Self-Similarity and the Beauty of Fractals
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Pictures of Julia and Mandelbrot Sets
Pictures of Julia and Mandelbrot Sets Wikibooks.org January 12, 2014 On the 28th of April 2012 the contents of the English as well as German Wikibooks and Wikipedia projects were licensed under Cre- ative Commons Attribution-ShareAlike 3.0 Unported license. An URI to this license is given in the list of figures on page 143. If this document is a derived work from the contents of one of these projects and the content was still licensed by the project under this license at the time of derivation this document has to be licensed under the same, a similar or a compatible license, as stated in section 4b of the license. The list of contributors is included in chapter Contributors on page 141. The licenses GPL, LGPL and GFDL are included in chapter Licenses on page 151, since this book and/or parts of it may or may not be licensed under one or more of these licenses, and thus require inclusion of these licenses. The licenses of the figures are given in the list of figures on page 143. This PDF was generated by the LATEX typesetting software. The LATEX source code is included as an attachment (source.7z.txt) in this PDF file. To extract the source from the PDF file, we recommend the use of http://www.pdflabs.com/tools/pdftk-the-pdf-toolkit/ utility or clicking the paper clip attachment symbol on the lower left of your PDF Viewer, selecting Save Attachment. After ex- tracting it from the PDF file you have to rename it to source.7z. -
Fractal 3D Magic Free
FREE FRACTAL 3D MAGIC PDF Clifford A. Pickover | 160 pages | 07 Sep 2014 | Sterling Publishing Co Inc | 9781454912637 | English | New York, United States Fractal 3D Magic | Banyen Books & Sound Option 1 Usually ships in business days. Option 2 - Most Popular! This groundbreaking 3D showcase offers a rare glimpse into the dazzling world of computer-generated fractal art. Prolific polymath Clifford Pickover introduces the collection, which provides background on everything from Fractal 3D Magic classic Mandelbrot set, to the infinitely porous Menger Sponge, to ethereal fractal flames. The following eye-popping gallery displays mathematical formulas transformed into stunning computer-generated 3D anaglyphs. More than intricate designs, visible in three dimensions thanks to Fractal 3D Magic enclosed 3D glasses, will engross math and optical illusions enthusiasts alike. If an item you have purchased from us is not working as expected, please visit one of our in-store Knowledge Experts for free help, where they can solve your problem or even exchange the item for a product that better suits your needs. If you need to return an item, simply bring it back to any Micro Center store for Fractal 3D Magic full refund or exchange. All other products may be returned within 30 days of purchase. Using the software may require the use of a computer or other device that must meet minimum system requirements. It is recommended that you familiarize Fractal 3D Magic with the system requirements before making your purchase. Software system requirements are typically found on the Product information specification page. Aerial Drones Micro Center is happy to honor its customary day return policy for Aerial Drone returns due to product defect or customer dissatisfaction. -
Copyright by Timothy Alexander Cousins 2016
Copyright by Timothy Alexander Cousins 2016 The Thesis Committee for Timothy Alexander Cousins Certifies that this is the approved version of the following thesis: Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media APPROVED BY SUPERVISING COMMITTEE: Supervisor: Hugh Daigle Maša Prodanović Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media by Timothy Alexander Cousins, B. S. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering The University of Texas at Austin August 2016 Dedication I would like to dedicate this to my parents, Michael and Joanne Cousins. Acknowledgements I would like to thank the continuous support of Professor Hugh Daigle over these last two years in guiding throughout my degree and research. I would also like to thank Behzad Ghanbarian for being a great mentor and guide throughout the entire research process, and for constantly giving me invaluable insight and advice, both for the research and for life in general. I would also like to thank my parents for the consistent support throughout my entire life. I would also like to acknowledge Edmund Perfect (Department of Earth and Planetary, University of Tennessee) and Jung-Woo Kim (Radioactive Waste Disposal Research Division, Korea Atomic Energy Research Institute) for providing Lacunarity MATLAB code used in this study. v Abstract Effect of Rough Fractal Pore-Solid Interface on Single-Phase Permeability in Random Fractal Porous Media Timothy Alexander Cousins, M.S.E. -
