Requirements for an Area Function

  Now, how do we define the area of a polygonal region in the hyperbolic plane? Should we define the area in the same way that we do in the Euclidean plane? If so, what are the minimum requirements for an area function of the Euclidean plane? Minimally it should satisfy the following: Let be the set of all polygonal regions in . An area function should be a function

such that

1. for every R;
2. if and intersect only in edges and vertices, then
   
   
3. if and are triangular regions with the same base and altitude, then .

If there is such a function , then by (2) and (3) it will satisfy

4.

if , then , because congruent triangles have the same bases and altitudes.

In Euclidean geometry we can show that this area function is unique and it must satisfy the formula for each triangular region, T, where b is the length of the base and h is the length of the altitude.

Theorem 17.8: There is no such function satisfying (1), (2), (3), and, hence, (4).

Proof: Consider the right angle , with . For each n, let be the point of such that . This gives a sequence of triangles

and a corresponding sequence of triangular regions

By condition (3) all the regions have the same ``area'' .

Now consider the defects of these triangles and let . For each n

Since the partial sums are bounded, we have that the infinite series,

is convergent. Therefore,

Hence for some n.

By Theorem 7 there is a point, B, between A and such that

Therefore by Bolyai's Theorem the regions T and determined by these triangles are equivalent by finite decomposition. By condition (4) this means that . But

Therefore,

Because and

this must be impossible.