thesis Graz Karl-Franzens-Universität Graz Walter Victoria Fraktalgeometrie Fraktale: Die geometrischen Elemente der Natur Die fraktale Geometrie gilt als relativ junge Disziplin der Mathematik. Deshalb ist es umso interessanter,diesen neuen Zugang zur Geometrie zu beleuchten. Die vorliegende Diplomarbeit soll,anhand von Beispielen verschiedener Errungenschaften und Entdeckungen der letzten Jahrzehnte,eine generelle Einführung in die Welt der Fraktale liefern. Viele davon beziehen sich auf Arbeitenvon Benoit B. Mandelbrot, der in den 1970er die fundamentalen Grundzüge der fraktalen Geometriegestaltete.Im zentralen Fokus dieser Arbeit stehen einige klassische Fraktale wie zum Beispiel die Cantor-Menge, das Sierpinski-Dreieck, diverse fraktale Kurven sowie die Mandelbrot-Menge und die Julia-Mengen. Diese fraktalen Objekte weisen eine Reihe von ungewöhnlichen und zugleich faszinierendenEigenschaften auf, die bis dato noch nicht vollständig geklärt werden konnten. Eine wesentlicheRolle spielt hier der Begriff der Selbstähnlichkeit, mit denen sich die Strukturen der Fraktale beschreibenlassen. Außerdem treten in vielen Bereichen der Natur und diversen Wissenschaftenbestimmte Zusammenhänge mit der fraktalen Geometrie auf, von denen einige am Ende dieser Arbeitnäher betrachtet werden. Fraktale Muster lassen sich im menschlichen Körper, in der Geologie,in der Chaostheorie und in vielen weiteren Wissenschaftszweigen finden. Ein großer Nutzen liegtdarin, dass mittels neuer Methoden aus der fraktalen Geometrie die Komplexität der Natur sehrgut modelliert werden kann und somit das Verständnis über deren Eigenschaften und Funktionenwächst. 2018 de Universitätsbibliothek Graz Hauptbibliothek, Signatur: II 807295 https://resolver.obvsg.at/urn:nbn:at:at-ubg:1-129578 77 Diplomarbeit conferencePaper ISBN 978-1-4503-6540-6 Proceedings of the 2018 International Conference on Computing and Big Data DOI 10.1145/3277104.3277119 Charleston SC USA ACM Cabutto Tyler A. Heeney Sean P. Ault Shaun V. Mao Guifen Wang Jin An Overview of the Julia Programming Language 2018-09-08 en DOI.org (Crossref) https://dl.acm.org/doi/10.1145/3277104.3277119 2024-06-09 17:59:28 87-91 ICCBD '18: 2018 International Conference on Computing and Big Data magazineArticle 8 Silesian Journal of Pure and Applied Mathematics Januszek Tomasz Pleszczyński Mariusz Comparative analysis of the efficiency of Julia language against the other classic programming languages 2018 https://yadda.icm.edu.pl/baztech/element/bwmeta1.element.baztech-c4339453-4519-4b92-a673-307638a50cb1/c/januszek_Silesian_J_Pure_Appl_Math_2018_8_1.pdf journalArticle Bezanson Jeff Karpinski Stefan Shah Viral Edelman Alan Julia Language Documentation en Zotero attachment Bezanson et al. - Julia Language Documentation.pdf https://readthedocs.org/projects/julia-wf/downloads/pdf/stable/ 2024-06-09 18:18:07 3 application/pdf book John Wiley & Sons Kenneth Falconer Fractal geometry: mathematical foundations and applications 2007 journalArticle Drakopoulos V. Mimikou N. Theoharis T. Fractals Mandelbrot and Julia sets Parallel implementation comparison Parallelism An overview of parallel visualisation methods for Mandelbrot and Julia sets We present a comparative study of simple parallelisation schemes for the most widely used methods for the graphical representation of Mandelbrot and Julia sets. The compared methods render the actual attractor or its complement. 2003 https://www.sciencedirect.com/science/article/pii/S0097849303001067 Number: 4 635-646 27 Computers & Graphics DOI https://doi.org/10.1016/S0097-8493(03)00106-7 4 ISSN 0097-8493 book MIT press Flake Gary William The computational beauty of nature: Computer explorations of fractals, chaos, complex systems, and adaptation 2000 journalArticle 15 IEEE Computer Graphics and Applications DOI 10.1109/38.364961 1 Monro D.M. Dudbridge F. Fractals Approximation algorithms Data structures Displays Graphics Particle measurements Rendering (computer graphics) Software algorithms Software performance Spirals Rendering algorithms for deterministic fractals 1995 Number: 1 32-41 attachment scholar.bib text/x-bibtex journalArticle Atella Anthony Rendering Hypercomplex Fractals en Zotero https://digitalcommons.ric.edu/cgi/viewcontent.cgi?article=1138&context=honors_projects