Currently maps refer to physical geography and topography and city points are placed by latitude and longitude. However the time it takes to travel between two cities is less and less closely tied to the physical locations of those cities and more to the available infrastructure used to travel.

The
nervous system homuncuous depicts a human body in a manner proportionate to nerve endings at a given site. The travel time homuncuous map positions cities closer or farther according to time of travel between them. Imagine a collection of points in 2D space, each named after a city. Their proximity would vary by rail, road, air and ship connections.

A very cool function would be if city travel time maps were done in real time. City sizes would pulse with rush hour, shrinking to tiny in the early morning hours and expanding massively as cuommuters clog the roads.

Travel time tube maphttp://www.tom-card...ravel_times/applet/ this map distorts to show the travel time from whichever station you click on. [stupop, Feb 25 2010]

Yes, it's good. It would mean there are times when parts of the Home Counties are further from Hounslow than some places on the continent of Europe. However, have you considered the question of distances relative to different points? It could be easier to get from Hounslow to Southall than Central London to Hounslow or Southall, but also harder to get from Hounslow to Slough than Central London to Slough (don't know if that's true, it's just an example). Would they be relative to a particular point?

well I like it. maybe I don't understand it exactly right but my Dad and I always have this discussion as to *how far somewhere is* and then someone answers in *time*. I don't know how you could really make the map as the variables are always different, but certainly there are generalities. So here's a real time bun.+

Relative to a particular point - no. Your 3 point example shows why it would take some computing power to position these points, and more as the number of points grows. A pilot project might be for major cities only. I am trying to picture what such a map would look like. I am not entirely sure this would be doable in 2D.

I have the solution, unfortunately it's not as cool as the amazing 3D maps you're probably visualizing. It's simple: provide scales on the map for each set of variables.

One mile:
|--------|

1m by plane:
|---|
1m by bi-plane
|-----|
1m by car:
|------------|
1m by car @ rush hour:
|------------------------|
1m by bike
|------------------------|
1m by foot:
|----//----|
1m by one foot:
|--------//--------|

It would simply be a matter of color coding the scales. Red arteries would correspond to a certain 1m scale by car and blue arteries would corrrespond to another 1m scale by car. Guh, I think I messed up my examples above. A faster rate of transportation would make the scale larger. So a mile by plane would be a longer scale and by foot a shorter scale. The way I have written it would be wrong.

I'm not sure you could do travel time because it's not reciprocal. That is, traveling from point A to B could take more time than traveling from B to A. (One way streets, non-symmetrical highway exits, whatever).

I'm not certain it could be done at all, even with advanced computing power. What could easily be done is similar items for specific routings. That is, get google directions, and it distorts the map to show the travel times rather than the distance involved.

I have a feeling it would end up having rather a lot of dimensions, or could maybe shift in size and shape according to what was at the centre. That is, however, doable.

//I am not entirely sure this would be doable in 2D.//

It isn't. To do it perfectly, and allow for all possibilities, it
needs N-1 dimensions, where N is the number of cities.

The problem is formally equivalent to a distance-geometry
algorithm used in solving molecular structures from NMR
data, and also used in constructing genome maps from a
collection of point-to-point distances.

In each of these cases, and in the city-time problem, the
distances A-B plus B-C need not equal the distance A-C.
So, if you have three cities, you need a triangle (2-D) to
represent all the distances. For four cities, a sort of
wonky tetrahedron (3-D), and so on.

How do you represent this in a useful way? One way would
be to show only three cities at a time on a 2-D map.

In the NMR and genome mapping applications, the answer
you want is either a 3D structure or a 1D (linear) map
(which you would get in the first place, if the data were
error-free). In these cases, the final solution is arrived at
by "squashing" the multidimensional map across its
thinnest dimension until that dimension vanishes, then
repeating with the next thinnest dimension, and so on
until you get down to either three or one dimensions as
required.

In this case, the multidimensionality is real instead of an
artefact of errors, so any loss of dimensionality will be an
untrue representation. But you could get a sort of "best
approximation" 2D map.

It's not even as simple as cities though is it, [MB]? Every route between any pair of locations in the city has two times associated with it. This reminds me of Julia sets being spun off points from the Mandelbrot Set, so maybe that's the answer - each point on the map has its own map associated with it for each time of day, making the map three-dimensional initially, then each route is associated with a further map. So you could draw a route and spin a map off from it maybe?

/could maybe shift in size and shape according to what was at the centre/

One could have a spyglass that you could slide across the map, with the remainder of the map shifting in scale according to spyglass position. There might have to be a limit to haow big the map could get, as it is a long way to anywhere if you are swimming in the middle of the Pacific.

I think it ends up at N*M dimensions, where N is the number of cities. (This term absorbs the length dimension required to actually represent the time, hence the lack of -1.) M is the maximum number of transportation options available. It might even end up as N^M if you consider multimodal options, but I'm not certain. Some of the dimensions drop out in regions where a transportation option is not available. Others end up masked when possible routes overlap, but they pretty much all need to be calculated.

//I think it ends up at N*M dimensions, where N is the
number of cities.//

Actually, I don't think that would work. However many
dimensions you use, the distance between a particular two
vertices in the net (in this case, the time between two
specified cities) can only have one value; the multiple
dimensions are there to accommodate the non-additive
distances between vertices. Or maybe I mean nodes.

No, that's the whole point. Regardless of the geometry, the
linear distance between two given points has a unique value.
Non-Euclidean geometry might mean that my nose is two
feet from my navel, and that my navel is a mile from my
toes; and that my nose is three miles from my toes; but the
distance from my nose to my navel has only one value.

Shirley though there are geometries where the distance from your nose to your navel is different to the distance from your navel to your nose? Is it possible for a geometry to be anisotropic, or dynamic?

That would be a non-commutative geometry, and I'm not an
expert on that.*

But in any case, the problem posed here is that the
"distance" (time) from A to B can have any one of several
different values (as can, presumably, the time from B to A).

*This statement is intended to imply, subliminally, that I am
an expert on everything else here.

This is an interesting idea, if a little complicated to solve.
Perhaps if the initial position was specified, then the map generated from there; it would require some fancy computer processing, and a massive database behind it all (possibly continuously updating).
Alternative concept: display the cost of getting from A to B, as that is often the most important factor (eg. it's cheaper for me to get (from Christchurch, NZ) to Melbourne than Auckland, even though Auckland is closer in both distance and travel time).