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The volume of a n-ball is the Lebesgue measure of this ball, which generalizes to any dimension the usual volume of a ball in 3-dimensional space. The volume of a n -ball of radius R is R n V n , {\displaystyle R^{n}V_{n},} where V n {\displaystyle V_{n}} is the volume of the unit n -ball , the n -ball of radius 1 .
A ball in n dimensions is called a hyperball or n-ball and is bounded by a hypersphere or (n−1)-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk , the area bounded by a circle .
An alternative given by Marsaglia is to uniformly randomly select a point = (,, …,) in the unit n-cube by sampling each independently from the uniform distribution over (,) , computing as above, and rejecting the point and resampling if (i.e., if the point is not in the -ball), and when a point in ...
A subset of R n is a null set if, for every ε > 0, it can be covered with countably many products of n intervals whose total volume is at most ε. All countable sets are null sets. If a subset of R n has Hausdorff dimension less than n then it is a null set with respect to n-dimensional Lebesgue measure.
In Euclidean space of dimensions, the ()-dimensional unit sphere is the set of all points (, …,) which satisfy the equation + + + = The open unit -ball is the set of all points satisfying the inequality
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as x n, while the surface area, being (n − 1)-dimensional, scales as x n−1.
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Dimensional analogy is the study of how (n − 1) dimensions relate to n dimensions, and then inferring how n dimensions would relate to (n + 1) dimensions. [20] The dimensional analogy was used by Edwin Abbott Abbott in the book Flatland, which narrates a story about a square that lives in a two-dimensional world, like the surface of a piece ...