attachment Atella - Rendering Hypercomplex Fractals.pdf https://digitalcommons.ric.edu/cgi/viewcontent.cgi?article=1138&context=honors_projects 2024-06-09 20:31:49 3 application/pdf journalArticle Danisch Simon Krumbiegel Julius Makie.jl: Flexible high-performance data visualization for Julia Makie.jl is a cross-platform plotting ecosystem for the Julia programming language (Bezanson et al., 2012), which enables researchers to create high-performance, GPU-powered, interactive visualizations, as well as publication-quality vector graphics with one unified interface. The infrastructure based on Observables.jl allows users to express how a visualization depends on multiple parameters and data sources, which can then be updated live, either programmatically, or through sliders, buttons and other GUI elements. A sophisticated layout system makes it easy to assemble complex figures. It is designed to avoid common difficulties when aligning nested subplots of different sizes, or placing colorbars or legends freely without spacing issues. Makie.jl leverages the Julia type system to automatically convert many kinds of input arguments which results in a very flexible API that reduces the need to manually prepare data. Finally, users can extend every step of this pipeline for their custom types through Julia’s powerful multiple dispatch mechanism, making Makie a highly productive and generic visualization system. 2021-09-01 en Makie.jl DOI.org (Crossref) https://joss.theoj.org/papers/10.21105/joss.03349 2024-06-09 20:36:58 http://creativecommons.org/licenses/by/4.0/ Number: 65 3349 6 Journal of Open Source Software DOI 10.21105/joss.03349 65 JOSS ISSN 2475-9066 attachment Danisch und Krumbiegel - 2021 - Makie.jl Flexible high-performance data visualiza.pdf https://joss.theoj.org/papers/10.21105/joss.03349.pdf 2024-06-09 20:36:56 3 application/pdf journalArticle Saupe Dietmar Efficient computation of Julia sets and their fractal dimension The computation of the fractal dimension is straightforward using the box-counting method. However, this approach may require very long computation times. If the Julia set is the connected common boundary of two or more basins of attraction, then a recursive version of the box-counting method can be made storage- and time-efficient. The method is also suitable for the computation of the Julia sets. We apply the method to verify a result of D. Ruelle regarding the dimension of Julia sets of R(z)= z2+c for small c∈C, to Newton's method for complex polynomials of degree 3 and to a sequence of Julia sets from the renormalization transformation for hierarchical lattices. We also discuss the computation of Julia sets and their information dimension by the inverse iteration method. In all examples tested we find that the information dimension is less than the fractal dimension. 1987 https://www.sciencedirect.com/science/article/pii/0167278987900248 Number: 3 358-370 28 Physica D: Nonlinear Phenomena DOI https://doi.org/10.1016/0167-2789(87)90024-8 3 ISSN 0167-2789 journalArticle Gaddis Michael E Zyda Michael J The Fractal Geometry of Nature; Its Mathematical Basis and Application to Computer Graphics Fractal Geometry is a recent synthesis of old mathematical constructs. It was first popularized by complex renderings of terrain on a computer graphics medium. Fractal geometry has since spawned research in many diverse scientific disciplines. Its rapid acceptance has been achieved due to its ability to model phenomena that defy discrete computation due to roughneas and discontinuities. With its quick acceptance has come problems. Fractal geometry is a misunderstood idea that is quickly becoming buried under grandiose terminology that serves no purpose. Its essence is induction using simple geometric constructs to transform initiating objects. The fractal objects that we create with this process often resemble natural phenomenon. The purpose of this work is to present fractal geometry to the graphics programmer as a simple workable technique. We hope to demystify the concepts of fractal geometry and make it available to all who are interested. en Zotero attachment Gaddis und Zyda - The Fractal Geometry of Nature; Its Mathematical B.pdf https://apps.dtic.mil/sti/tr/pdf/ADA165185.pdf 2024-06-09 20:54:11 3 application/pdf book CRC Press Addison Paul S Fractals and chaos: an illustrated course 1997 bookSection ISBN 0-470-86412-5 Encyclopedia of Computer Science GBR John Wiley and Sons Ltd. Saupe Dietmar Fractals Much scientific research of the past has analyzed human-made machines and the physical laws that govern their operation. The success of science relies on the predictability of the underlying experiments. Euclidean geometry-based on lines, circles, etc.