The Fractal Dimension of Islamic and Persian Four-Folding Gardens
Edinburgh Research Explorer The fractal dimension of Islamic and Persian four-folding gardens Citation for published version: Patuano, A & Lima, MF 2021, 'The fractal dimension of Islamic and Persian four-folding gardens', Humanities and Social Sciences Communications, vol. 8, 86. https://doi.org/10.1057/s41599-021-00766-1 Digital Object Identifier (DOI): 10.1057/s41599-021-00766-1 Link: Link to publication record in Edinburgh Research Explorer Document Version: Publisher's PDF, also known as Version of record Published In: Humanities and Social Sciences Communications General rights Copyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s) and / or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Take down policy The University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorer content complies with UK legislation. If you believe that the public display of this file breaches copyright please contact [email protected] providing details, and we will remove access to the work immediately and investigate your claim. Download date: 06. Oct. 2021 ARTICLE https://doi.org/10.1057/s41599-021-00766-1 OPEN The fractal dimension of Islamic and Persian four- folding gardens ✉ Agnès Patuano 1 & M. Francisca Lima 2 Since Benoit Mandelbrot (1924–2010) coined the term “fractal” in 1975, mathematical the- ories of fractal geometry have deeply influenced the fields of landscape perception, archi- tecture, and technology. Indeed, their ability to describe complex forms nested within each fi 1234567890():,; other, and repeated towards in nity, has allowed the modeling of chaotic phenomena such as weather patterns or plant growth. -
4.3 Discovering Fractal Geometry in CAAD
4.3 Discovering Fractal Geometry in CAAD Francisco Garcia, Angel Fernandez*, Javier Barrallo* Facultad de Informatica. Universidad de Deusto Bilbao. SPAIN E.T.S. de Arquitectura. Universidad del Pais Vasco. San Sebastian. SPAIN * Fractal geometry provides a powerful tool to explore the world of non-integer dimensions. Very short programs, easily comprehensible, can generate an extensive range of shapes and colors that can help us to understand the world we are living. This shapes are specially interesting in the simulation of plants, mountains, clouds and any kind of landscape, from deserts to rain-forests. The environment design, aleatory or conditioned, is one of the most important contributions of fractal geometry to CAAD. On a small scale, the design of fractal textures makes possible the simulation, in a very concise way, of wood, vegetation, water, minerals and a long list of materials very useful in photorealistic modeling. Introduction Fractal Geometry constitutes today one of the most fertile areas of investigation nowadays. Practically all the branches of scientific knowledge, like biology, mathematics, geology, chemistry, engineering, medicine, etc. have applied fractals to simulate and explain behaviors difficult to understand through traditional methods. Also in the world of computer aided design, fractal sets have shown up with strength, with numerous software applications using design tools based on fractal techniques. These techniques basically allow the effective and realistic reproduction of any kind of forms and textures that appear in nature: trees and plants, rocks and minerals, clouds, water, etc. For modern computer graphics, the access to these techniques, combined with ray tracing allow to create incredible landscapes and effects. -
6.5 the Recursion Theorem
6.5. THE RECURSION THEOREM 417 6.5 The Recursion Theorem The recursion Theorem, due to Kleene, is a fundamental result in recursion theory. Theorem 6.5.1 (Recursion Theorem, Version 1 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. For every total recursive function f, there is some n such that ϕn = ϕf(n). The recursion Theorem can be strengthened as follows. Theorem 6.5.2 (Recursion Theorem, Version 2 )Letϕ0,ϕ1,... be any ac- ceptable indexing of the partial recursive functions. There is a total recursive function h such that for all x ∈ N,ifϕx is total, then ϕϕx(h(x)) = ϕh(x). 418 CHAPTER 6. ELEMENTARY RECURSIVE FUNCTION THEORY A third version of the recursion Theorem is given below. Theorem 6.5.3 (Recursion Theorem, Version 3 ) For all n ≥ 1, there is a total recursive function h of n +1 arguments, such that for all x ∈ N,ifϕx is a total recursive function of n +1arguments, then ϕϕx(h(x,x1,...,xn),x1,...,xn) = ϕh(x,x1,...,xn), for all x1,...,xn ∈ N. As a first application of the recursion theorem, we can show that there is an index n such that ϕn is the constant function with output n. Loosely speaking, ϕn prints its own name. Let f be the recursive function such that f(x, y)=x for all x, y ∈ N. 6.5. THE RECURSION THEOREM 419 By the s-m-n Theorem, there is a recursive function g such that ϕg(x)(y)=f(x, y)=x for all x, y ∈ N. -
