The Representation of Dynamics in Imaginary and Multiple Dimensions

Abstract

I n classical representational art, the aim is to produce an image either in two spatial dimensions, such as in a painting, or in three spatial dimensions, such as in a sculpture. The image is not an accurate copy; it is a personal interpretation of a real or fictitious world structure. This structure may be static, as in a still life, or it may represent a flow, as in van Gogh's night sky-this introduces the element of time. Such representations are not 'imitation photographs'; they can combine views from different perspectives, they can be almost abstract, in an attempt to capture the essence of the subject. Abstract art is an extreme case, where the relations of forms and colours are explored, irrespective of any representational context. The representation of dynamics in nonlinear science poses special problems. Scientists are often poor artists; the essence of dynamics is movement, not static structures; and while the human visual system is geared for interpreting moving objects in twoand three-dimensional space, the dimension of the 'space' of nonlinear dynamics is usually considerably higher than three. This 'state space', within which flows occur, consists of the independent variables, which may be changing. Appropriate choice of the variables, so that they may be represented at right angles (i.e. are orthogonal), gives the state space properties similar to those of real physical space; thus many of the problems-choosing a perspective, representing motion and reducing a high-order space to a flat image (a picture, or an image on a cathode ray tube)-are similar to the problems solved by representational artists. Producing a computer image from nonlinear dynamics is not an algorithmic process; rather, it is a process of choice, of both how and what to illustrate. This personal interpretation is important. The problem is further compounded by the fact that some of the state spaces are complex (a mixture of real and imaginary parts). Different directions mean different things. As an illustration, consider the computer graphics exhibition Sch6nheit im chaos [1], which primarily deals with amazing images produced by the simple, iterative, nonlinear mapping