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Regular polytopes serve as building stones for fractal constructions, think of the fractal \(n\)-gons in 2 dimensions or self-similar sets arising from the platonic solids like Sierpinski's pyramid. In four-dimensional space exist six regular polytopes. Beside constructions generalizing the five solids in three dimensions there exists a self-dual 24-cell. We will present several fractal constructions based on the 24-cell and other regular polyhedra in four dimensions.
Fractal constructions in four dimensions cannot be illustrated but we can visualise their three-dimensional intersections with hyperplanes. We call such an intersection a slice. To illustrate slices we were using the cutting plane method. This method can be applied to a class of self-similar sets generated by 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.
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