Extending the growing point language to self-organise patterns in three dimensions

Abstract

The Growing Point Language (GPL) is used to engineer the emergent behaviour of an amorphous computer. GPL patterns are topologically one-dimensional objects, regardless of the dimension of the space in which the system exists. A crude length measure in GPL means that GPL patterns also have a geometric character to them. One of the constructs defined in GPL (diatropisim), directs a growing point to propagate tangentially to the level curve of a spatial distribution called a pheromone. In 2-dimensions, tangent spaces are 1-dimensional and therefore diatropism is reasonably well defined. However, in 3-dimensions (and higher) diatropismis no longer confined to 1-dimension, which means that some programs whose behaviour was well understood in 2-dimensional systems, become less so in higher dimensions. We argue that the predictability of the geometric properties of a GPL program in 3-dimensions can be completely recovered. We support this argument with the presentation of a program that given a centre point, a direction, and a radius will generate a circular path in the plane containing the centre, that is normal to the given direction. We provide quantitative data from a single run to illustrate how well the geometric objectives can be achieved.