The solid intercepted between two parallel planes is a "zone."^ The arc of the equinoctial intercepted between two meridians is called the "longitude" of the city.

^ The upper arc of the epicycle intercepted between those two stations is called "direction," and when the planet is there it is called "direct."

^ It likewise happens that the portion intercepted between two points equidistant from the beginning of Capricorn is always left below the horizon.

.The geometry of the sphere was studied by the Greeks; Euclid, in book xii.^ The study of spheres is basic to terrestrial geography and is one of the principal areas of Euclidean geometry and elliptic geometry .
• sphere (mathematics) -- Britannica Online Encyclopedia 11 January 2010 4:53 UTC www.britannica.com [Source type: FILTERED WITH BAYES]

^ Index/Ghostly Greek Geometry: Sphere .
• B Fuller Master Index: Sphere 11 January 2010 4:53 UTC www.buckminster.info [Source type: Academic]

of his .Elements, discusses various properties of the sphere, and in book xiii.^ Elements, discusses various properties of the sphere, and in book xiii.

he shows how to inscribe the five regular polyhedra within it. .But with the sole exception of proving that the volumes of spheres are in the triplicate ratio of their diameters, a theorem probably due to Eudoxus, no mention is made of its mensuration.^ The Aleph was probably two or three centimeters in diameter, but universal space was contained inside it, with no diminution in size.
• Untitled Document 11 January 2010 4:53 UTC www.cooper.edu [Source type: Original source]

^ Attachments: homology of the sphere (Derivation) by mathcam area of the -sphere (Derivation) by CWoo volume of the -sphere (Derivation) by CWoo intersection of sphere and plane (Theorem) by pahio .
• PlanetMath: sphere 11 January 2010 4:53 UTC planetmath.org [Source type: Academic]

^ The ETERNAL SPHERE is a 12" diameter sphere made of sturdy, high-fired stoneware; one of the most durable of man-made materials.

This subject was investigated by Archimedes, who, by his "method of exhaustions," derived the principal results. .He showed that the surface of a segment is equal to the area of the circle whose radius equals the distance from the vertex to the base of the segment; that the surface of the entire sphere is equal to the curved surface of the circumscribing cylinder, and to four times the area of a great circle of the sphere; and that the volume is twothirds that of the circumscribing cylinder.^ Volume of a sphere Sphere geometry Area of a sphere Surface area of a s...
• Sphere Definition | Definition of Sphere at Dictionary.com 11 January 2010 4:53 UTC dictionary.reference.com [Source type: Reference]

^ Planes, spheres, circles, and great circles.
• The Geometry of the Sphere 1. 11 January 2010 4:53 UTC math.rice.edu [Source type: Original source]

^ Example: To the nearest tenth, what is the volume and surface area of a sphere having a radius of 4cm?
• Space figures and basic solids 11 January 2010 4:53 UTC www.mathleague.com [Source type: Academic]

.To Zenodorus (c. 200100 B.C.) is due the important problem in maxima and minima that for a given surface the sphere is the solid of maximum volume.^ Volume of a sphere Sphere geometry Area of a sphere Surface area of a s...
• Sphere Definition | Definition of Sphere at Dictionary.com 11 January 2010 4:53 UTC dictionary.reference.com [Source type: Reference]

^ The surface area of a sphere and volume of the ball of radius are given by .

^ The volume of a sphere is given as V sphere = 4/3 p r 3 , where r is the radius of the sphere.
• CELL AGGREGATION AND SPHERE PACKING 11 January 2010 4:53 UTC www.tiem.utk.edu [Source type: Academic]

Calling the radius r, and denoting by the ratio of the circumference to the diameter of a circle, the volume is 31rr 3, and the surface 41rr2.
.Archimedes gave his results in the treatise IIepi Ti j c aOaipas Kai roD KUXLvbpov: he left unfinished the problem of dividing a sphere into segments whose volumes are in a given ratio.^ The volume of a sphere is given as V sphere = 4/3 p r 3 , where r is the radius of the sphere.
• CELL AGGREGATION AND SPHERE PACKING 11 January 2010 4:53 UTC www.tiem.utk.edu [Source type: Academic]

^ Archimedes gave his results in the treatise IIepi Ti j c aOaipas Kai roD KUXLvbpov: he left unfinished the problem of dividing a sphere into segments whose volumes are in a given ratio.

^ But with the sole exception of proving that the volumes of spheres are in the triplicate ratio of their diameters, a theorem probably due to Eudoxus, no mention is made of its mensuration .

.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 by the Arabian mathematician and astronomer, Al Kuhi.^ A solution by means of the parabola and hyperbola was given by Dionysodorus of Amisus ( c.

^ 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 by the Arabian mathematician and astronomer, Al Kuhi.

