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I was looking at the answers to Why square units? and I felt that the discussion was bogged down by all the talk about circles instead of showing what's special about the unit square area, so I'd like to ask a follow-up question:

Suppose we took the area of a unit circular sector with $\theta=1 \ \text{radian}$ and used that as the unit of area instead. Let's call this unit of area "slice". What would the disadvantages be compared to using the unit square?

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If you think tiling is an issue, take note that a $\text{slice}$ is equal to half a unit square, so you could just do the usual square tiling but with each square cut in half in whatever way you want. Remember: an area is an area regardless of shape. You can visualize a $\text{slice}$ as a rectangle, or as an isosceles right triangle, or whatever. I chose the radian sector definition above simply because the radian is a fundamental aspect of two-dimensional geometry, arguably more than any specific shape.

With the $\text{slice}$ you could still use any existing area formula that uses unit squares, provided you multiply the area by two:

$\text{area of a triangle in slices}=\text{base}*\text{height}$

$\text{area of a circle in slices}=\tau r^2$

$\text{area of a rectangle in slices}=2*\text{length}*\text{width}$

We could extend this idea to higher dimensions. For an alternative unit of volume, we could use the volume of a unit sphere sector whose cap has a surface area of $1 \ \text{slice}$. We could call this a $\text{core}$. (Side note: The solid angle of that spherical sector can replace the $\text{steradian}$.) The unit sphere's volume would be $4\tau \ \text{cores}$. The unit cube's volume would be $6 \ \text{cores}$. A pyramid's volume would be equal to the area of its base (in $\text{slices}$) times its height.

Further questions: For a unit of $n$-volume based on an $n$-ball sector analogous to those described above, would the unit $n$-cube generally have an $n$-volume of $n!$ in that unit? And would an $n$-pyramid have an $n$-volume equal to the $(n-1)$-volume of its "base" times its height in the same unit?

Edit: If I'm right (I'm not a mathematician so it's not easy for me to verify), then as I said in the comments below, using radian-based $n$-volume units will allow the simplest space-filling polytopes (i.e. the family of simplexes that are generalizations of Hill tetrahedra and with unit base and unit height) to have an $n$-volume of $1$, which is pretty neat. It's definitely nice to get rid of the ugly $\frac{1}{n!}$ factor in simplex n-volume formulas.

Francis Ocoma
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    You ask what the disadvantages would be. What (on earth) would the advantages be? Also, your discussion of tiling is confusing, since tiling is definitely a property of a specific shape rather than a quantity of area. The fact that squares tile the plane is used constantly when developing the general concept of area out of the area of a square. – Greg Martin Jul 05 '22 at 17:09
  • @GregMartin, well, there is something that makes a bit more sense when using the "slice": The simplest shape with n-volume=1 is no longer the unit n-cube, but rather it would be the family of simplexes whose base and height is one. – Francis Ocoma Jul 05 '22 at 22:57
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    By what measure does that make more sense? – Greg Martin Jul 05 '22 at 22:59
  • Why would it make more sense? Well, a simplex is the simplest of shapes in any dimension, so doesn't it deserve that property? – Francis Ocoma Jul 05 '22 at 23:03
  • I think it's cool that the simplest space-filling n-polytopes get to have a unit n-volume in this system. – Francis Ocoma Jul 05 '22 at 23:13
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    If you use Hill tetrahedra (rather than slices) to define areas and volumes, I think you can reproduce all the results the we get from squares and cubes in a very similar way. The problematic part is when you make the area of a slice your starting point rather than something to be found. Note that you still haven't really answered the student's question in either case, because the area of a circle will still be irrational. – David K Jul 06 '22 at 13:36
  • @DavidK I wasn't trying to answer why a circle's area is irrational. I was trying to answer "Why square units?" with "Well, let's look for a good alternative, shall we?" :) – Francis Ocoma Jul 06 '22 at 13:53

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I personally feel there wouldn't be any difference from the theoretical point of view because of what you said about 'area being just area' but that needs further justification. The reason is that there is a result in measure theory(read Theorem 2.20(d) from Rudin's Real and Complex Analysis) that roughly states that any 'reasonable'(again for details read the text, but roughly properties like area staying same under translation, area of two disjoint sets being the sum of each of them and so on..) definition of length/area/volume is just the 'usual'(Lebesgue measure) times a constant. The Lebesgue Measure just sets the area of the unit square to 1. So if you take your unit of measurement as the slice, without the border for technical reasons, you could cover the unit square with countably many of them(pick the slices with rational centers), and get some finite value for the area. Then other areas are just the same constant times unit square, so nothing has 'essentially' changed except a scaling factor. Why not just set it to 1?

Mr.Flaze
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  • It's very unlikely that one can cover the unit square in the manner described. – Greg Martin Jul 05 '22 at 23:00
  • It is a theorem that you can write any open set in $\mathbb{R^2}$ as a countable union of rectangles; just pick slices that are bit bigger than than these rectangles and you could cover it pretty much the same way; getting an 'outer measure' if you take infimum over all such possible coverings. That would be same as its 'area' – Mr.Flaze Jul 06 '22 at 03:49
  • To emphasize, my definition of a "slice" is the area of the sector of a unit circle with central angle of 1 radian. A unit square's area will be exactly 2 slices. I'm not sure why you talk vaguely of "covering the unit square with countably many of them". There's exactly two of them in a unit square. Or am I misunderstanding something? – Francis Ocoma Jul 06 '22 at 12:40
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    @FrancisOcoma How do you know there are exactly two slices in a square? In your diagram you equate the two shaded areas without proof. Of course we know that they are equal when the definition of area is based on the unit square, but you said you don't want to use the unit square to define area. – David K Jul 06 '22 at 13:13
  • @DavidK I suppose I could instead define a slice as the area of any triangle whose base and height are equal to 1 to make things easier for a layman like me. It's trivial to see that a unit square is made of two right triangles both of which are 1 slice. You can also use triangles to prove that the unit circle sector of 1 radian is 1 slice. So, are you saying I got things backwards? – Francis Ocoma Jul 06 '22 at 13:43