# What comes after length, area and volume? [duplicate]

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?

• 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. – Blue Mar 15 '16 at 9:17
• @Blue I thought there had to be similar question asked about this, but it seem like a more general question than mine. – Frank Vel Mar 15 '16 at 9:26
• 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. – dotancohen Mar 15 '16 at 12:47
• 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. – 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.

• 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? – Frank Vel Mar 15 '16 at 9:20
• @FrankVel benjaminblonder.org/papers/2014_GEB.pdf Does this help? – S.C.B. Mar 15 '16 at 10:01
• 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. – MichaelS Mar 15 '16 at 12:49
• Four-dimensional hypervolume is sometimes referred to as "bulk" – SuperJedi224 Mar 15 '16 at 14:33
• Can we just use volume if we don't mind about the dimentions? – 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.

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