 Just as a stereographic projection can project a sphere's surface to a plane, it can also project a 3-sphere's surface into 3-space. This image shows three coordinate directions projected to 3-space: parallels (red), meridians (blue) and hypermeridians (green). Due to the conformal property of the stereographic projection, the curves intersect each other orthogonally (in the yellow points) as in 4D. All of the curves are circles: the curves that intersect <0,0,0,1> have an infinite radius (= straight line). In mathematics, a hypersphere is a higher dimensional analog of the usual 2-sphere. It can be constructed by gluing two Euclidean spaces together and can be embedded in Euclidean space as the locus of points a fixed distance from the origin. Image File history File links Sphere-wireframe. ...
Image File history File links Sphere-wireframe. ...
In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ...
Image File history File links No higher resolution available. ...
Stereographic projection of a circle of radius R onto the x axis. ...
Meridian is used in perimetry and in specifying visual fields. ...
In mathematics, a conformal map is a function which preserves angles. ...
Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
For other uses, see sphere (disambiguation). ...
Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...
A hypersphere of dimension n is called an n-sphere and denoted  . It is an n-dimensional manifold and can be embedded in Euclidean (n+1)-space. On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...
Euclidean coordinates in (n+1)-space The set of points in (n+1)-space: (x1,x2,x3,...,xn + 1) that define an n-sphere, ( ) is represented by the equation:  where C is a center point, and r is the radius.
The above hypersphere exists in n + 1-dimensional Euclidean space and is an example of an n-manifold. On a sphere, the sum of the angles of a triangle is not equal to 180° (see spherical trigonometry). ...
n-ball The space enclosed by an n-sphere is called an (n+1)-ball. An (n+1)-ball is closed if it included the equality, and open otherwise. In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U...
Specifically:
- A 1-ball, a line segment, is the interior of a (0-sphere).
- A 2-ball, a disk, is the interior of a circle (1-sphere).
- A 3-ball, an ordinary ball, is the interior of a sphere (2-sphere).
- A 4-ball, is the interior of a 3-sphere, etc.
The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...
In geometry, a disk is the region in a plane contained inside of a circle. ...
Circle illustration This article is about the shape and mathematical concept of circle. ...
In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ...
A sphere is a symmetrical geometrical object. ...
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ...
Notation Labelling hyperspheres with the dimensionality of the surface (as used in this article) is the convention common in mathematical use. Potentially confusingly, some authors use the dimensionality of the containing space to label hyperspheres.[1] Thus what most call a 1-sphere (a regular circle in a plane), others term a 2-sphere (reflecting the dimensionality of the plane in which it lies). Circle illustration This article is about the shape and mathematical concept of circle. ...
Two intersecting planes in three-dimensional space In mathematics, a plane is a two-dimensional manifold or surface that is perfectly flat. ...
Hyperspherical volume The hyperdimensional volume of the space which a (n − 1)-sphere encloses (the n-ball) is:  where Γ is the gamma function. (For even n,  ; for odd n,  , where n!! denotes the double factorial.) The Gamma function along part of the real axis In mathematics, the Gamma function is an extension of the factorial function to complex numbers. ...
In mathematics, the factorial of a natural number n is the product of the positive integers less than or equal to n. ...
The "surface area" of this (n-1)-sphere is
 The following relationships hold between the hyperspherical surface area and volume:   The interior of a hypersphere, that is the set of all points whose distance from the centre is less than R, is called a hyperball, or if the hypersphere itself is included, a closed hyperball.
Hyperspherical volume - some examples For small values of n, the volumes, Vn , of the unit n-ball (R = 1) are: -
If the dimension n, is not limited to integral values, the hypersphere volume is a continuous function of n with a global maximum for the unit sphere in "dimension" n = 5.2569464... where the "volume" is 5.277768... It has a hypervolume of 1 when n = 0 or when n = 12.76405... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ...
In geometry, a disk is the region in a plane contained inside of a circle. ...
In mathematics, a ball is the inside of a sphere; both concepts apply not only in the three-dimensional space but also for lower and higher dimensions, and for metric spaces in general. ...
In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
A graph illustrating local min/max and global min/max points In mathematics, a point x* is a local maximum of a function f if there exists some ε > 0 such that f(x*) ≥ f(x) for all x with |x-x*| < ε. Stated less formally, a local maximum...
The hypercube circumscribed around the unit n-sphere has an edge length of 2 and hence a volume of 2n; the ratio of the volume of the hypersphere to its circumscribed hypercube decreases monotonically as the dimension increases.
This apparently strange behavior of the unit n-sphere volume can be understood by assigning units of length to each dimension. It then becomes clearly meaningless to compare the unit-sphere volumes in different n's, just as it is meaningless to compare a length to an area. A meaningful comparison is obtained by using a dimensionless measure of the volume, such as the ratio of the hypersphere and its circumscribed hypercube volumes. Using this measure restores the intuitively normal behavior of a monotonic decline in the volume as the dimension increases.
Hyperspherical coordinates We may define a coordinate system in an n-dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate  , and  angular coordinates  . If  are the Cartesian coordinates, then we may define This article describes some of the common coordinate systems that appear in elementary mathematics. ...
      While the inverse transformations can be derived from those above:    Note that last angle φn − 1 has a range of 2π while the other angles have a range of π. This range covers the whole sphere.
The hyperspherical volume element will be found from the Jacobian of the transformation: In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...
  and the above equation for the volume of the hypersphere can be recovered by integrating: 
Stereographic projection Just as a two dimensional sphere embedded in three dimensions can be mapped onto a two-dimensional plane by a stereographic projection, an n-dimensional hypersphere can be mapped onto an n-dimensional hyperplane by the n-dimensional version of the stereographic projection. For example, the point ![[x,y,z]](http://upload.wikimedia.org/math/9/6/1/96155b1bc331454ce48057c12f9596a1.png) on a two-dimensional sphere of radius 1 maps to the point ![[x,y,z] mapsto left[frac{x}{1-z},frac{y}{1-z}right]](http://upload.wikimedia.org/math/0/b/1/0b1878f3547966654bf4d9f02e4dc1dc.png) on the  plane. In other words: Stereographic projection of a circle of radius R onto the x axis. ...
Likewise, the stereographic projection of a hypersphere  of radius 1 will map to the n-1 dimensional hyperplane  perpendicular to the  axis as: ![[x_1,x_2,ldots,x_n] mapsto left[frac{x_1}{1-x_n},frac{x_2}{1-x_n},ldots,frac{x_{n-1}}{1-x_n}right]](http://upload.wikimedia.org/math/a/8/4/a84d0bc1b3aa403d237a1dcef32927d9.png) bcgfhgffhhgfhhhhhh hhhhhghf hfgdhgfdhgfhgfhgfhgf fghgfdhgdhgfhgfhggh</smalhgfhgfhgfhfl> blockqugote> hg References - David W. Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, second edition, 2001, [2] (Chapter 20: 3-spheres and hyperbolic 3-spaces.)
- Jeffrey R. Weeks, The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds, 1985, (Chapter 14: The Hypersphere)
- Exploring Hyperspace with the Geometric Product
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