Why are rectangular cartograms hard to generate?

Question

I was reading this paper on rectangular cartograms - http://ac.els-cdn.com/S0925772106000770/1-s2.0-S0925772106000770-main.pdf?_tid=1ecf274e-cb1e-11e3-b916-00000aacb361&acdnat=1398281777_8c46d847188d279ee8c5527d9804739e . In the abstract of the paper the author mentions that "rectangular cartograms are quite hard to generate because the area specifications for each rectangle may make it impossible to realize correct adjacencies between the regions and so hamper the intuitive understanding of the map". I am not able to figure out what is meant by this statement.Can someone explain precisely what this means.

Answer 1

If some maps are presented as a rectangular cartogram, it may be impossible for a region adjacent to another in the original map to also be adjacent in the rectangular cartogram.

Consider the Google Chrome logo. How would you change each colour to a rectangle of the same area, and still have every element adjacent to every other element? You could put them in a rectangular grid, like the Windows logo, but then you have some elements with only corners touching and not edge to edge adjacency. More complicated situations may not even allow touching corners.

EDIT: Question: Would this example work with a circular cartogram?

This was asked in a comment, but I'll answer it here to show the calculations involved.

The three outer colours would be circles of the same size. For each one to touch the other two they would be arranged such that their centres formed an equilateral triangle with a side length equal to the diameter of one circle. The blue circle would have to be in the centre of this triangle and the exact size to fill the space. Too small and it can only touch two other circles. Too large and it'll push the triangle apart.

Radius of an outer circle = $r_{o}$

Triangle side length = $L = 2r_{0}$

Distance from centre of triangle to vertex = $\frac{L}{\sqrt{3}} = \frac{2r_{0}}{\sqrt{3}}$

Radius of inner circle = $r_{i} = \frac{2r_{0}}{\sqrt{3}} - r_{0} = \frac{2-\sqrt{3}}{\sqrt{3}}r_{0} \approx 0.155r_{0}$

For area,

$$A_{i} = \pi r_{i}^2 = \pi (\frac{2-\sqrt{3}}{\sqrt{3}}r_{0})^2 = \pi \frac{4-4\sqrt{3}+3}{3}r_{o}^2$$

Getting it as a relation,

$$A_{o}/A_{i} = \frac{\pi r_{o}^2}{\pi \frac{4-4\sqrt{3}+3}{3}r_{o}^2} = \frac{3}{4-4\sqrt{3}+3} \approx 41.785$$

The area of one of the outer 3 sections would have to be nearly 42 times the size of the inner blue circle. Visual inspection of the Chrome logo reveals this to not be the case.

Answer 2

It is not possible to represent Luxemburg, Belgium, France, and Germany by rectangles such that each is adjacent to all others.