An Introduction to Apophysis © Clive Haynes MMXX
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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. -
Iterated Function System Quasiarcs
CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 21, Pages 78–100 (February 3, 2017) http://dx.doi.org/10.1090/ecgd/305 ITERATED FUNCTION SYSTEM QUASIARCS ANNINA ISELI AND KEVIN WILDRICK Abstract. We consider a class of iterated function systems (IFSs) of contract- ing similarities of Rn, introduced by Hutchinson, for which the invariant set possesses a natural H¨older continuous parameterization by the unit interval. When such an invariant set is homeomorphic to an interval, we give necessary conditions in terms of the similarities alone for it to possess a quasisymmetric (and as a corollary, bi-H¨older) parameterization. We also give a related nec- essary condition for the invariant set of such an IFS to be homeomorphic to an interval. 1. Introduction Consider an iterated function system (IFS) of contracting similarities S = n {S1,...,SN } of R , N ≥ 2, n ≥ 1. For i =1,...,N, we will denote the scal- n n ing ratio of Si : R → R by 0 <ri < 1. In a brief remark in his influential work [18], Hutchinson introduced a class of such IFSs for which the invariant set is a Peano continuum, i.e., it possesses a continuous parameterization by the unit interval. There is a natural choice for this parameterization, which we call the Hutchinson parameterization. Definition 1.1 (Hutchinson, 1981). The pair (S,γ), where γ is the invariant set of an IFS S = {S1,...,SN } with scaling ratio list {r1,...,rN } is said to be an IFS path, if there exist points a, b ∈ Rn such that (i) S1(a)=a and SN (b)=b, (ii) Si(b)=Si+1(a), for any i ∈{1,...,N − 1}. -
Annotated List of References Tobias Keip, I7801986 Presentation Method: Poster
Personal Inquiry – Annotated list of references Tobias Keip, i7801986 Presentation Method: Poster Poster Section 1: What is Chaos? In this section I am introducing the topic. I am describing different types of chaos and how individual perception affects our sense for chaos or chaotic systems. I am also going to define the terminology. I support my ideas with a lot of examples, like chaos in our daily life, then I am going to do a transition to simple mathematical chaotic systems. Larry Bradley. (2010). Chaos and Fractals. Available: www.stsci.edu/~lbradley/seminar/. Last accessed 13 May 2010. This website delivered me with a very good introduction into the topic as there are a lot of books and interesting web-pages in the “References”-Sektion. Gleick, James. Chaos: Making a New Science. Penguin Books, 1987. The book gave me a very general introduction into the topic. Harald Lesch. (2003-2007). alpha-Centauri . Available: www.br-online.de/br- alpha/alpha-centauri/alpha-centauri-harald-lesch-videothek-ID1207836664586.xml. Last accessed 13. May 2010. A web-page with German video-documentations delivered a lot of vivid examples about chaos for my poster. Poster Section 2: Laplace's Demon and the Butterfly Effect In this part I describe the idea of the so called Laplace's Demon and the theory of cause-and-effect chains. I work with a lot of examples, especially the famous weather forecast example. Also too I introduce the mathematical concept of a dynamic system. Jeremy S. Heyl (August 11, 2008). The Double Pendulum Fractal. British Columbia, Canada. -
Fractal Dimension of Self-Affine Sets: Some Examples
FRACTAL DIMENSION OF SELF-AFFINE SETS: SOME EXAMPLES G. A. EDGAR One of the most common mathematical ways to construct a fractal is as a \self-similar" set. A similarity in Rd is a function f : Rd ! Rd satisfying kf(x) − f(y)k = r kx − yk for some constant r. We call r the ratio of the map f. If f1; f2; ··· ; fn is a finite list of similarities, then the invariant set or attractor of the iterated function system is the compact nonempty set K satisfying K = f1[K] [ f2[K] [···[ fn[K]: The set K obtained in this way is said to be self-similar. If fi has ratio ri < 1, then there is a unique attractor K. The similarity dimension of the attractor K is the solution s of the equation n X s (1) ri = 1: i=1 This theory is due to Hausdorff [13], Moran [16], and Hutchinson [14]. The similarity dimension defined by (1) is the Hausdorff dimension of K, provided there is not \too much" overlap, as specified by Moran's open set condition. See [14], [6], [10]. REPRINT From: Measure Theory, Oberwolfach 1990, in Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, numero 28, anno 1992, pp. 341{358. This research was supported in part by National Science Foundation grant DMS 87-01120. Typeset by AMS-TEX. G. A. EDGAR I will be interested here in a generalization of self-similar sets, called self-affine sets. In particular, I will be interested in the computation of the Hausdorff dimension of such sets. -
Bachelorarbeit Im Studiengang Audiovisuelle Medien Die