April 22 7.1 Recursion Theorem
CSE 431 Theory of Computation Spring 2014 Lecture 7: April 22 Lecturer: James R. Lee Scribe: Eric Lei Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. They may be distributed outside this class only with the permission of the Instructor. An interesting question about Turing machines is whether they can reproduce themselves. A Turing machine cannot be be defined in terms of itself, but can it still somehow print its own source code? The answer to this question is yes, as we will see in the recursion theorem. Afterward we will see some applications of this result. 7.1 Recursion Theorem Our end goal is to have a Turing machine that prints its own source code and operates on it. Last lecture we proved the existence of a Turing machine called SELF that ignores its input and prints its source code. We construct a similar proof for the recursion theorem. We will also need the following lemma proved last lecture. ∗ ∗ Lemma 7.1 There exists a computable function q :Σ ! Σ such that q(w) = hPwi, where Pw is a Turing machine that prints w and hats. Theorem 7.2 (Recursion theorem) Let T be a Turing machine that computes a function t :Σ∗ × Σ∗ ! Σ∗. There exists a Turing machine R that computes a function r :Σ∗ ! Σ∗, where for every w, r(w) = t(hRi; w): The theorem says that for an arbitrary computable function t, there is a Turing machine R that computes t on hRi and some input. Proof: We construct a Turing Machine R in three parts, A, B, and T , where T is given by the statement of the theorem. -
The Implications of Fractal Fluency for Biophilic Architecture
JBU Manuscript TEMPLATE The Implications of Fractal Fluency for Biophilic Architecture a b b a c b R.P. Taylor, A.W. Juliani, A. J. Bies, C. Boydston, B. Spehar and M.E. Sereno aDepartment of Physics, University of Oregon, Eugene, OR 97403, USA bDepartment of Psychology, University of Oregon, Eugene, OR 97403, USA cSchool of Psychology, UNSW Australia, Sydney, NSW, 2052, Australia Abstract Fractals are prevalent throughout natural scenery. Examples include trees, clouds and coastlines. Their repetition of patterns at different size scales generates a rich visual complexity. Fractals with mid-range complexity are particularly prevalent. Consequently, the ‘fractal fluency’ model of the human visual system states that it has adapted to these mid-range fractals through exposure and can process their visual characteristics with relative ease. We first review examples of fractal art and architecture. Then we review fractal fluency and its optimization of observers’ capabilities, focusing on our recent experiments which have important practical consequences for architectural design. We describe how people can navigate easily through environments featuring mid-range fractals. Viewing these patterns also generates an aesthetic experience accompanied by a reduction in the observer’s physiological stress-levels. These two favorable responses to fractals can be exploited by incorporating these natural patterns into buildings, representing a highly practical example of biophilic design Keywords: Fractals, biophilia, architecture, stress-reduction, -
Rendering Hypercomplex Fractals Anthony Atella [email protected]
Rhode Island College Digital Commons @ RIC Honors Projects Overview Honors Projects 2018 Rendering Hypercomplex Fractals Anthony Atella [email protected] Follow this and additional works at: https://digitalcommons.ric.edu/honors_projects Part of the Computer Sciences Commons, and the Other Mathematics Commons Recommended Citation Atella, Anthony, "Rendering Hypercomplex Fractals" (2018). Honors Projects Overview. 136. https://digitalcommons.ric.edu/honors_projects/136 This Honors is brought to you for free and open access by the Honors Projects at Digital Commons @ RIC. It has been accepted for inclusion in Honors Projects Overview by an authorized administrator of Digital Commons @ RIC. For more information, please contact [email protected]. Rendering Hypercomplex Fractals by Anthony Atella An Honors Project Submitted in Partial Fulfillment of the Requirements for Honors in The Department of Mathematics and Computer Science The School of Arts and Sciences Rhode Island College 2018 Abstract Fractal mathematics and geometry are useful for applications in science, engineering, and art, but acquiring the tools to explore and graph fractals can be frustrating. Tools available online have limited fractals, rendering methods, and shaders. They often fail to abstract these concepts in a reusable way. This means that multiple programs and interfaces must be learned and used to fully explore the topic. Chaos is an abstract fractal geometry rendering program created to solve this problem. This application builds off previous work done by myself and others [1] to create an extensible, abstract solution to rendering fractals. This paper covers what fractals are, how they are rendered and colored, implementation, issues that were encountered, and finally planned future improvements. -
