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245
literatur.bib
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@ -0,0 +1,245 @@
@book{addisonFractalsChaosIllustrated1997,
title = {Fractals and Chaos: An Illustrated Course},
author = {Addison, Paul S},
year = {1997},
publisher = {CRC Press}
}
@article{atellaRenderingHypercomplexFractals,
title = {Rendering {{Hypercomplex Fractals}}},
author = {Atella, Anthony},
langid = {english}
}
@phdthesis{bezanson2015,
title = {Abstraction in {{Technical Computing}}},
author = {Bezanson, Jeffrey Werner},
year = {2015},
school = {MASSACHUSETTS INSTITUTE OF TECHNOLOGY}
}
@article{bezansonJuliaLanguageDocumentation,
title = {Julia {{Language Documentation}}},
author = {Bezanson, Jeff and Karpinski, Stefan and Shah, Viral and Edelman, Alan},
langid = {english}
}
@inproceedings{cabuttoOverviewJuliaProgramming2018,
title = {An {{Overview}} of the {{Julia Programming Language}}},
booktitle = {Proceedings of the 2018 {{International Conference}} on {{Computing}} and {{Big Data}}},
author = {Cabutto, Tyler A. and Heeney, Sean P. and Ault, Shaun V. and Mao, Guifen and Wang, Jin},
year = {2018},
month = sep,
pages = {87--91},
publisher = {ACM},
address = {Charleston SC USA},
doi = {10.1145/3277104.3277119},
urldate = {2024-06-09},
isbn = {978-1-4503-6540-6},
langid = {english}
}
@misc{carbonellePYPLPopularityProgramming2023,
title = {{{PYPL}} ({{Popularity}} of {{Programming Language}}) {{Index}}},
author = {Carbonelle, Pierre},
year = {2023},
urldate = {2024-06-12}
}
@misc{christPlotsJlUser2022,
title = {Plots.Jl -- a User Extendable Plotting {{API}} for the Julia Programming Language},
author = {Christ, Simon and Schwabeneder, Daniel and Rackauckas, Christopher and Borregaard, Michael Krabbe and Breloff, Thomas},
year = {2022},
month = jun,
number = {arXiv:2204.08775},
eprint = {2204.08775},
primaryclass = {cs},
publisher = {arXiv},
urldate = {2024-06-10},
abstract = {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.},
archiveprefix = {arxiv},
langid = {english},
keywords = {Computer Science - Graphics,I.3.3},
note = {Comment: 22 pages, 6 figures, 6 code listings},
file = {/home/roman/snap/zotero-snap/common/Zotero/storage/Q44MJU9D/Christ et al. - 2022 - Plots.jl -- a user extendable plotting API for the.pdf}
}
@article{danischMakieJlFlexible2021,
title = {Makie.Jl: {{Flexible}} High-Performance Data Visualization for {{Julia}}},
shorttitle = {Makie.Jl},
author = {Danisch, Simon and Krumbiegel, Julius},
year = {2021},
month = sep,
journal = {Journal of Open Source Software},
volume = {6},
number = {65},
pages = {3349},
issn = {2475-9066},
doi = {10.21105/joss.03349},
urldate = {2024-06-09},
abstract = {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.},
copyright = {http://creativecommons.org/licenses/by/4.0/},
langid = {english}
}
@inproceedings{decanTopologyPackageDependency2016,
title = {On the Topology of Package Dependency Networks: A Comparison of Three Programming Language Ecosystems},
shorttitle = {On the Topology of Package Dependency Networks},
booktitle = {Proccedings of the 10th {{European Conference}} on {{Software Architecture Workshops}}},
author = {Decan, Alexandre and Mens, Tom and Claes, Maelick},
year = {2016},
month = nov,
pages = {1--4},
publisher = {ACM},
address = {Copenhagen Denmark},
doi = {10.1145/2993412.3003382},
urldate = {2024-06-13},
isbn = {978-1-4503-4781-5},
langid = {english}
}
@article{drakopoulosOverviewParallelVisualisation2003,
title = {An Overview of Parallel Visualisation Methods for {{Mandelbrot}} and {{Julia}} Sets},
author = {Drakopoulos, V. and Mimikou, N. and Theoharis, T.},
year = {2003},