Bachelorarbeit im Studiengang Audiovisuelle Medien Die Nutzbarkeit von Fraktalen in VFX Produktionen vorgelegt von Denise Hauck an der Hochschule der Medien Stuttgart am 29.03.2019 zur Erlangung des akademischen Grades eines Bachelor of Engineering Erst-Prüferin: Prof. Katja Schmid Zweit-Prüfer: Prof. Jan Adamczyk Eidesstattliche Erklärung Name: Vorname: Hauck Denise Matrikel-Nr.: 30394 Studiengang: Audiovisuelle Medien Hiermit versichere ich, Denise Hauck, ehrenwörtlich, dass ich die vorliegende Bachelorarbeit mit dem Titel: „Die Nutzbarkeit von Fraktalen in VFX Produktionen“ selbstständig und ohne fremde Hilfe verfasst und keine anderen als die angegebenen Hilfsmittel benutzt habe. Die Stellen der Arbeit, die dem Wortlaut oder dem Sinn nach anderen Werken entnommen wurden, sind in jedem Fall unter Angabe der Quelle kenntlich gemacht. Die Arbeit ist noch nicht veröffentlicht oder in anderer Form als Prüfungsleistung vorgelegt worden. Ich habe die Bedeutung der ehrenwörtlichen Versicherung und die prüfungsrechtlichen Folgen (§26 Abs. 2 Bachelor-SPO (6 Semester), § 24 Abs. 2 Bachelor-SPO (7 Semester), § 23 Abs. 2 Master-SPO (3 Semester) bzw. § 19 Abs. 2 Master-SPO (4 Semester und berufsbegleitend) der HdM) einer unrichtigen oder unvollständigen ehrenwörtlichen Versicherung zur Kenntnis genommen. Stuttgart, den 29.03.2019 2 Kurzfassung Das Ziel dieser Bachelorarbeit ist es, ein Verständnis für die Generierung und Verwendung von Fraktalen in VFX Produktionen, zu vermitteln. Dabei bildet der Einblick in die Arten und Entstehung der Fraktale -
Apophysis :– an Intermediate User Guide
Apophysis :– An Intermediate User Guide By Carl Skepper (aka 2B2H) April 2006 Contents Introduction 3 Working with the Editor 4 Using my Metallica script 10 Falling in love with Julia (and Julia ‘n’) 15 Tiling (prepare to be disappointed) 24 Use of blur 31 Adding Colour to your flames 37 Introduction Welcome to my intermediate user guide to using the great freeware application called Apophysis , also affectionately known as ‘Apo’. I first downloaded this software on December 5 th 2005 and I have been hooked on it ever since. The variety of fractals it is able to produce is only limited by your creativity but saying that, your creativity amounts to nothing if you fail to persevere despite the lack of documentation out there. The definitive starter guides are those by Lance and for scripting, by Datagram. Links to both (and much more useful stuff) can be found at The Fractal Farm website : http://www.woosie.net/fracfan/viewtopic.php?t=15 These guides help you with the GUI and offer some basic advice on creating fractals. This guide is different. Together we will create specific flames so you can see how they are done. By doingthis I hope you will gain a better understandiing of Apo and use the knowledge to create your own fractal wonders ☺ It is not intended to teach you the basics of the GUI, those are covered in the docs linked at The Fractal Farm. It will show you a few tricks that you may not be aware of. I do not profess to be any kind of expert with the software. -
On Bounding Boxes of Iterated Function System Attractors Hsueh-Ting Chu, Chaur-Chin Chen*
Computers & Graphics 27 (2003) 407–414 Technical section On bounding boxes of iterated function system attractors Hsueh-Ting Chu, Chaur-Chin Chen* Department of Computer Science, National Tsing Hua University, Hsinchu 300, Taiwan, ROC Abstract Before rendering 2D or 3D fractals with iterated function systems, it is necessary to calculate the bounding extent of fractals. We develop a new algorithm to compute the bounding boxwhich closely contains the entire attractor of an iterated function system. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Fractals; Iterated function system; IFS; Bounding box 1. Introduction 1.1. Iterated function systems Barnsley [1] uses iterated function systems (IFS) to Definition 1. A transform f : X-X on a metric space provide a framework for the generation of fractals. ðX; dÞ is called a contractive mapping if there is a Fractals are seen as the attractors of iterated function constant 0pso1 such that systems. Based on the framework, there are many A algorithms to generate fractal pictures [1–4]. However, dðf ðxÞ; f ðyÞÞps Á dðx; yÞ8x; y X; ð1Þ in order to generate fractals, all of these algorithms have where s is called a contractivity factor for f : to estimate the bounding boxes of fractals in advance. For instance, in the program Fractint (http://spanky. Definition 2. In a complete metric space ðX; dÞ; an triumf.ca/www/fractint/fractint.html), we have to guess iterated function system (IFS) [1] consists of a finite set the parameters of ‘‘image corners’’ before the beginning of contractive mappings w ; for i ¼ 1; 2; y; n; which is of drawing, which may not be practical. -
Math Morphing Proximate and Evolutionary Mechanisms
Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales. -
Iterated Function System
IARJSET ISSN (Online) 2393-8021 ISSN (Print) 2394-1588 International Advanced Research Journal in Science, Engineering and Technology ISO 3297:2007 Certified Vol. 3, Issue 8, August 2016 Iterated Function System S. C. Shrivastava Department of Applied Mathematics, Rungta College of Engineering and Technology, Bhilai C.G., India Abstract: Fractal image compression through IFS is very important for the efficient transmission and storage of digital data. Fractal is made up of the union of several copies of itself and IFS is defined by a finite number of affine transformation which characterized by Translation, scaling, shearing and rotat ion. In this paper we describe the necessary conditions to form an Iterated Function System and how fractals are generated through affine transformations. Keywords: Iterated Function System; Contraction Mapping. 1. INTRODUCTION The exploration of fractal geometry is usually traced back Metric Spaces definition: A space X with a real-valued to the publication of the book “The Fractal Geometry of function d: X × X → ℜ is called a metric space (X, d) if d Nature” [1] by the IBM mathematician Benoit B. possess the following properties: Mandelbrot. Iterated Function System is a method of constructing fractals, which consists of a set of maps that 1. d(x, y) ≥ 0 for ∀ x, y ∈ X explicitly list the similarities of the shape. Though the 2. d(x, y) = d(y, x) ∀ x, y ∈ X formal name Iterated Function Systems or IFS was coined 3. d x, y ≤ d x, z + d z, y ∀ x, y, z ∈ X . (triangle by Barnsley and Demko [2] in 1985, the basic concept is inequality). -
