Großkopf, P. (2020). Intersecting the twin dragon with rational lines [Diploma Thesis, Technische Universität Wien]. reposiTUm. https://doi.org/10.34726/hss.2020.67762
E104 - Institut für Diskrete Mathematik und Geometrie
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Date (published):
2020
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Number of Pages:
83
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Keywords:
Fraktale; Automaten; Hausdorff Dimension
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Fractals; Automaten; Hausdorff dimension
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Abstract:
The Twin Dragon is a certain well known compact, connected subset of the plane. It appears in the radix representation of complex numbers in base -1+i and its boundary is a fractal with Hausdorff dimension 1.5236... . It is expected that intersecting a (Borel) fractal in the plane with a straight line reduces its Hausdorff dimension by 1, which holds for a family of lines of positive Lebesgue measure. Although this theorem applies to the Twin Dragon, all intersections for which the Hausdorff measure is known lie in the exceptional null set. Following techniques of Akiyama and Scheicher using Büchi automata it is possible to analyze further rational lines. To understand the given problem an introduction into fractal geometry is given including Hausdorff dimension, box-counting dimension, self-similarity, canonical number systems, self-similar tiles and Büchi automata.
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