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Hyperbolische Strukturen in Kreisinversionsfraktalen by Kraut, Gunther

Hyperbolische Strukturen in Kreisinversionsfraktalen

by Gunther Kraut
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Book cover type: Paperback
  • ISBN13: 9783668077362
  • Binding: Paperback
  • Subject: N/A
  • Publisher: Grin Verlag
  • Publisher Imprint: Grin Verlag
  • Publication Date:
  • Pages: 116
  • Original Price: USD 67.9
  • Language: German
  • Edition: N/A
  • Item Weight: 159 grams
  • BISAC Subject(s): Geometry / General

Diplomarbeit aus dem Jahr 2007 im Fachbereich Mathematik - Geometrie, Note: 1,3, Technische Universität München (Fakultät für Mathematik), Sprache: Deutsch, Abstract: In den 70er Jahren führte Benoît Mandelbrot den Begriff Fraktal ob der "gebrochenen Dimension" dieser Gebilde ein. Allerdings hat sich bis heute noch keine einheitliche Definition durchgesetzt, die alle Objekte, die klarerweise als Fraktale angesehen werden sollten, umfasst. Es wird am häufigsten verlangt, dass die fraktale Dimension - oftmals die Hausdorff-Dimension - eines Fraktals grö er ist als seine topologische Dimension. Zudem wird Selbstähnlichkeit beziehungsweise Skaleninvarianz gefordert. Fraktale können auf unterschiedliche Weise erzeugt werden. Es besteht jedoch immer die Notwendigkeit, zwischen kurzer Rechenzeit und detailgenauer Darstellung abzuwägen. Ein exaktes Verständnis für die Struktur des jeweiligen Fraktals ermöglicht es, beides zu vereinen. In der ersten Abbildung der Arbeit ist ein Fraktal zu sehen, das durch Kreisinversionen an acht symmetrisch angeordneten Kreisen entsteht. Das sichtbare fraktale Muster ist eine Approximation der Grenzpunkte des durch die Kreisinversionen definierten iterierten Funktionensystems. Im Rahmen dieser Diplomarbeit wird ein Programm entwickelt, das es ermöglicht, direkt die Grenzpunkte - das hei t den Orbit eines bestimmten Startkreises unter der Transformationsgruppe - zu erzeugen. Dies soll nicht zufällig, sondern exakt bis zu einer festgelegten Genauigkeit erfolgen. Die Genauigkeit bezieht sich dabei auf die Grö e der Kreise, die die Grenzpunktmenge bilden. Das im Fraktal entstehende Muster verändert sich sehr stark in Abhängigkeit von dem Radius des mittig liegenden Kreises. Wie entsprechende spezielle Radien numerisch bestimmt werden können und in welchen Situationen der Orbit disjunkt ist, ist wesentlicher Inhalt dieser Arbeit. In Kapitel 2 werden zunächst die mathematischen Grundlagen des projektiven Raumes CP1 sowie der Kreisinversion er

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