The Logic of Recursive Equations Author(S): A
The Logic of Recursive Equations Author(s): A. J. C. Hurkens, Monica McArthur, Yiannis N. Moschovakis, Lawrence S. Moss, Glen T. Whitney Source: The Journal of Symbolic Logic, Vol. 63, No. 2 (Jun., 1998), pp. 451-478 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2586843 . Accessed: 19/09/2011 22:53 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org THE JOURNAL OF SYMBOLIC LOGIC Volume 63, Number 2, June 1998 THE LOGIC OF RECURSIVE EQUATIONS A. J. C. HURKENS, MONICA McARTHUR, YIANNIS N. MOSCHOVAKIS, LAWRENCE S. MOSS, AND GLEN T. WHITNEY Abstract. We study logical systems for reasoning about equations involving recursive definitions. In particular, we are interested in "propositional" fragments of the functional language of recursion FLR [18, 17], i.e., without the value passing or abstraction allowed in FLR. The 'pure," propositional fragment FLRo turns out to coincide with the iteration theories of [1]. Our main focus here concerns the sharp contrast between the simple class of valid identities and the very complex consequence relation over several natural classes of models. -
Connectivity Calculus of Fractal Polyhedrons
Pattern Recognition 48 (2015) 1150–1160 Contents lists available at ScienceDirect Pattern Recognition journal homepage: www.elsevier.com/locate/pr Connectivity calculus of fractal polyhedrons Helena Molina-Abril a,n, Pedro Real a, Akira Nakamura b, Reinhard Klette c a Universidad de Sevilla, Spain b Hiroshima University, Japan c The University of Auckland, New Zealand article info abstract Article history: The paper analyzes the connectivity information (more precisely, numbers of tunnels and their Received 27 December 2013 homological (co)cycle classification) of fractal polyhedra. Homology chain contractions and its combi- Received in revised form natorial counterparts, called homological spanning forest (HSF), are presented here as an useful 6 April 2014 topological tool, which codifies such information and provides an hierarchical directed graph-based Accepted 27 May 2014 representation of the initial polyhedra. The Menger sponge and the Sierpiński pyramid are presented as Available online 6 June 2014 examples of these computational algebraic topological techniques and results focussing on the number Keywords: of tunnels for any level of recursion are given. Experiments, performed on synthetic and real image data, Connectivity demonstrate the applicability of the obtained results. The techniques introduced here are tailored to self- Cycles similar discrete sets and exploit homology notions from a representational point of view. Nevertheless, Topological analysis the underlying concepts apply to general cell complexes and digital images and are suitable for Tunnels Directed graphs progressing in the computation of advanced algebraic topological information of 3-dimensional objects. Betti number & 2014 Elsevier Ltd. All rights reserved. Fractal set Menger sponge Sierpiński pyramid 1. Introduction simply-connected sets and those with holes (i.e. -
S27 Fractals References
S27 FRACTALS S27 Fractals References: Supplies: Geogebra (phones, laptops) • Business cards or index cards for building Sierpinski’s tetrahedron or Menger’s • sponge. 407 What is a fractal? S27 FRACTALS What is a fractal? Definition. A fractal is a shape that has 408 What is a fractal? S27 FRACTALS Animation at Wikimedia Commons 409 What is a fractal? S27 FRACTALS Question. Do fractals necessarily have reflection, rotation, translation, or glide reflec- tion symmetry? What kind of symmetry do they have? What natural objects approximate fractals? 410 Fractals in the world S27 FRACTALS Fractals in the world A fractal is formed when pulling apart two glue-covered acrylic sheets. 411 Fractals in the world S27 FRACTALS High voltage breakdown within a block of acrylic creates a fractal Lichtenberg figure. 412 Fractals in the world S27 FRACTALS What happens to a CD in the microwave? 413 Fractals in the world S27 FRACTALS A DLA cluster grown from a copper sulfate solution in an electrodeposition cell. 414 Fractals in the world S27 FRACTALS Anatomy 415 Fractals in the world S27 FRACTALS 416 Fractals in the world S27 FRACTALS Natural tree 417 Fractals in the world S27 FRACTALS Simulated tree 418 Fractals in the world S27 FRACTALS Natural coastline 419 Fractals in the world S27 FRACTALS Simulated coastline 420 Fractals in the world S27 FRACTALS Natural mountain range 421 Fractals in the world S27 FRACTALS Simulated mountain range 422 Fractals in the world S27 FRACTALS The Great Wave o↵ Kanagawa by Katsushika Hokusai 423 Fractals in the world S27 FRACTALS Patterns formed by bacteria grown in a petri dish.