^ It is useful as a teaching tool and as an analog computer for solving various astronomical problems to a crude degree of accuracy.
• Armillary Sphere 11 January 2010 4:53 UTC www.humboldt.edu [Source type: FILTERED WITH BAYES]

.In analytical geometry, the equation to the sphere takes the forms x 2 +y 2 +z 2 =a 2, and r=a, the first applying to rectangular Cartesian co-ordinates, the second to polar, the origin being in both cases at the centre of the sphere.^ The equation of a sphere of radius centered at the origin is given in Cartesian coordinates by .

^ In analytical geometry, the equation to the sphere takes the forms x 2 +y 2 +z 2 =a 2, and r=a, the first applying to rectangular Cartesian co-ordinates, the second to polar, the origin being in both cases at the centre of the sphere.

^ If r, r i be the radii of two spheres, d the distance between the centres, and 0 the angle at which they intersect, then d2=r2+ r12 2rr l cos ¢ hence 2rr 1 cos =d2r2 - r22.

.If the centre be (a, a, y), the Cartesian equation becomes (x - a) 2 l3)2 + (z - y)2 = a2; consequently the general equation is x2+y2 -}- z 2 + 2Ax+ 2By+2Cz+D =o, and it is readily shown that the co-ordinates of the centre are (-A, -B, -C), and the radius A2+B2+C2-D. A sphere can therefore be described so as to satisfy four given conditions.^ How to create the protocols for three dimensional organisational charts based on Fuller’s omnidirectional geometry (radiating out from the centre of the sphere) and not the traditional x,y and z cartesian dimensions?
• Open Sphere - P2P Foundation 11 January 2010 4:53 UTC p2pfoundation.net [Source type: FILTERED WITH BAYES]

^ In the role-playing game Mage the Ascension , the nine Sphere s are the terms used OOC to describe a given mage 's capacity to manipulate reality.

^ In order to co-ordinate and support the work of the Sphere project teams and to provide a flexible interface between the project teams and the Executive and Board a Sphere Integration Team (SIT) has been formed.
• Internet Society (ISOC) : Sphere Project 11 January 2010 4:53 UTC wiki.chapters.isoc.org [Source type: FILTERED WITH BAYES]

.Systems of spheres have characters analogous to those of systems of circles.^ The components and properties of a sphere are analogous to those of a circle.
• sphere (mathematics) -- Britannica Online Encyclopedia 11 January 2010 4:53 UTC www.britannica.com [Source type: FILTERED WITH BAYES]

^ Placing vertices at points having those circles as horizons forms a polytope with all edges tangent to the sphere.
• The Geometry Junkyard: Sphere Packing 11 January 2010 4:53 UTC www.ics.uci.edu [Source type: Reference]

^ Those have a right horizon and right sphere whose zenith is on the equinoctial, since their horizon is a circle passing through the poles of the world cutting the equinoctial at right angles, wherefore it is called "right horizon" and "right sphere."

.If r, r i be the radii of two spheres, d the distance between the centres, and 0 the angle at which they intersect, then d2=r2+ r12 2rr l cos ¢ hence 2rr 1 cos =d2r2 - r22. This function is named the "power" of the two spheres, and it is important in the investigation of systems of spheres.^ (The effect of the variation in distance between the Earth and the Sun caused by the Earth's elliptical orbit is of little consequence because of the power of the oceans to store heat.

^ Thus it's clear that once several distance functions have been defined on the same set the expression "the distance between two points" becomes ambiguous.
• Objects distant and near from Interactive Mathematics Miscellany and Puzzles 11 January 2010 4:53 UTC www.cut-the-knot.org [Source type: FILTERED WITH BAYES]

^ The simple function of this transforming sphere makes it an extremely addictive desktop fidget toy -- one which will be hard for your co-workers to resist picking up.

.If the sphere r, degenerate to a point, the function 2rr 1 cos 0 has the limit d 2 - r2; this is the square of the tangent to the sphere from the point, and is named the "power of the sphere at the point," or the "power of the point with respect to the sphere."^ Placing vertices at points having those circles as horizons forms a polytope with all edges tangent to the sphere.
• The Geometry Junkyard: Sphere Packing 11 January 2010 4:53 UTC www.ics.uci.edu [Source type: Reference]

^ The simple function of this transforming sphere makes it an extremely addictive desktop fidget toy -- one which will be hard for your co-workers to resist picking up.

