# What are some practical usages of computing volume in $n$ dimensions?

I am not sure if this is really a mathematical problem, but I know how to find volume of say a sphere in n dimensions, but after coming to realize how to do this, i just don't get what would be the practical usages of the result?

Like for instance, why would it be of any interest to find volumes of objects in multi dimensions, are there any practical reasoning or applications to this instead of looking from a purely mathematical point of view?

I know for a fact many mathematical theorems dont have any practical usage but could be just a mathematically interesting idea and possibly lead to new mathematical techniques in the future, but i was just curious what applications n dimensional volumes might have.

• Such computations directly come up often in statistical mechanics and quantum field theory, and establishing what we mean by notions of volume in $\mathbb{R}^n$ is important to being able to define the corresponding concept in more general settings, such as manifolds, and these notions are fundamental to descriptions of nature such as general relativity. More straightforwardly, one must define volume before one can define integration (more true of the Lebesgue integral, but you still need volumes of boxes for the Riemann integral), and practically every corner of science uses integration. – jawheele May 9 at 18:25
• Among other things, see Part 2. Statistical Applications in Kendall's A Course in the Geometry of $n$ Dimensions. – Dave L. Renfro May 9 at 18:32
• The volume of the $n$ sphere has obvious applications. But the volume of the regular 600-cell polytope in four dimensions? I think that one's just for fun. – David K May 9 at 18:37

The real question here seems to me to be "What is the use of $$n$$ dimensions?" The history of mathematics from the $$19$$th century onwards has tremendous applications for $$n$$ dimensional space in physics. See, for example, Wikipedia article Phase space. One important idea is that given any list of $$n$$ numbers, it can be interpreted as the coordinates of a point in an $$n$$ dimensional space. This allows the use of geometric concepts such a volume and hyperplanes. This is the conceptual basis for Data science and Machine learning.