journal = {Computers \& Graphics},
volume = {27},
number = {4},
pages = {635--646},
issn = {0097-8493},
doi = {10.1016/S0097-8493(03)00106-7},
abstract = {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.},
keywords = {Fractals,Mandelbrot and Julia sets,Parallel implementation comparison,Parallelism}
}
@book{flakeComputationalBeautyNature2000,
title = {The Computational Beauty of Nature: {{Computer}} Explorations of Fractals, Chaos, Complex Systems, and Adaptation},
author = {Flake, Gary William},
year = {2000},
publisher = {MIT press}
}
@article{gaddisFractalGeometryNature,
title = {The {{Fractal Geometry}} of {{Nature}}; {{Its Mathematical Basis}} and {{Application}} to {{Computer Graphics}}},
author = {Gaddis, Michael E and Zyda, Michael J},
abstract = {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.},
langid = {english}
}
@article{januszekComparativeAnalysisEfficiency2018,
title = {Comparative Analysis of the Efficiency of {{Julia}} Language against the Other Classic Programming Languages},
author = {Januszek, Tomasz and Pleszczy{\'n}ski, Mariusz},
year = {2018},
journal = {Silesian Journal of Pure and Applied Mathematics},
volume = {8}
}
@book{kennethFractalGeometryMathematical2007,
title = {Fractal Geometry: Mathematical Foundations and Applications},
author = {Kenneth, Falconer},
year = {2007},
publisher = {John Wiley \& Sons}
}
@article{krantzFractalGeometry,
title = {Fractal Geometry},
author = {Krantz, Steven G},
langid = {english}
}
@misc{MATLABFileexchange,
title = {{{MATLAB Fileexchange}}},
urldate = {2024-06-12}
}
@article{monroRenderingAlgorithmsDeterministic1995,
title = {Rendering Algorithms for Deterministic Fractals},
author = {Monro, D.M. and Dudbridge, F.},
year = {1995},
journal = {IEEE Computer Graphics and Applications},
volume = {15},
number = {1},
pages = {32--41},
doi = {10.1109/38.364961},
keywords = {Approximation algorithms,Data structures,Displays,Fractals,Graphics,Particle measurements,Rendering (computer graphics),Software algorithms,Software performance,Spirals},
file = {/home/roman/snap/zotero-snap/common/Zotero/storage/FRJNGNDE/scholar.bib}
}
@article{perkel2019,
title = {Julia: Come for the Syntax, Stay for the Speed},
shorttitle = {Julia},
author = {Perkel, Jeffrey M.},
year = {2019},
month = aug,
journal = {Nature},
volume = {572},
number = {7767},
pages = {141--142},
issn = {0028-0836, 1476-4687},
doi = {10.1038/d41586-019-02310-3},
urldate = {2024-06-14},
copyright = {http://www.springer.com/tdm},
langid = {english},
file = {/home/roman/snap/zotero-snap/common/Zotero/storage/QPIWTYK8/Perkel - 2019 - Julia come for the syntax, stay for the speed.pdf}
}
@book{prusinkiewicz,
title = {The Algorithmic Beauty of Plants},
author = {Prusinkiewicz, Przemyslaw and Lindenmayer, Aristid},
file = {/home/roman/snap/zotero-snap/common/Zotero/storage/ULXJ7GC2/abop.pdf}
}
@misc{PyPiPythonPackage,
title = {{{PyPi}} ({{Python Package Index}})},
urldate = {2024-04-17}
}
@article{saupeEfficientComputationJulia1987,
title = {Efficient Computation of {{Julia}} Sets and Their Fractal Dimension},
author = {Saupe, Dietmar},
year = {1987},
journal = {Physica D: Nonlinear Phenomena},
volume = {28},
number = {3},
pages = {358--370},
issn = {0167-2789},
doi = {10.1016/0167-2789(87)90024-8},
abstract = {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{$\in$}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.}
}
@incollection{saupeFractals2003,
title = {Fractals},
booktitle = {Encyclopedia of {{Computer Science}}},
author = {Saupe, Dietmar},
year = {2003},
pages = {725--732},
publisher = {{John Wiley and Sons Ltd.}},
address = {GBR},
abstract = {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.},
