Imagining a Complex World
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Fractals.Pdf
Fractals Self Similarity and Fractal Geometry presented by Pauline Jepp 601.73 Biological Computing Overview History Initial Monsters Details Fractals in Nature Brownian Motion L-systems Fractals defined by linear algebra operators Non-linear fractals History Euclid's 5 postulates: 1. To draw a straight line from any point to any other. 2. To produce a finite straight line continuously in a straight line. 3. To describe a circle with any centre and distance. 4. That all right angles are equal to each other. 5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles. History Euclid ~ "formless" patterns Mandlebrot's Fractals "Pathological" "gallery of monsters" In 1875: Continuous non-differentiable functions, ie no tangent La Femme Au Miroir 1920 Leger, Fernand Initial Monsters 1878 Cantor's set 1890 Peano's space filling curves Initial Monsters 1906 Koch curve 1960 Sierpinski's triangle Details Fractals : are self similar fractal dimension A square may be broken into N^2 self-similar pieces, each with magnification factor N Details Effective dimension Mandlebrot: " ... a notion that should not be defined precisely. It is an intuitive and potent throwback to the Pythagoreans' archaic Greek geometry" How long is the coast of Britain? Steinhaus 1954, Richardson 1961 Brownian Motion Robert Brown 1827 Jean Perrin 1906 Diffusion-limited aggregation L-Systems and Fractal Growth -
Waltz's Theory of Theory
WALTZ’S THEORY OF THEORY 201 Waltz’s Theory of Theory Ole Wæver Abstract Waltz’s 1979 book, Theory of International Politics, is the most infl uential in the history of the discipline. It worked its effects to a large extent through raising the bar for what counted as theoretical work, in effect reshaping not only realism but rivals like liberalism and refl ectivism. Yet, ironically, there has been little attention paid to Waltz’s very explicit and original arguments about the nature of theory. This article explores and explicates Waltz’s theory of theory. Central attention is paid to his defi nition of theory as ‘a picture, mentally formed’ and to the radical anti-empiricism and anti-positivism of his position. Followers and critics alike have treated Waltzian neorealism as if it was at bottom a formal proposition about cause–effect relations. The extreme case of Waltz being so victorious in the discipline, and yet being so consistently misinterpreted on the question of theory, shows the power of a dominant philosophy of science in US IR, and thus the challenge facing any ambitious theorising. The article suggests a possible movement of fronts away from the ‘fourth debate’ between rationalism and refl ectivism towards one of theory against empiricism. To help this new agenda, the article introduces a key literature from the philosophy of science about the structure of theory, and particularly about the way even natural science uses theory very differently from the way IR’s mainstream thinks it does – and much more like the way Waltz wants his theory to be used. -
Infinite Perimeter of the Koch Snowflake And
The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area Yaroslav D. Sergeyev∗ y Abstract The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Nu- merical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA and EU. It is revealed in the paper that at infinity the snowflake is not unique, i.e., different snowflakes can be distinguished for different infinite numbers of steps executed during the process of their generation. It is then shown that for any given infinite number n of steps it becomes possible to calculate the exact infinite number, Nn, of sides of the snowflake, the exact infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn, of the Koch snowflake as the result of multiplication of the infinite Nn by the infinitesimal Ln. It is established that for different infinite n and k the infinite perimeters Pn and Pk are also different and the difference can be in- finite. It is shown that the finite areas An and Ak of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite n and k and the difference An − Ak results to be infinitesimal. Finally, snowflakes con- structed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed. -
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. -
The Oxford Democrat
