572 lines
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<dcterms: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.</dcterms:abstract>
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<dcterms: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∈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.</dcterms:abstract>
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<bib:pages>358-370</bib:pages>
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<prism:volume>28</prism:volume>
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<prism:number>3</prism:number>
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<dcterms: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.</dcterms:abstract>
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<dc:title>Fractals and chaos: an illustrated course</dc:title>
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<dc:date>1997</dc:date>
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<dc:title>Fractals</dc:title>
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<dcterms: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.</dcterms:abstract>
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<dc:date>2003</dc:date>
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<bib:pages>725–732</bib:pages>
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