isbn = {0-470-86412-5}
}
@article{smithFractalGeometryHistory2011,
title = {Fractal {{Geometry}}: {{History}} and {{Theory}}},
author = {Smith, Geri},
year = {2011},
month = apr
}
@phdthesis{walterFraktaleGeometrischenElemente2018,
type = {{Diplomarbeit}},
title = {{Fraktale: Die geometrischen Elemente der Natur}},
author = {Walter, Victoria},
year = {2018},
address = {Graz},
abstract = {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 Einf{\"u}hrung in die Welt der Fraktale liefern. Viele davon beziehen sich auf Arbeitenvon Benoit B. Mandelbrot, der in den 1970er die fundamentalen Grundz{\"u}ge 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 ungew{\"o}hnlichen und zugleich faszinierendenEigenschaften auf, die bis dato noch nicht vollst{\"a}ndig gekl{\"a}rt werden konnten. Eine wesentlicheRolle spielt hier der Begriff der Selbst{\"a}hnlichkeit, mit denen sich die Strukturen der Fraktale beschreibenlassen. Au{\ss}erdem treten in vielen Bereichen der Natur und diversen Wissenschaftenbestimmte Zusammenh{\"a}nge mit der fraktalen Geometrie auf, von denen einige am Ende dieser Arbeitn{\"a}her betrachtet werden. Fraktale Muster lassen sich im menschlichen K{\"o}rper, in der Geologie,in der Chaostheorie und in vielen weiteren Wissenschaftszweigen finden. Ein gro{\ss}er Nutzen liegtdarin, dass mittels neuer Methoden aus der fraktalen Geometrie die Komplexit{\"a}t der Natur sehrgut modelliert werden kann und somit das Verst{\"a}ndnis {\"u}ber deren Eigenschaften und Funktionenw{\"a}chst.},
langid = {ngerman},
lccn = {Universit{\"a}tsbibliothek Graz Hauptbibliothek, Signatur: II 807295},
school = {Karl-Franzens-Universit{\"a}t Graz},
keywords = {Fraktalgeometrie}
}
@misc{zotero-27,
title = {{{CRAN}} Contributed Packages},
urldate = {2024-06-12}
}

95
seminararbeit.tex
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@ -45,7 +45,89 @@ Am Ende einfügen
% Unterunterabschnitte mit \subsubsection
% -------------------------------------------------------
\section{Einleitung}
Die Disziplin der Fraktalen Geometrie arbeitet mit mathematischen Objekten, die sich auf eine vielfältige und künstlerische Weise visualisieren lassen.
Schon im 19. Jahrhundert beschäftigten sich Mathematiker
Die Grundlagenarbeit der fraktalen Geometrie beruht auf Werken des Mathematikers Gaston Julia und seinem Konkurrenten Pierre Fatou anfangs des 20. Jahrhunderts \autocite{walterFraktaleGeometrischenElemente2018}.
Diese Errungenschaften griff der Mathematiker Benoît Mandelbrot um die 1970er Jahre in seinem Buch \textit{The Fractal Geometry of Nature} wieder auf und verlieh dem Bereich der fraktalen Geometrie an wachsender Popularität. \autocite{smithFractalGeometryHistory2011}
Die Begründer Gaston und Julia besaßen zur damaligen Zeit noch keine Computer, mit denen Sie ihren Forschungsobjekte darstellen konnten. \autocite{walterFraktaleGeometrischenElemente2018}
In seinem Werk bedient sich Mandelbrot an einer Vielzahl von Visualisierungen für die untersuchten Fraktale.
Die resultierenden Bilder wecken aufgrund der Ästhetik auch das Interesse vieler Nicht-Mathematiker sich mit dem Themengebiet zu befassen
\autocite{smithFractalGeometryHistory2011}.
Mit dem Lauf der Zeit
\section{Hintergrund}
\section{Die Programmiersprache Julia}
Ein Team aus unabhängigen Entwicklern entschied sich 2009 den Startschuss für die Entwicklung von Julia zu setzen. Bis zur ersten Veröffentlichung von Julia verliefen 3 weitere Jahre .
Im August 2018 wurde die Veröffentlichung von Julia 1.0.0 bekanntgegeben \autocite{januszekComparativeAnalysisEfficiency2018}.\\
Das Ziel von Julia ist das \textit{Zwei Sprachen Problem} zu lösen.
