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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.

Xavier Stanton
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3 Answers3

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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.

RandomMathGuy
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  • I see, but then why even give special names to the 2nd and 3rd power? Also, it was just a pattern I was thinking about, sort of connecting exponentiation to dimensional geometry. – Xavier Stanton Dec 07 '18 at 03:21
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    @XavierStanton Historical reasons mostly. The Greeks did a lot of geometry in two and three dimensions, so we usually have special names there. Also, life has three dimensions, and paper has two. As to why the tesseract has a name, I am unsure, but it is probably something very esoteric and not at all meaningful to the math community. – RandomMathGuy Dec 07 '18 at 03:23
  • I personally prefer the term "hypercube" to tesseract myself but that's a matter of personal taste. – PrincessEev Dec 07 '18 at 03:25
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    Oh, and quickly Googling, the origin of the term per Wikipedia - "According to the Oxford English Dictionary, the word tesseract was coined and first used in 1888 by Charles Howard Hinton in his book A New Era of Thought, from the Greek τέσσερεις ακτίνες (téssereis aktines, "four rays"), referring to the four lines from each vertex to other vertices.[4] In this publication, as well as some of Hinton's later work, the word was occasionally spelled "tessaract"." – PrincessEev Dec 07 '18 at 03:25
  • @EeveeTrainer For sure, but only because of Cube 2: Hypercube – RandomMathGuy Dec 07 '18 at 03:26
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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.

PrincessEev
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  • I said the same thing to the other comment, why have the terms squaring and cubing then? – Xavier Stanton Dec 07 '18 at 03:22
  • Because it's easier to think in the lower dimensions, resulting in much easier to understand geometric analogues. "Square" and "cube" are also relatively easy to say, as well as very familiar terms, whereas most people aren't familiar with a tessaract (unless you want to talk about the Marvel Comic Universe), just as one example - and at higher dimensions even that reference goes away. – PrincessEev Dec 07 '18 at 03:24
  • LOL at the Marvel reference – Xavier Stanton Dec 07 '18 at 03:25
  • Also, I will admit, after the tesseract, there's no special names for the 5-hypercube, 6-hypercube, etc. – Xavier Stanton Dec 07 '18 at 03:27
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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.

BallBoy
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