–is the tool to describe spatial relations, where differential equations are essential in the study of motion and growth. However, natural shapes such as mountains, clouds or trees do not fit well into this framework. The understanding of these phenomena has undergone a fundamental change in the last two decades. Fractal geometry, as conceived by Mandelbrot, provides a mathematical model for many of the seemingly complex forms found in nature. One of Mandelbrot's key observations has been that these forms possess a remarkable statistical invariance under magnification. This may be quantified by a fractal dimension, a number that agrees with our intuitive understanding of dimension but need not be an integer. These ideas may also be applied to time-variant processes. 2003 725–732 journalArticle Krantz Steven G Fractal geometry en Zotero attachment Krantz - Fractal geometry.pdf https://www.mimuw.edu.pl/~pawelst/rzut_oka/Zajecia_dla_MISH_2011-12/Lektury_files/Math.%20Intelligencer%201989%20Krantz.pdf 2024-06-09 21:15:37 3 application/pdf journalArticle Smith Geri Fractal Geometry: History and Theory 26.04.2011 https://www.marywood.edu/programs/resources/math-research-geri-smith-fractal.pdf preprint arXiv Christ Simon Schwabeneder Daniel Rackauckas Christopher Borregaard Michael Krabbe Breloff Thomas Computer Science - Graphics I.3.3 Plots.jl -- a user extendable plotting API for the julia programming language There are plenty of excellent plotting libraries. Each excels at a different use case: one is good for printed 2D publication figures, the other at interactive 3D graphics, a third has excellent LATEX integration or is good for creating dashboards on the web. 2022-06-17 en arXiv.org http://arxiv.org/abs/2204.08775 2024-06-10 16:24:55 arXiv:2204.08775 [cs] arXiv:2204.08775 Comment: 22 pages, 6 figures, 6 code listings attachment Christ et al. - 2022 - Plots.jl -- a user extendable plotting API for the.pdf https://arxiv.org/pdf/2204.08775 2024-06-10 16:24:52 1 application/pdf computerProgram CRAN contributed packages https://cran.r-project.org/web/packages/ 2024-06-12 computerProgram PyPi (Python Package Index) https://pypi.org/ 2024-04-17 computerProgram MATLAB Fileexchange https://de.mathworks.com/matlabcentral/fileexchange/ 2024-06-12 computerProgram Carbonelle Pierre PYPL (Popularity of Programming Language) Index 2023 2024-06-12 conferencePaper ISBN 978-1-4503-4781-5 Proccedings of the 10th European Conference on Software Architecture Workshops DOI 10.1145/2993412.3003382 Copenhagen Denmark ACM Decan Alexandre Mens Tom Claes Maelick On the topology of package dependency networks: a comparison of three programming language ecosystems 2016-11-28 en On the topology of package dependency networks DOI.org (Crossref) https://dl.acm.org/doi/10.1145/2993412.3003382 2024-06-13 08:27:45 1-4 ECSAW '16: European Conference on Software Architecture Workshops book Prusinkiewicz Przemyslaw Lindenmayer Aristid The algorithmic beauty of plants attachment abop.pdf http://algorithmicbotany.org/papers/abop/abop.pdf 2024-06-13 12:37:15 1 application/pdf journalArticle Perkel Jeffrey M. Julia: come for the syntax, stay for the speed 8/2019 en Julia DOI.org (Crossref) https://www.nature.com/articles/d41586-019-02310-3 2024-06-14 14:57:17 http://www.springer.com/tdm 141-142 572 Nature DOI 10.1038/d41586-019-02310-3 7767 Nature ISSN 0028-0836, 1476-4687 attachment Perkel - 2019 - Julia come for the syntax, stay for the speed.pdf https://media.nature.com/original/magazine-assets/d41586-019-02310-3/d41586-019-02310-3.pdf 2024-06-14 14:57:14 1 application/pdf thesis MASSACHUSETTS INSTITUTE OF TECHNOLOGY Bezanson Jeffrey Werner Abstraction in Technical Computing 2015 https://dspace.mit.edu/handle/1721.1/99811 literatur