### Applicant

Prof. Dr. Jörg Arndt

Technische Hochschule Nürnberg Georg Simon Ohm

Wassertorstraße 10, 90489 Nürnberg

### Project Overview

Three-dimensional curves with fractal envelopes exist. However, such a curve is yet to be seen. A first approach is analyzing the spatial regions in which such curves reside. These can be defined with iterated function systems (IFS). The linear parts of the affine transformation are all an identical rotated dilation. This is equivalent to a strict self-similarity of the spatial region, not just a self-affinity. The end points of the line segments of the curves lie on a crystal lattice. The search will be restricted to cubic lattices and spatial regions of genus zero. Furthermore, it shall be examined whether and how properties of plane curves can be transferred to space-filling curves. Curves that fill other spatial regions than the cube are of particular interest. This project aims to give first examples of three-dimensional curves in these spatial regions. For each region found numerous curves exist; these are the Eulerian paths in the implied graph inside the spatial region.

Finally, automatons (Lindenmayer systems) are to be determined, which, as in the two-dimensional case allow for a pleasant way of displaying the curves.