Why is the area of a square $a^2$ and volume of a cube $a^3$?

Question

How area and volume emerge from lower dimensions

This might sound as a silly question but I think there is some hidden complexity behind it. Esentially, my question is how is it possible that:

1. multiplication of something with essentially no area grants you an object with an area.

2. multiplication of something with essentially no volume grants you an object with a volume.

The thing is, even if I acknowledge that line segment could have some nonzero but really small area (for example a wooden toothpick), why its according-times multiplication would give me an area of a square? If I put these toothpicks together, they still won't fill the appropriate square (see illustration):

I know, that if I discretize it and assign some fundamental "area value" it would make sense, for example having 4 cabbages in a row and 4 rows in a total gives me 16 cabbages... BUT how to make a sense of it for the case with toothpicks etc.?

Thank you very much!

Answer 1

This is an intuitive answer.

The area of a square of side length $a$ is $a\cdot a$ because the square area may be subdivided into a number of exactly $a\cdot a$ smaller squares that are neither overlapping nor gapped and each small square has a sidelength of 1 and an undetermined area, which we choose to call “unit of area”.

Undetermined as it is, all smaller squares have the same area because they are congruent.

The division of a segment of length a into a number of “a” segments of length $\frac{1}{a}$ is not exactly possible in the real world, however in mathematical terms is possible to be done exactly.

The fact that we call the undetermined area as “unit of area” might seem unrigurous but is not. We simply count in terms of that small area which we regard as a reference.

Same logic applies to volumes.

Answer 2

You can start by simply defining length on open intervals $(a,b)$ to be $b-a$. Then, in general for cubes $(a_i,b_i)^n\subseteq \mathbb R^n$ define volume to be $\prod (b_i-a_i).$ The Lebesgue theory will then capture the intuition we have for these ideas, with the added benefit that it will also treat non-intuitive situations that arise. Another approach, for example here, uses linear functionals to develop the theory, which is perhaps more direct; and it leads to the same results.

Answer 3

If we consider infinitesimal strips of width $x$ from one side $ dA= x \cdot dx $, so to start with area did exist in its differential form. Area did not arise from nowhere.

Integrating,

$$ A = x^2 $$

and similarly for the volume.

Answer 4

...multiplication of something with essentially no area grants you an object with an area...multiplication of something with essentially no volume grants you an object with a volume...

Maybe you're having trouble because your intuitive understanding of "dimension" so naturally connects to common physical length/area/volume examples that it is hard to realize that there is actually a conceptual abstraction happening.

---

To show how completely abstract "dimensions" really are I'm going to give an example that is entirely unrelated to your length/area/volume examples... the working-world concept of a man-hour.

If you think about it, a man-hour (or rather a worker-hour) is a 2-dimensional concept that combines the two very distinct 1-dimensional concepts of "worker(s)" and "time" into a single new concept. Can 2-workers fold 100 paper airplanes in an hour? Well then 6-workers could do the same job in 20 minutes, or 1-worker alone could do the job in two hours.

It's a pretty convenient concept, but it's also nice here in that we can understand how it is clearly an abstraction. We can intuit that workers and time are completely different categories that each contribute to the concept; and we can also understand that a question like "How many 'workers' are in an 'hour'?" shows a muddied understanding of how those two categories are distinct.

---

Getting back to the length/area/volume examples that you were originally asking about we could maybe decide to think of them in that same kind of abstract way as "worker-hours": is it an "area" or just a "width-length", a "volume" or a "width-length-height"?

There is something strange about how width, length, and height can all be measured with the exact same kinds of basic "distance" units, and it is also a bit amazing that we can just "rotate" things and say "I'm calling this the direction of length and that the direction of width now." and have it work out. But none of that changes the fact that the abstract concept of a "width-length-height" is something distinct from the concepts of a "height" or "width-length".

And lastly, the reason why it goes $a$, $a^2$, $a^3$ is simply because we have special words for shapes when the sides are the same. Any old rectangle could be $a \cdot b$ but if we say that $b=a$ we have $a \cdot b \to a^2$ and call it a square; same thing for 3D box shapes $a \cdot b \cdot c$ if all sides are the same $a=b=c$ just gives $a \cdot b \cdot c \to a^3$ and we call it a cube.

Answer 5

In Euclid’s geometry you get a square only by constructing it on a given straight line segment (Elements I, 46), not by multiplying a line. Numbers are not involved at all. By the time you get to Descartes’ geometry, which makes use of algebra, “squaring” a line yields not a square (area) but another line, a third proportional, with an arbitrary line playing the role of “unit”. Similarly, “multiplication” of a line by a line yields a fourth proportional, i.e. another line, not a rectangle. For Descartes as for Euclid, only magnitudes of the same can have a ratio to one another.

So it seems a square is not produced by numerically “squaring” a line, nor a rectangle by multiplying two lines, nor a cube by numerically “cubing” a line. Squares and rectangles and cubes arise only by geometric construction, traditionally with straight-edge and compass. But if we are given a numerical measure of the line(s),we can calculate the measure of the square, rectangle, or cube by taking the “square” or product or “cube” of the number(s). E.g. if line $a$ is two unit-lines in length, then the square built on that line is $2^2=4$ unit-squares in area, and the cube is $2^3=8$ unit-cubes in volume.