I can state without much explanation that the London Underground map is a remarkable invention. But divorcing the map from the geography of the overground leads to misconceptions (like thinking it's a good idea to go two stops and change once to get to a station you could have walked to in two minutes).
So, my flatmates and I were talking on the way into work (God, we're so young professional) about how to encode additional information in the map. Is it possible to indicate by, say, the colour intensity of the station the (overground) proximity to other stations? Hm. Well.
Consider a true map of London. Now consider crumpling this map so that it's all scrunched up, but a top down view is the same as the tube map, but on a different scale. Leaving aside whether this transform is possible, this yields what we're after. As long as the crumpled true map only bends on a station (ie no peaks or trough on a line between nearest stations), then we could say 0% colour intensity was at the lowest point of the crumpled map and 100% was at the top, and show this on the tube map. Task achieved.
["But how would you read it?" I hear you cry. The greater the colour intensity difference the further apart the stations. You also take the distance on the tube map into account: the closer the colour intensities the closer the true distance is to that of the tube map. I've a feeling it would actually be quite intuitive.]
But is this possible? I can think about it like this: Imagine the tube map as a mesh of triangles (each vertex being a station), and the same for the true map (essentially we're joining each station to its six nearest). As long as each station does not cross a line between two other closest stations then we can transform this mesh into whatever we like. So it is.
Maybe. I'm not sure about the last step -- but shifting questions onto other questions. I like this.