 # Measures

Volume, for example, is measured in three-dimensional spaces. Area is measured in two-dimensional spaces. What are some measures used in higher dimensions?

Where are you getting these higher dimensional spaces from?

Mathematicians do play with higher dimensional spaces as theoretical constructs, but they just use volume along with the appropriate exponent. Physicists do the same for studying the extra dimensions of string theory.

But since no actual measurements of any higher dimensional spaces are possible, no common words are necessary to distinguish them.

I’ve seen the word ‘hypervolume’ used when talking about higher-dimensional spaces.

IIRC, flux is the measure of a volume passing through a known area.

There’s actually a whole branch of mathematics known as “measure theory”. Here’s a quick intro.

Basically, you first take a set and define what subsets are “measurable”. The whole space and the empty subset should be measurable. The union and intersection of countable families (finite or only as infinite as the integers are) of measurable sets should be measurable, as well as the complement of a measurable set.

Now, for each set you define its measure, which is a positive number. The measure of the union of a family of disjoint measurable sets should be the sum of the measures of the individual sets.

For R[sup]n[/sup] (“normal” n-dimensional space), there is a well-defined measure which starts by assigning the measure of a rectangular prism to be the product of the lengths of its edges. The cube of points with coordinates all between 0 and 1 should have measure 1, for instance. Then you chop up measurable spaces into rectangular prisms, calculate their measures, and add them up.

That’s the same thing as flow, i.e. gallons per minute, cubit feet per second, etc.

Well, flux is measured with a capacitor, so why not in farads?