Thursday, 20 May 2010

Define Your Space

Traditionally, the recipe for jugged hare begins with the instruction; ‘First catch your
hare’. There is a lot to be said for a recipe like this. Not the least of which is that it
doesn’t miss the obvious. With this example in mind perhaps the first instruction for
Intense Seeing has to be, define your space of exploration. In order to make this instruction
widely useful we should not be limited to thinking about the three-dimensional,
Euclidean, space we are used to navigating around in everday life. In fact it is useful
to learn a few visualisation and conceptual tricks from physicists, who are very used
to manipulating spaces that are different from 3D space. In particular the concept of
a phase space repays consideration, this is an idea that was originally developed by
the brilliant American theoretical physicist Josiah Willard Gibbs (1839-1903), but it’s
general approach is very widely used.

In maths and physics a phase space is the space of all possible states of a physical
system, with each possible state of the system corresponding to one unique point in
the phase space. This mapping of what a physical system is to a single point in a
high-dimensional space is a flexible and powerful concept and by using the word ‘state’
physicists do not simply mean the spatial positions of all of the objects in the system
of interest, these would occupy a physical space or configuration space, but also their
velocities or momenta. These two sets of quantities allow a physicist to understand not
only the initial state of the system but also allow them to follow the evolution of the
system over time. In phase space a changing set of positions and momenta track out a
path over time, to produce a distinctive ’phase portrait’ of the dynamics of the system.
In any scientific or artistic exploration then it pays to have, or develop, a sense of
the shape and scale of the phase space of interest. In order to keep the following free of
mathematics, I will not explain phase space in the strict manner that physicists use the
expression, butt rather try and widen the concept to signify a high-dimensional space
that encompasses the entire range of exploration, taking into account ‘dimensions’ that
are not spatial; cultural, temporal, intellectual.
This is best illustrated with an example inspired by the lifelong work of Ed Ricketts
in the tide-pools of the Pacific coast.

The figure (a) shows a simplified map of the coastline of the Monterey peninsula
on the Pacific coast in California. Using this map we can define any point along
the coastline between Monterey (M) and Carmel-by-the-Sea (C) by giving the linear
distance along the coast line in kilometres. For example, take the point p which is 3.4
kilometres from Monterey. Zooming into this point we show a 200 metre stretch of the
shoreline facing Spanish Bay. The hatched regions shows the extent of the inter-tidal,
or littoral, region of this stretch of beach, which has an area of about ?? m2. Zooming
in again to the line a − b and looking at this line as a cross-section through the beach we see in (c) the topography of the shoreline and the mean sea level. During the
normal ebb and flow of the tides the sea level rises and falls around this mean level
and maps out on the shore an intertidal area that reflects both the range of high and
low tides and the local topography of the shoreline. Now in (d) we show a phase space
representation of the rise and fall of the sea level at the point p. The position of the
point is a one-dimensional distance and this is the x co-ordinate measured in kilometres
and the sea level height is the h co-ordinate measured in metres. This is a new way of
looking at the state of the sea level on the Monterey coast. Each and every point in this
new two-dimensional phase space represents one particular ’state’ of the interaction
between sea level and the coastline of the Monterey peninsula. We can also define
volume of interst within this space – for example, the grey bar shown is the volume of
the phase space represented by the beach at Spanish Bay, over the course of the full
cycle of the tides.
Using this phase space representation we can conceptually interogate the Monterey
peninsula coastline in a number of different ways by describing different ’volumes of
exploration’ within the phase space (they are strictly areas but we soon extend to 3
dimensions and higher in which volume is a better term to use).

Thursday, 6 May 2010

The Decision Tree

Tuesday, 4 May 2010

Street Fighting Mathematics

This has literally just been published by MIT press. 
 
Street-Fighting Mathematics

The Art of Educated Guessing and Opportunistic Problem Solving

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