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The length of the unit is 1.
The area of the unit square is 1.
The volume of the unit cube is 1.
The $\color{red}{???}$ of the unit tesseract is 1.
The $\color{blue}{???}$ of the unit 5-cube is 1.

So this is a fill in the blanks-question, but I've been wondering if there's a common term for this? I'd imagine it'd be "n-volume", but I cannot seem to find any uses of that term, so I'm guessing there is another term?


marked as duplicate by colormegone, John B, Alekos Robotis, Daniel W. Farlow, user228113 Mar 16 '16 at 0:58

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    $\begingroup$ Related (duplicate?): "What is the general term for concepts like length, area and volume?". I recall looking through old-school geometry texts where "content" was the preferred general term (with "$n$-content" the specific $n$-dimensional term), so it's what I tend to use. $\endgroup$ – Blue Mar 15 '16 at 9:17
  • $\begingroup$ @Blue I thought there had to be similar question asked about this, but it seem like a more general question than mine. $\endgroup$ – Frank Vel Mar 15 '16 at 9:26
  • $\begingroup$ I've heard the term תפיסה used to describe the space time occupied by an object: volume * timeInExistence. That might translate to "holding" or "capture" in English, but note that one of the dimensions was the time dimension, not another spatial dimension. $\endgroup$ – dotancohen Mar 15 '16 at 12:47
  • $\begingroup$ A common theme is called measure theory. In this language length of unit is the induced Lebesgue measure from the real line and so on. $\endgroup$ – DBS Mar 15 '16 at 14:09

The term "hypervolume" is in use, but that doesn't tell you the dimension. I don't see any trouble using "$4$-volume" to describe the amount of space the unit tesseract occupies.

  • $\begingroup$ I couldn't find any information on wolframmathworld nor wikipedia, and wiktionary claims it's simply the volume of an n-dimensional object. Is there anywhere I could read about this? $\endgroup$ – Frank Vel Mar 15 '16 at 9:20
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    $\begingroup$ @FrankVel benjaminblonder.org/papers/2014_GEB.pdf Does this help? $\endgroup$ – S.C.B. Mar 15 '16 at 10:01
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    $\begingroup$ A bit more verbose, but "nD hyper-unit" works too. So "4D hyper-volume", "5D hyper-cube", etc. This is generally extensible to any number of dimensions or types of thing you're extrapolating to so many dimensions. $\endgroup$ – MichaelS Mar 15 '16 at 12:49
  • $\begingroup$ Four-dimensional hypervolume is sometimes referred to as "bulk" $\endgroup$ – SuperJedi224 Mar 15 '16 at 14:33
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    $\begingroup$ Can we just use volume if we don't mind about the dimentions? $\endgroup$ – Ooker Mar 15 '16 at 18:17

The term Tensor is appropriate. A square is a tensor of rank 2 , a cube is a tensor of rank 3 , and so one. Specifically a tensor of rank 4 and 5 are the answer.

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    $\begingroup$ Isn't a tensor of rank 2 is a vector? $\endgroup$ – Ooker Mar 15 '16 at 18:19
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    $\begingroup$ This isn't true and it doesn't answer the question. $\endgroup$ – Deusovi Mar 15 '16 at 18:24
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    $\begingroup$ @Ooker a tensor of rank one is a vector, one of rank 2 would be a matrix. $\endgroup$ – tox123 Mar 15 '16 at 20:18

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