${x^4}$ as "tesseracting" a number $x$

Question

So, this strange thought popped up into my head. You know how we call ${x^2}$ squaring due to the fact that what you're essentially doing is finding the area of a square with side length $x$? The same goes for cubing. Saying ${x^3}$ is really going to give you the volume of a cube with side length $x$. Now, what if I tried to coin a term that would take this -ing pattern to another level with ${x^4}$? This would technically give me the 4D volume, per se, of a tesseract(a 4D cube). So, couldn't this really be thought of as "tesseracting" a number?

In fact, I may have a deduction/thought. Saying ${x^n}$ could just be thought of as n-cubing a number. A square can be thought of as a cube in 2D. As in, a cube with only length and height, no depth. So, saying ${x^2}$ can be seen as 2-cubing, or squaring a number. The same goes for ${x^3}$. You are 3-cubing, or just cubing, a number. So it seems this n-cubing pattern holds. So, why not extend it to the tesseract? Why isn't ${x^4}$ just thought of as 4-cubing or "tesseracting"? The pattern I thought of would still hold.

Also, do you mind going easy with the criticisms? I'm not trying to sound like a you-know-what, but just keep in mind I'm extremely amateur and only in 11th grade. And since it seems like this is original to my thoughts, I'm a bit overexcited about this thought.

Answer 1

Mostly this naming convention breaks down for two reasons:

1) We don't have easy to remember fancy names for every dimension of products of the unit interval.

2) Saying "$n$-cubing" is potentially ambiguous and sounds very awkward when $n=3$. The typical "to the $n$" rolls off of the tongue and leaves no room for doubt.

Answer 2

I mean, it's legitimate, but people find thinking in higher dimensions - some even in just three - really difficult. So in a sense I believe it has a nice geometric intuition to it, but people just don't like thinking in these higher dimensions unless necessary. It might also be a remnant from a time when thinking about higher dimensions just was brushed off as being nonsensical or irrelevant or unimportant.

It also just doesn't roll that nicely off the tongue, personally. I'd sooner just say "to the fourth power" than "tessaracted" or "4-cubed". Of course, that's anecdotal and moreso a matter of personal taste.

Answer 3

I think this is a really good way to bring geometric intuition into exponentiation. I'm not sure "tesseract" is universally the term for the "4-cube," but "4-cubing" seems great.