Slices through fractals and applications in physics and life sciences

Regular polytopes serve as building stones for fractal constructions, see for instance the illustrations below. In $\mathbb R^4$ exist six regular polytopes: Beside constructions generalising the five platonic solids, there exists a self-dual 24-cell. We will present several fractal constructions based on the 24-cell and other regular polyhedra in three and four dimensions. Fractal constructions in $\mathbb R^4$ cannot be illustrated, but we can visualise their three-dimensional intersections with hyperplanes. We call such an intersection a slice. To illustrate slices we used the cutting plane method. This method can be applied to a class of self-similar sets generated from homotheties with scaling factor an inverse of a Pisot unit $\beta$ and translations in $\mathbb Q(\beta)^n$. From the algebraic point of view our method generalises a result on $\beta$-representations stated by Schmidt in 1979.
The second part of the talk is contributed to application examples of fractals in physics and life sciences. Interactions of fractal structures and electromagnetic waves promise new technologies, e.g. for producing energy or for cloaking. On the other hand, concepts of fractal geometry can be applied in order to describe biological tissues and their cells.