Tom,

I'll leave the defence of the axial line to those that use it, as you
seem to be approaching the inevitable collision of single analytic
measure versus the whole of social dynamics, but I can start to answer
one of your questions:

> Alasdair writes that "there is nothing wrong with there being more
> than one solution, or indeed, calling several different solutions all
> "good" axial maps."     Again, can t we be a bit more definite about
> what types of axial map are possible? (leaving aside confused ones);
> permissible maps reflect the possible variations of mechanism.

If you apply VGA to a system it is possible to obtain *all* the sight
lines that exist in that system, subject to two provisos: the resolution
at which you conduct the VG analysis, and the level of detail that you
model the system (including whether you look at permeability /
visibility or some other combination of parameters).

This gives you the complete set of possible axial maps.  There are a lot
of them.  Some figures: in my model of the Tate at 0.75m there are
24,252 locations.  There are 1,631,652 connections, that is 1,631,652
possible axial lines.  (This includes duplicates).  I make the number of
possible maps in the region of 300,000,000,000.

I tried working out a program to reduce these senisibly, but it's kind
of difficult: some of these can be immediately discounted due to the
fact that they cover in their entirety the same space that another
longer line covers in its entirety.  There are further reductions until
at some point I had to turn to an optimisation algorithm to get a
'minimum depth' map out.

All this wore my computer out, and it still had the problem of
resolution: in order to cover alcoves I would end up with extra lines.
Questions such as "when is an alcove part of a wider 'convex' space and
when is it a true space in its own right?" arose.

Thus, I approached the conclusion I wrote to you: if we are going to get
a better definition, I don't think it's going to come out of better
mathematics, but better understanding of the social scientists who draw
the lines.  Since there are great claims for the axial lines, perhaps
there is some underlying social feature that they capture (where did all
that anti-structuralist argument go...), and if we believe that, let's
find out what these people are drawing.

Some postscripts:

> Kuhn refuted Popper?

Very drole.

> by the way Alasdair, when did air cease to be a fluid?

Oops... now I remember what was wrong with that argument (this doesn't
alter the fact that CFD is assumption laden).  Whether or not people do
act as fluids is an interesting question --- there's a paper by Helbing,
D., Farkas, I. & Vicsek, T. Simulating dynamical features of escape
panic.  Nature 407, 487-490 (2000) which considers exactly these sorts
of questions.

Alasdair