This question comes out of a simple wish for graphical displays of people (say members of the Wikimedia chapter for whom I now work) that will display demographic clustering. Obviously in typical developed countries if you place one dot per person you just see blobs for major cities. The usual way is to pull out a map of London (say) on a larger scale. But say that I just want one map.
The following would be a crude thing, but then the idea is just to be able to tell something at a glance. Most countries have various local regions on a fairly small scale. The UK has postcodes, and the initial segment of three to four characters defines regions small enough for most national mapping purposes. The question is motivated by the need to take such postcodes (e.g. NW1) as basic regions. We are going to "round" people to the centroid of their postcode region. Say the ultimate display will be discs at those points of variable colour, or radius.
So now we get the geometry. The centroids of the postcode regions are just some points, and we can define Voronoi polygons for them. (The Earth can be flat: having spherical geometry would be fine, too). What I want is a transformation that forces the Voronoi polygons (large and small) to become tesselating regular hexagons (all of the same area). Such maps seem often to be used, e.g. for electoral maps.
I don't even know the correct language here. We are going to be asking for a non-linear, discontinuous (in general) "map projection", not at all unique and not preserving adjacency. But what I want is something like an existence theorem: is there is a way to do this, that can be computed (once and for all) so that big cities with their small-area polygons are expanded in a fairly sensible way?
If you think about rolling out lumpy dough into a pancake of uniform height, you can perhaps see what I mean. How would it roll out, therefore, under a purely vertical pressing motion? That picture has to be reconciled with the "honeycomb" answer.
Any illumination or references would be very acceptable.
Edit: To clarify: My real question is how to best to show demographic clustering by rescaling of the underlying map, neither creating artefacts, nor hiding genuine clusters. I don't have a good formulation but maybe for smoothed out versions it is like a Radon-Nikodym derivative? But I'm starting with postal data.