The Oxford Democrat. NUMBER 24 VOLUME 80. SOUTH PARIS, MAINE, TUESDAY, JUNE 17, 1913. where hit the oat "Hear Hear The drearily. "That's you look out for that end of the business. to an extent. low Voted Out the Saloons called ye! ye! poll· D. *ΆΚΚ. increase the aupply quite They he said. on | are now closed." Everyone drew a aigb nail on the head," "Mouey. ▲11 I want you to do 1b to pass this AMONG THE FARMERS. Some Kansas experiment· show an in- The following extract from λ lettei Auctioneer, DON'T HURRY OR WORRY of and went borne to aopper, ex- the mean and dirty thing that can here note." Lictjnsed crease after a crop of clover was turned 'rom Mrs. Benj. H. Fiab of Santa Bar relief, — who ate theirs world—that's «"»«· At Meals Follows. TH* cept tbe election board, the best man In the "Colonel Tod hunter," replied tlie »IU>. Dyspepsia "SPKED PLOW." under. The yield of corn was Increased >ara, Calif., may be of interest bott whip âû0TH and oat of pail· and boxes, and afterwards Thurs." "the Indorsement and the col- Moderate- 20 bushel· an acre, oats 10 bushels, rom the temperance and tbe suffrage Colonel the trouble. banker, Tera» I went to A serene mental condition and time began counting votes. aleep other man's mon- a Correspondence on practical agricultural topics potatoes 30 bushel*. itand point. "Ifs generally the lateral make this note good, and it's to chow your food is more la bearing them count. thoroughly aollcHed. -
Fractal Expressionism—Where Art Meets Science
Santa Fe Institute. February 14, 2002 9:04 a.m. Taylor page 1 Fractal Expressionism—Where Art Meets Science Richard Taylor 1 INTRODUCTION If the Jackson Pollock story (1912–1956) hadn’t happened, Hollywood would have invented it any way! In a drunken, suicidal state on a stormy night in March 1952, the notorious Abstract Expressionist painter laid down the foundations of his masterpiece Blue Poles: Number 11, 1952 by rolling a large canvas across the oor of his windswept barn and dripping household paint from an old can with a wooden stick. The event represented the climax of a remarkable decade for Pollock, during which he generated a vast body of distinct art work commonly referred to as the “drip and splash” technique. In contrast to the broken lines painted by conventional brush contact with the canvas surface, Pollock poured a constant stream of paint onto his horizontal canvases to produce uniquely contin- uous trajectories. These deceptively simple acts fuelled unprecedented controversy and polarized public opinion around the world. Was this primitive painting style driven by raw genius or was he simply a drunk who mocked artistic traditions? Twenty years later, the Australian government rekindled the controversy by pur- chasing the painting for a spectacular two million (U.S.) dollars. In the history of Western art, only works by Rembrandt, Velazquez, and da Vinci had com- manded more “respect” in the art market. Today, Pollock’s brash and energetic works continue to grab attention, as witnessed by the success of the recent retro- spectives during 1998–1999 (at New York’s Museum of Modern Art and London’s Tate Gallery) where prices of forty million dollars were discussed for Blue Poles: Number 11, 1952. -
Constitutional Patriotism” the Answer?
Democracy’s Identity Problem: Is “Constitutional Patriotism” the Answer? Clarissa Rile Hayward NOVEMBER 2006, PAPER NUMBER 27 © 2006 Unpublished by Clarissa Rile Hayward The Occasional Papers of the School of Social Science are versions of talks given at the School’s weekly Thursday Seminar. At these seminars, Members present work-in-progress and then take questions. There is often lively conversation and debate, some of which will be included with the papers. We have chosen papers we thought would be of interest to a broad audience. Our aim is to capture some part of the cross-disciplinary conversations that are the mark of the School’s programs. While Members are drawn from specific disciplines of the social sciences—anthropology, economics, sociology and political science—as well as history, philosophy, literature and law, the School encourages new approaches that arise from exposure to different forms of interpretation. The papers in this series differ widely in their topics, methods, and disciplines. Yet they concur in a broadly humanistic attempt to under- stand how, and under what conditions, the concepts that order experience in different cultures and societies are produced, and how they change. Clarissa Rile Hayward is Associate Professor of Political Science at Ohio State University, where she teaches political theory in the political science department. Her research and teach- ing interests include democratic theory; theories of identity, power, and justice; urban politics; and political aspects of education. Hayward was a Member of the School of Social Science and a National Endowment for the Humanities Fellow at IAS in 2005-2006. During her year at the Institute, she worked on a book manuscript, tentatively titled Cities and Citizens, which explores the ways that democratic state actors shape political identities through institutions that racialize and privatize urban space. -
