[Math] Definition of area

Definitionseuclidean-geometrymg.metric-geometryplane-geometryreference-request

I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts in a not boring way? Say in such a way that not all the students will sleep on the lecture?

Comments.

  • The real problem is to prove existence, the uniqueness is easy.

  • Using integral does not seem to be a good idea.

  • There is an approach where you write the formula for area and then proving its properties. I do not like it since it moves you to discrete geometry which is completely irrelevant and the ideas used nearly useless anywhere else (so no reason to learn this stuff). [See for example "Geometry: A Metric Approach with Models" by Millman and Parker.]

  • The method with measuring grid (cutting everything into small squares and counting) looks much better. One can consider this method as an introduction to integral. There is only one technical statement which has to be proved — if you rotate square then its area does not change. The only problem is that it is not generalizable to say absolute plane or sphere…

  • One may define the area as a limit of $\varepsilon^2\cdot N_\varepsilon$, where $N_\varepsilon$ denotes the maximal number of points in the figure one distance $>\varepsilon$ from each other. The only hard part is to prove existence of the limit $\varepsilon^2\cdot N_\varepsilon$ for say polygons. (You can exchange the limit to ultralimit — this way everything works smoothly, but I do not want to sell my soul just to get a def of area…)

Best Answer

There is an intuitive approach to area, based on the fact that polygons $P, P'$ have the same area if and only if they are equidecomposable (that is, one may be cut into pieces and reassembled to form the other).

The first three pages of this note sketch a "motivic" approach to the definition of area, for polygons. Namely, one defines $K(\text{Poly})$ to be the free Abelian group generated by plane polygons $P$, subject to the following two relations:

  1. $[P]=[P']$ if $P$ is congruent to $P'$
  2. $[P]=[P_1]+[P_2]$ if $P$ may be cut into polygons $P_1$ and $P_2$.

An easy exercise (sketched in the note I link to) shows that $[P]=[P']$ if and only if $P$ and $P'$ have the same area, so $K(Poly)\simeq \mathbb{R}$. But even better, one can define the area of a polygon $P$ as its class $[P]$ in $K(\text{Poly})$.

Indeed, for many reasonable classes of subsets of the plane, one may extend this definition to assign to such a set a class in $K(\text{Poly})$. For example, say a sequence of classes $[P_i]$ in $K(\text{Poly})$ converges to $[P]$ if there exists a representative $[A]-[B]$ for $[P-P_i]$ with both $A$ and $B$ contained in $[0, \epsilon_i]\times [0, \epsilon_i]$, with $\epsilon_i\to 0$.

Suppose $X$ is a subset of the plane so that there is a sequence of polygons $P_i$, such that the symmetric difference $(X\cup P_i)-(X\cap P_i)$ is contained in a polygon $Q_i$. Suppose further that $[Q_i]=[Q'_i]$ and $Q'_i\subset [0, \epsilon_i]\times [0,\epsilon_i]$, with $\epsilon_i\to 0$. Then we assign to $X$ the class $$\lim_{i\to \infty} [P_i]$$ if it exists. It's not hard to check (geometrically!) that this assignment is well-defined.

NB: This approach does not work to define volume in $\mathbb{R}^n, n>2$. Indeed, Dehn showed that there are many polyhedra with the same volume that are not equidecomposable.

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