Für das Angehen von technischen Problemen werden ist populärste Ansatz zwei Sprachen zu verwenden. Eine Sprache mit leichter Syntax auf hoher Ebene wird in Kombination mit einer Sprache auf niedriger Abstraktionsebene verwendet. Ein bekanntes Beispiel ist die Python-Bibliothek \textit{NumPy}, die in C geschrieben ist.
\autocite{bezanson2015}.
\section{Kriterien}
Im folgenden Abschnitt betrachten wir die ausgewählten Kriterien:
\begin{itemize}
\item Performance
\item Parallelisierbarkeit
\item Verfügbarkeit Softwarepakete
\item Entwicklungsumgebungen
\end{itemize}
Tests werden durchgeführt auf <Prozessor> <Betriebssystem> <Arbeitsspeicher> und <Julia-Version>
\subsection{Performance}
\subsection{Parallelisierbarkeit}
\subsection{Softwarepakete}
Neben den Standardbibliotheken, die mit der Installation einer Programmiersprache mitgeliefert werden, steht Nutzern die Möglichkeit offen eigene Softwarepakete zu entwickeln.
Um die Pakete der Öffentlichkeit zur Verfügung zu stellen, können diese in Paketverzeichnisse im Internet hochgeladen werden. \autoref{table:packages} zeigt eine Auswahl von Verzeichnissen für Sprachen aus dem Bereich des wissenschaftlichen Berechnens.
\begin{table}[ht!]
\centering
\renewcommand{\arraystretch}{1.1}
\caption{Paketanzahl nach Sprache und Paketverzeichnis}
\label{table:packages}
\begin{footnotesize}
\begin{tabular}{llrr}
\toprule
Sprache & Paketverzeichnis & PYPL Rank & Anzahl Pakete\\
\midrule
Python3 & \acs{PyPi} & 1 & 548.169\\
R & \acs{CRAN} & 6 & 20.919\\
MATLAB & FileExchange & 14 & 11.693\\
Julia & JuliaRegistry & 24 & 10.968\\
\bottomrule
\end{tabular}
\end{footnotesize}
\end{table}
Nach dem \ac{PYPL} Index wird Python als populärste Programmiersprache evaluiert. Julia hingegen belegt hingegen Platz 24 \autocite{carbonellePYPLPopularityProgramming2023}.
Diese hohe Popularität erklärt weshalb die Anzahl zwischen Paketen auf \ac{PyPi}, dem Python Package Index, und \textit{JuliaRegistry} sich dem Verhältnis 50:1 nähert.
Aufgrund des relativ jungen Alters, mit Veröffentlichung von Julia im Jahr 2012 \autocite{cabuttoOverviewJuliaProgramming2018} und der niedrigeren Popularität, besitzt Julia im Vergleich zu den restlichen ausgewählten Programmiersprachen die geringste Anzahl an Paketen.
Kann trotz geringerer Paketanzahl eine diverse Anzahl an Fraktalen visualisiert werden?
Wir betrachten folgende Auswahl an klassischen Fraktalen, nach \autocite{walterFraktaleGeometrischenElemente2018}.
\begin{itemize}
\item Cantor Menge
\item Sierpinski-Dreieck
\item Koch-Kurve
\item Raumfüllende Kurven
\item Mandelbrot-Menge und Julia Mengen
\end{itemize}
\subsection{Entwicklungsmgebungen}
\section{Ergebnis}
\section{Ausblick}
@ -58,14 +140,9 @@ Die Disziplin der Fraktalen Geometrie arbeitet mit mathematischen Objekten, die
% Umbruch zu verhindern
% Sie müssen die Abkürzungen selbst alphabetisch sortieren!
\begin{acronym}[IEEE]
\acro{A2A}{Application-to-Application}
\acro{ABK}{Abkürzung}
\acro{ACL}{Acess Control List}
\acro{ACM}{Association of Computing Machinery}
\acro{AES}{Advanced Encryption Standard}
\acro{IEEE}{Institute of Electrical and Electronics Engineers}
\acro{ISO}{International Organization for Standardization}
\acro{PDF}{Portable Document Format}
\acro{PyPi}{Python Package Index}
\acro{PYPL}{PopularitY of Programming Language}
\acro{CRAN}{Comprehensive R Archive Network}
\end{acronym}
% Literaturverzeichnis