^ And even if you happen to know and like your particular garbage man, at one point or another we all have limits to our sphere of monkey concern.
• What is the Monkeysphere? | Cracked.com 11 January 2010 4:53 UTC www.cracked.com [Source type: Original source]

.Two spheres intersect in a plane, and the equation to a system of spheres which intersect in a common circle is x 2 + y 2 + z 2 +2Ax -fD = o, in which A varies from sphere to sphere, and D is constant for all the spheres, the plane yz being the plane of intersection, and the axis of x the line of centres.^ Placing vertices at points having those circles as horizons forms a polytope with all edges tangent to the sphere.
• The Geometry Junkyard: Sphere Packing 11 January 2010 4:53 UTC www.ics.uci.edu [Source type: Reference]

^ Shows a working party train system Also a starting of a vehicle system :) The moogle is a person, the two pacman sprites - (actually a jeep and a plane) are vehicles...

^ Given two points on a sphere, the shortest path on the surface of the sphere which connects them (the sphere geodesic ) is an arc of a circle known as a great circle .

.Corresponding to the radical centre of three circles, it may be shown that four spheres have a radical centre, i.e. that there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts ,the four spheres at right angles; this "orthotomic" sphere corresponds to the orthogonal circle of a system of circles.^ Points on a circle are identified with the corresponding central angle.
• Objects distant and near from Interactive Mathematics Miscellany and Puzzles 11 January 2010 4:53 UTC www.cut-the-knot.org [Source type: FILTERED WITH BAYES]

^ There are four points of note on the chart above.
• The Celestial Sphere 11 January 2010 4:53 UTC wind.cc.whecn.edu [Source type: General]

^ The regions of an EQ partition have been proven to have small diameter, in the sense that there exists a constant C(dim) such that the maximum diameter of the regions of an N region EQ partition of S^dim is bounded above by C(dim)*N^(-1/dim).
• Recursive Zonal Equal Area Sphere Partitioning Toolbox 11 January 2010 4:53 UTC eqsp.sourceforge.net [Source type: Academic]

.The investigation of triangles and other figures drawn upon the surface of a sphere is all-important in the sciences of astronomy, geodesy and geography.^ Home The Project The Rules The Gallery Post Submission   " Sphere : [sfeer, noun] a space or solid enclosed by a surface, all points of which are equidistant from the center."
• CGSphere The Project 11 January 2010 4:53 UTC www.cgsphere.com [Source type: General]

^ Figure 5: The mapping between planar texture triangles and spherical triangles on the lit sphere.
• Lit Sphere Shading 11 January 2010 4:53 UTC www.cs.utah.edu [Source type: Reference]

^ Above all, it’s his proactive leadership that has made the most important impact on Sphere.

.In astronomy, we are principally concerned with the orientation of points on a sphere - the so-called celestial sphere - with regard to certain planes and points within the sphere; this subject is treated in the article Astronomy (Spherical). In "geodesy," and the cognate subject "figure of the earth," the matter of greatest moment with regard to the sphere is the determination of the area of triangles drawn on the surface of a sphere - the so-called "spherical triangles"; this is a branch of trigonometry, and is studied under the name of spherical trigonometry.^ The poles of this sphere, called the celestial poles, correspond with the projections of the earth's poles onto the sphere.
• The Celestial Sphere 11 January 2010 4:53 UTC wind.cc.whecn.edu [Source type: General]

^ Even today, basic observational astronomy is taught using the celestial sphere as an analog for the real sky.
• SPACE.com -- The Celestial Sphere 11 January 2010 4:53 UTC www.space.com [Source type: General]

^ Doctrine of the sphere , applications of the principles of spherical trigonometry to the properties and relations of the circles of the sphere , and the problems connected with them, in astronomy and geography , as to the latitudes and longitudes , distance and bearing , of places on the earth , and the right ascension and declination , altitude and azimuth , rising and setting , etc, of the heavenly bodies; spherical geometry .
• Sphere - definition from Biology-Online.org 11 January 2010 4:53 UTC www.biology-online.org [Source type: FILTERED WITH BAYES]

.In mathematical geography the problem of representing the surface of a sphere on a plane is of fundamental importance; this subject is treated in the article MAP.^ We represented electro magnets and magnets in a liner arrangement, but you can see 3-D animation work in conducting sphere surfaces .
• AirShip Technologies Group - Delorean Maglev Track Sphere 11 January 2010 4:53 UTC www.airshiptg.org [Source type: FILTERED WITH BAYES]

# Simple English

File:Sphere wireframe 10deg
An image of a sphere.

A sphere is a shape in space that is like the surface of a ball, usually the terms ball and sphere are used alike, but in maths, the precise (exact) definition only allows points in the 3 dimensional space which are uniformly and symmetrically allocated at a fixed length called radius of the sphere.

Examples of these are basketballs, superballs, and playground balls.

In other places sphere means the earth.

A sphere is the 3 dimensional analogue of a circle.

## Volume

$\!V = \frac\left\{4\right\}\left\{3\right\}\pi r^3$

# Citable sentences

Up to date as of December 14, 2010

Here are sentences from other pages on Sphere, which are similar to those in the above article.