Post-Truth Politics and Richard Rorty's Postmodernist Bourgeois Liberalism
Ash Center Occasional Papers Tony Saich, Series Editor Something Has Cracked: Post-Truth Politics and Richard Rorty’s Postmodernist Bourgeois Liberalism Joshua Forstenzer University of Sheffield (UK) July 2018 Ash Center for Democratic Governance and Innovation Harvard Kennedy School Ash Center Occasional Papers Series Series Editor Tony Saich Deputy Editor Jessica Engelman The Roy and Lila Ash Center for Democratic Governance and Innovation advances excellence and innovation in governance and public policy through research, education, and public discussion. By training the very best leaders, developing powerful new ideas, and disseminating innovative solutions and institutional reforms, the Center’s goal is to meet the profound challenges facing the world’s citizens. The Ford Foundation is a founding donor of the Center. Additional information about the Ash Center is available at ash.harvard.edu. This research paper is one in a series funded by the Ash Center for Democratic Governance and Innovation at Harvard University’s John F. Kennedy School of Government. The views expressed in the Ash Center Occasional Papers Series are those of the author(s) and do not necessarily reflect those of the John F. Kennedy School of Government or of Harvard University. The papers in this series are intended to elicit feedback and to encourage debate on important public policy challenges. This paper is copyrighted by the author(s). It cannot be reproduced or reused without permission. Ash Center Occasional Papers Tony Saich, Series Editor Something Has Cracked: Post-Truth Politics and Richard Rorty’s Postmodernist Bourgeois Liberalism Joshua Forstenzer University of Sheffield (UK) July 2018 Ash Center for Democratic Governance and Innovation Harvard Kennedy School Letter from the Editor The Roy and Lila Ash Center for Democratic Governance and Innovation advances excellence and innovation in governance and public policy through research, education, and public discussion. -
Cybernetics and Contingency, Codes and Programs : an Account of Social System Thinking in Law and Legal Theory Today
ORBIT-OnlineRepository ofBirkbeckInstitutionalTheses Enabling Open Access to Birkbeck’s Research Degree output Cybernetics and contingency, codes and programs : an account of social system thinking in law and legal theory today https://eprints.bbk.ac.uk/id/eprint/40396/ Version: Full Version Citation: Bryson, Giulia (2019) Cybernetics and contingency, codes and programs : an account of social system thinking in law and legal theory today. [Thesis] (Unpublished) c 2020 The Author(s) All material available through ORBIT is protected by intellectual property law, including copy- right law. Any use made of the contents should comply with the relevant law. Deposit Guide Contact: email CYBERNETICS AND CONTINGENCY, CODES AND PROGRAMS: AN ACCOUNT OF SOCIAL SYSTEM THINKING IN LAW AND LEGAL THEORY TODAY GIULIA BRYSON DOCTORATE OF PHILOSOPHY (PHD) IN LAW 2018 LAW DEPARTMENT, BIRKBECK COLLEGE, UNIVERSITY OF LONDON 1 I hereby declare that the work presented in this thesis is my own, except where explicit reference is made to the work of others. Giulia Bryson 2 ABSTRACT The thesis discusses aspects of current Social Systems Theory, with the main attention devoted both to the level of the compassing social system society and to that of function systems, especially law. Throughout, I refer to the version of social systems theory developed and presented as theory of social autopoiesis in Niklas Luhmann's mature work, while a lim- ited but important part of the thesis will explain this choice and serve as a comparative and genealogical guideline. Central will be the notion and idea of what Luhmann calls a Contingency Formula — term that both func- tions as a problem outline and that indicates how the problem can be solved, within the context of the Legal System. -
Transformations in Sirigu Wall Painting and Fractal Art
TRANSFORMATIONS IN SIRIGU WALL PAINTING AND FRACTAL ART SIMULATIONS By Michael Nyarkoh, BFA, MFA (Painting) A Thesis Submitted to the School of Graduate Studies, Kwame Nkrumah University of Science and Technology in partial fulfilment of the requirements for the degree of DOCTOR OF PHILOSOPHY Faculty of Fine Art, College of Art and Social Sciences © September 2009, Department of Painting and Sculpture DECLARATION I hereby declare that this submission is my own work towards the PhD and that, to the best of my knowledge, it contains no material previously published by another person nor material which has been accepted for the award of any other degree of the University, except where due acknowledgement has been made in the text. Michael Nyarkoh (PG9130006) .................................... .......................... (Student’s Name and ID Number) Signature Date Certified by: Dr. Prof. Richmond Teye Ackam ................................. .......................... (Supervisor’s Name) Signature Date Certified by: K. B. Kissiedu .............................. ........................ (Head of Department) Signature Date CHAPTER ONE INTRODUCTION Background to the study Traditional wall painting is an old art practiced in many different parts of the world. This art form has existed since pre-historic times according to (Skira, 1950) and (Kissick, 1993). In Africa, cave paintings exist in many countries such as “Egypt, Algeria, Libya, Zimbabwe and South Africa”, (Wilcox, 1984). Traditional wall painting mostly by women can be found in many parts of Africa including Ghana, Southern Africa and Nigeria. These paintings are done mostly to enhance the appearance of the buildings and also serve other purposes as well. “Wall painting has been practiced in Northern Ghana for centuries after the collapse of the Songhai Empire,” (Ross and Cole, 1977). -
Constructivism Meets Critical Realism
466572EJT20210.1177/1354066112466572European Journal of International Relations2012Fiaz EJIR Article European Journal of International Relations Constructivism meets critical 2014, Vol. 20(2) 491 –515 © The Author(s) 2012 realism: Explaining Pakistan’s Reprints and permissions: sagepub.co.uk/journalsPermissions.nav state practice in the DOI: 10.1177/1354066112466572 ejt.sagepub.com aftermath of 9/11 Nazya Fiaz National Defence University, Islamabad, Pakistan Abstract This article investigates the theoretical added-value of critical realist incursions into International Relations constructivism. While constructivism focuses on providing multi- causal explanations, its conceptual horizon and subsequent methodological framework fundamentally obscure and limit the opportunity to conceptualize social dialectic and multi-causality in world politics. In this respect, a critical realist meta-theory can provide constructivism with a greatly expanded conceptual framework that transcends material–ideational divisions, and a framework that is able to envision more clearly the process of social dialectic. Second, critical realism affords a methodological diversity that can withstand simultaneous constructivist investigations into the material, agential, ideational, or structural. Using the synthesis of critical realism and constructivism, I illustrate by way of example by employing Pakistan’s participation in the ‘war on terror’ as a case study. While constructivism can show that Pakistan’s role in the war on terror was preceded by a legitimizing narrative, it is a critical realist depth analysis that sheds new light on how a complex social reality was achieved through the convergence of multi-causal explanatory factors. Keywords Agency, constructivism, critical realism, discourse, meta-theory, multi-causality, Pakistan, social dialectic, structure, war on terror, world politics Corresponding author: Nazya Fiaz, Department of International Relations, National Defence University, Islamabad, Pakistan. -
WHY the KOCH CURVE LIKE √ 2 1. Introduction This Essay Aims To
p WHY THE KOCH CURVE LIKE 2 XANDA KOLESNIKOW (SID: 480393797) 1. Introduction This essay aims to elucidate a connection between fractals and irrational numbers, two seemingly unrelated math- ematical objects. The notion of self-similarity will be introduced, leading to a discussion of similarity transfor- mations which are the main mathematical tool used to show this connection. The Koch curve and Koch island (or Koch snowflake) are thep two main fractals that will be used as examples throughout the essay to explain this connection, whilst π and 2 are the two examples of irrational numbers that will be discussed. Hopefully,p by the end of this essay you will be able to explain to your friends and family why the Koch curve is like 2! 2. Self-similarity An object is self-similar if part of it is identical to the whole. That is, if you are to zoom in on a particular part of the object, it will be indistinguishable from the entire object. Many objects in nature exhibit this property. For example, cauliflower appears self-similar. If you were to take a picture of a whole cauliflower (Figure 1a) and compare it to a zoomed in picture of one of its florets, you may have a hard time picking the whole cauliflower from the floret. You can repeat this procedure again, dividing the floret into smaller florets and comparing appropriately zoomed in photos of these. However, there will come a point where you can not divide the cauliflower any more and zoomed in photos of the smaller florets will be easily distinguishable from the whole cauliflower.