EXPLANATION
The invention of the quaternion is a curious episode in the history of pedestrian culture. On 16 October 1843 mathematician William Rowan Hamilton made a Sunday afternoon stroll through Dublin with his wife, when all of sudden: " An electric circuit seemed to close; and a spark flashed forth". This spark was a revolutionary formula that moved mathematics into four dimension & that became known as the quaternion: 'a new System of Imaginaries'. Hamilton in utter joy could not resist the temptation to scratch the formula into the limestone of the Brougham bridge which for these reason became a place for nerd-pilgrimage.
To a psychogeographer the particularities of the birth of the quaternion is not a freak incident, it only proves the fundamental concept that walking is a necessity for creativity. That this peripatetic brain-chemistry applies even more so for understanding non-Euclidian geometry is a fact that should appeal to everybody doing work in the mathematics of urban space & place, a field that still has to make some major breakthroughs.
QUE PASA?
A dimension is
a geometric way of referring to a quantity that can change (a variable). The
number of dimensions of an object is the number of directions it points in.
From Hamilton's own words we laymen can deduce a little of what is means for
an object to point in multiple dimensions. A quaternion contains a 'real'
part which is the set of three variables that contain the position of an object
in the dimensions we take for granted; usually expressed as: x, y, z. The
4th dimension is the 'imaginary' part, which is 
more
understandable when you call it a vector. Hamilton described this dimension
as 'forward' & 'backwards'. Imagine in your minds eye a cube placed in the
abstraction of a desolate 3D mathematical space. Suppose you want to transport
this cube in it's entirely +5 in every dimension. In 3D you can do this by
adding the required amount for every dimension in turn, an operation that
can easily result in what game-developers call Gimbals-lock, an error in matrix
computation. By using quaternions it becomes possible to move the cube along
the fourth dimensional axis: this causes its movement in one part through
the first three dimensions. Remember that just like a square is waferthin
in our third dimension so is our cube in the fourth dimension. Once you get
the idea you actually haven't gained one new dimension, but an infinite amount
of them: all possible directions the cube can move in. The novelty of the
quaternion was that it enabled mathematicians for the first time to calculate
this dimension from within the system itself. You could perhaps also say that
the fourth dimension contains metadata about the dimensions 'below' it. That
psychogeographers are interested in dimensions that can only been understand
as movement is no small wonder.
Generative psychogeography understood within it's own system is a two dimensional system. Rules like "second left, first right, third right, repeat" when executed on a mathematical grid always loop themselves invariable. But when applied in a city this hardly ever happens, so seen form inside the algorithm something goes wrong outside of it because the system itself is flawlessly executing it's rules. This dimension that distorts the loop is the city. In it's original function as tool for urban explorations, this was precisely the point of using these algorithms, but to a .walk application that doesn't use swarm tactics (random encounters between a large amount of participants) this randomness in the urban system is seriously threatening .walk functionality because it needs interaction at predictable intervals. This urban interference in a straight forward .walk application might at best result in a severe decrease in processing speed, but it might also yield to situations in which 2+2 does not always equals 4. To prevent the trajectories of the .walk participants to become unpredictable we propose to employ quaternion-tactics to guide psychogeographers through the randomness of the city grid & bring them back to where they began. The space (the number of turns) between the actual position & the position where the walk started could be deployed as buffer-space for pending operations, thus killing 2 birds with one stone.
The generative quaternion .walk executed during Transmediale 04 will try to come to grasp with these fundamental technical problems in the .walk project that as yet prevent .walk to be a truly functional computer that relies on hardwired interaction predictability. By providing the city shape as meta-information within the system in order to guide the algorithm to where it started the processing speed will make a steep curve upwards .
http://www.wired.com/news/technology/0,1282,55918,00.html http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/ http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Letters/BroomeBridge.html http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern2/Quatern2.html http://mathworld.wolfram.com/QuaternionGroup.html http://skal.planet-d.net/demo/matrixfaq.htm