Do a Circle and a Square Measure Up?

Lately I have been thinking the following. How are a circle and a square related? It all started with a simple question: Do the areas representing integration by parts add up? Here I explain this question, and write how that question leads to the one heading this note.

If you draw v(u) on a v-u graph, the area from u(1), v(1) to u(2), v(2) using straight lines perpendicular to the axes can be calculated in two ways, one multiplying base times height of the square that ends in the second point, and subtracting the area of the square determined by the axes and the first point, i.e., u(2)v(2)-u(1)v(1), or drawing rectangles with base Δu (Δu is the difference between two u values)and height v, adding these up from point 1 to point 2; then doing the same with u and v exchanged. Graphically it is clear that they add up to the same area.

Algebraically one has Leibniz´s rule and the fundamental theorem of calculus, to prove the same result.

Now we come to the circle and the square.

Using the construction described above, one ends up with a very crooked line. The lines of the rectangles for constant u and v values are the steps and the heights of the steps of a ladder.

There is no problem for the area calculation, but if one choses to calculate the length of the crooked line (you can see the crooked line in Is it a Fractal?) one gets in trouble. The line gets close to a line that joins the bents in the crooked line into a line of different length than the crooked line.

The squarecircle above is an example, the length of the square is different than the length of the line defining the circle. One is 2πr, and the other 4 √2 r.

The circle radius is r.

Obviously these two numbers are different, and there lies the problem.