History of Sphere

	Revision 8 . . (edit) December 14, 2001 6:37 am by Zundark [reformat area and volume]
	Revision 7 . . (edit) November 20, 2001 10:37 pm by LA2
	Revision 6 . . (edit) August 28, 2001 12:08 am by Zundark [link]
	Revision 2 . . (edit) August 17, 2001 6:41 am by (logged).176.164.xxx [*fixed a contradiction in dimensional numbering.]

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Difference (from prior major revision) (minor diff, author diff)

Changed: 3c3

More precisely, a sphere is the set of points in 3-dimensional Euclidean space 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.

More precisely, a sphere is the set of points in 3-dimensional Euclidean space 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.

Changed: 5c5,8

This can be generalized to other dimensions. For any natural number n, an n-sphere 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. A 3-sphere is therefore an ordinary sphere, while a 2-sphere is a circle and a 1-sphere is a pair of points. An n-sphere is an example of a compact n-manifold.

This can be generalized to other dimensions. For any natural number n, an n-sphere 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. A 2-sphere is therefore an ordinary sphere, while a 1-sphere is a circle and a 0-sphere is a pair of points. An n-sphere is an example of a compact n-manifold. The surface area of a sphere of radius r is 4πr2, and its volume is 4πr3/3.