Symmetry breaking in fields as a methodology for three-dimensional fractal form generation

A new method for generating three-dimensional fractals is described. As a starting point, a two-dimensional iterated function system is considered and its attractor is constructed. Then, a linear tree with endpoints coinciding with points on the attractor is defined. Subsequently, fields are associated with all points on the two-dimensional attractor, and by averaging also with all lower level branches of the tree. These fields, which are fractals themselves, define curvature for the branches of the tree. Different field definitions are discussed, resulting in a wide class of non-linear three-dimensional fractals. If the iterated function system from which the construction starts has symmetry, then the symmetry of the initial linear tree can be broken in a controlled way by appropriate choice of the field definition.