Michael Angelo Gomez – Exegesis
CCA1103 – Creativity: Theory, Practice, and History 1 CCA1103 – Creativity: Theory, Practice, and History Fractal Imaging: A Mini Exegesis by Michael Angelo Gomez 10445917 CCA1103 – Creativity: Theory, Practice, and History Project Exegesis Christopher Mason 2 CCA1103 – Creativity: Theory, Practice, and History Table of Contents I. Introduction ......................................................................................................................... 4 II. Purpose and Application.................................................................................................... 4 III. Theoretical Context .......................................................................................................... 6 IV. The Creative Process ..................................................................................................... 10 V. Conclusion ....................................................................................................................... 18 VI. Appendices ..................................................................................................................... 19 VII. References .................................................................................................................... 21 3 CCA1103 – Creativity: Theory, Practice, and History I. Introduction Upon the completion of my proposed creative project, a number of insights have been unearthed in light of the perception of my work now that it has reached its final stage of presentation and display. This reflection -
The Aesthetics and Fractal Dimension of Electric Sheep
International Journal of Bifurcation and Chaos, Vol. 18, No. 4 (2008) 1243–1248 c World Scientific Publishing Company THE AESTHETICS AND FRACTAL DIMENSION OF ELECTRIC SHEEP SCOTT DRAVES Spotworks, 2261 Market St #158, San Francisco, CA 94114, USA RALPH ABRAHAM Mathematics Department, University of California, Santa Cruz, CA 95064, USA PABLO VIOTTI Politics Department, University of California, Santa Cruz, CA 95064, USA FREDERICK DAVID ABRAHAM Blueberry Brain Institute, 1396 Gregg Hill Road, Waterbury Center, VT 05677, USA JULIAN CLINTON SPROTT Physics Department, University of Wisconsin, Madison, WI 53706-1390, USA Received January 22, 2007; Revised May 31, 2007 Physicist Clint Sprott demonstrated a relationship between aesthetic judgments of fractal images and their fractal dimensions [1993]. Scott Draves, aka Spot, a computer scientist and artist, has created a space of images called fractal flames, based on attractors of two-dimensional iterated function systems. A large community of users run software that automatically downloads animated fractal flames, known as “sheep”, and displays them as their screen-saver. The users may vote electronically for the sheep they like while the screen-saver is running. In this report we proceed from Sprott to Spot. The data show an inverted U-shaped curve in the relationship between aesthetic judgments of flames and their fractal dimension, confirming and clarifying earlier reports. Keywords: Fractal dimensions; electric sheep. 1. Introduction thus combines the Electric Sheep of Draves and the fractal aesthetics of Sprott. This is a report on a new study of aesthetic judg- The Electric Sheep home page is available ments made by a large community participating from electricsheep.org. -
An Introduction to the Mandelbrot Set
An introduction to the Mandelbrot set Bastian Fredriksson January 2015 1 Purpose and content The purpose of this paper is to introduce the reader to the very useful subject of fractals. We will focus on the Mandelbrot set and the related Julia sets. I will show some ways of visualising these sets and how to make a program that renders them. Finally, I will explain a key exchange algorithm based on what we have learnt. 2 Introduction The Mandelbrot set and the Julia sets are sets of points in the complex plane. Julia sets were first studied by the French mathematicians Pierre Fatou and Gaston Julia in the early 20th century. However, at this point in time there were no computers, and this made it practically impossible to study the structure of the set more closely, since large amount of computational power was needed. Their discoveries was left in the dark until 1961, when a Jewish-Polish math- ematician named Benoit Mandelbrot began his research at IBM. His task was to reduce the white noise that disturbed the transmission on telephony lines [3]. It seemed like the noise came in bursts, sometimes there were a lot of distur- bance, and sometimes there was no disturbance at all. Also, if you examined a period of time with a lot of noise-problems, you could still find periods without noise [4]. Could it be possible to come up with a model that explains when there is noise or not? Mandelbrot took a quite radical approach to the problem at hand, and chose to visualise the data.