The first submission illustrates some solutions found for the unconstrained MDS-error function. It illustrates how the present method integrates linear and non-linear elements. The linear elements are the side-lines of the hypercubes. In most of the illustrations, the edges were partially filled. As is the case for ordinary MDS, they are not necessarily plane surfaces. But most of the non-linearity in the representations is due to the trees. Three examples of bare non-linear trees were included in the illustration. At the bottom, two views are shown of a hypercube based house that was made with different software, in which the user can choose within bounds the location of the edges.

Photographic paper, 16.5" x 23.5"

The second submission illustrates some solutions obtained for the constrained MDS-error function. The hypercube visualizations are such that they have horizontal ground floor. The origin of the tree is in the same plane. The trees were permitted to partially grow branches below this plane. Constraining MDS gives a different error surface, again with a large wealth of local minima which correspond to aesthetically pleasing solutions. The red 'blossoms' in the middle picture correspond to four branches of very high curvature. Some forms shown are rotations of each other.

This submission illustrates the type of trees used, but for ternary bifurcation, and for eight branching levels. The tree was initialized linearly, with endpoints coinciding with points on the Sierpinski triangle. Branches were provided of fractally defined weights. These defined curvature for all branches at all levels. In the main figure, only branches of the five highest levels were drawn.

Philip Van Loocke is a Mathematician and Professor of Philosophy at the University of Ghent (Belgium), Dept of Philosophy, Art and Consciousness Research Group

Statement about my art:

"The artwork combines two lines of research. First, three dimensional fractal trees were provided with angular parameters which define the way in which they curl. Second, a multi-dimensional scaling (MDS) approach was used to obtain visualizations of high-dimensional objects. In the combination of both methods, the vertices of hypercube visualizations were located at the endpoints of the trees. The trees were bent by a stochastic algorithm until the MDS-error function reached a local minimum. The error function defines a landscape over a high-dimensional parameter space, with much more local minima than in case of ordinary MDS. From the artist point of view, this is an advantage, since many local minima correspond to configurations of aesthetic quality. The trees were initialized in a symmetric way, so that the search algorithm corresponds to a partial symmetry breaking process. In view of possible design relevance, situations in which constraints were added to the MDS-error function were studied. For instance, also solutions for runs in which particular vertices were constrained to be located in the same plane are illustrated."