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Properties and Generalizations of the Fibonacci Word Fractal Exploring Fractal Curves José L
The Mathematica® Journal Properties and Generalizations of the Fibonacci Word Fractal Exploring Fractal Curves José L. Ramírez Gustavo N. Rubiano This article implements some combinatorial properties of the Fibonacci word and generalizations that can be generated from the iteration of a morphism between languages. Some graphic properties of the fractal curve are associated with these words; the curves can be generated from drawing rules similar to those used in L-systems. Simple changes to the programs generate other interesting curves. ‡ 1. Introduction The infinite Fibonacci word, f = 0 100 101 001 001 010 010 100 100 101 ... is certainly one of the most studied words in the field of combinatorics on words [1–4]. It is the archetype of a Sturmian word [5]. The word f can be associated with a fractal curve with combinatorial properties [6–7]. This article implements Mathematica programs to generate curves from f and a set of drawing rules. These rules are similar to those used in L-systems. The outline of this article is as follows. Section 2 recalls some definitions and ideas of combinatorics on words. Section 3 introduces the Fibonacci word, its fractal curve, and a family of words whose limit is the Fibonacci word fractal. Finally, Section 4 generalizes the Fibonacci word and its Fibonacci word fractal. The Mathematica Journal 16 © 2014 Wolfram Media, Inc. 2 José L. Ramírez and Gustavo N. Rubiano ‡ 2. Definitions and Notation The terminology and notation are mainly those of [5] and [8]. Let S be a finite alphabet, whose elements are called symbols. A word over S is a finite sequence of symbols from S. -
POV-Ray Reference
POV-Ray Reference POV-Team for POV-Ray Version 3.6.1 ii Contents 1 Introduction 1 1.1 Notation and Basic Assumptions . 1 1.2 Command-line Options . 2 1.2.1 Animation Options . 3 1.2.2 General Output Options . 6 1.2.3 Display Output Options . 8 1.2.4 File Output Options . 11 1.2.5 Scene Parsing Options . 14 1.2.6 Shell-out to Operating System . 16 1.2.7 Text Output . 20 1.2.8 Tracing Options . 23 2 Scene Description Language 29 2.1 Language Basics . 29 2.1.1 Identifiers and Keywords . 30 2.1.2 Comments . 34 2.1.3 Float Expressions . 35 2.1.4 Vector Expressions . 43 2.1.5 Specifying Colors . 48 2.1.6 User-Defined Functions . 53 2.1.7 Strings . 58 2.1.8 Array Identifiers . 60 2.1.9 Spline Identifiers . 62 2.2 Language Directives . 64 2.2.1 Include Files and the #include Directive . 64 2.2.2 The #declare and #local Directives . 65 2.2.3 File I/O Directives . 68 2.2.4 The #default Directive . 70 2.2.5 The #version Directive . 71 2.2.6 Conditional Directives . 72 2.2.7 User Message Directives . 75 2.2.8 User Defined Macros . 76 3 Scene Settings 81 3.1 Camera . 81 3.1.1 Placing the Camera . 82 3.1.2 Types of Projection . 86 3.1.3 Focal Blur . 88 3.1.4 Camera Ray Perturbation . 89 3.1.5 Camera Identifiers . 89 3.2 Atmospheric Effects . -
Pattern, the Visual Mathematics: a Search for Logical Connections in Design Through Mathematics, Science, and Multicultural Arts
Rochester Institute of Technology RIT Scholar Works Theses 6-1-1996 Pattern, the visual mathematics: A search for logical connections in design through mathematics, science, and multicultural arts Seung-eun Lee Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Lee, Seung-eun, "Pattern, the visual mathematics: A search for logical connections in design through mathematics, science, and multicultural arts" (1996). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. majWiif." $0-J.zn jtiJ jt if it a mtrc'r icjened thts (waefii i: Tkirr. afaidiid to bz i d ! -Lrtsud a i-trio. of hw /'' nvwte' mfratWf trawwi tpf .w/ Jwitfl- ''','-''' ftatsTB liovi 'itcj^uitt <t. dt feu. Tfui ;: tti w/ik'i w cWii.' cental cj lniL-rital. p-siod. [Jewy anJ ./tr.-.MTij of'irttion. fan- 'mo' that. I: wo!'.. PicrdoTrKwe, wiifi id* .or.if'iifer -now r't.t/ry summary Jfem2>/x\, i^zA 'jX7ipu.lv z&ietQStd (Xtrma iKc'i-J: ffocub. The (i^piicaiom o". crWx< M Wf flrtu't. fifesspwn, Jiii*iaiin^ jfli dtfifi,tmi jpivxei u3ili'-rCi luvjtv-tzxi patcrr? ;i t. drifters vcy. Symmetr, yo The theoretical concepts of symmetry deal with group theory and figure transformations. Figure transformations, or symmetry operations refer to the movement and repetition of an one-, two , and three-dimensional space. (f_8= I lhowtuo irwti) familiar iltjpg . -
FACTA UNIVERSITATIS UNIVERSITY of NIŠ ISSN 0354-4605 (Print) ISSN 2406-0860 (Online) Series Architecture and Civil Engineering COBISS.SR-ID 98807559 Vol
CMYK K Y M C FACTA UNIVERSITATIS UNIVERSITY OF NIŠ ISSN 0354-4605 (Print) ISSN 2406-0860 (Online) Series Architecture and Civil Engineering COBISS.SR-ID 98807559 Vol. 16, No 2, 2018 Contents Milena Jovanović, Aleksandra Mirić, Goran Jovanović, Ana Momčilović Petronijević NIŠ OF UNIVERSITY EARTH AS A MATERIAL FOR CONSTRUCTION OF MODERN HOUSES ..........175 FACTA UNIVERSITATIS Đorđe Alfirević, Sanja Simonović Alfirević CONSTITUTIVE MOTIVES IN LIVING SPACE ORGANISATION .......................189 Series Ivana Bogdanović Protić, Milena Dinić Branković, Milica Igić, Milica Ljubenović, Mihailo Mitković ARCHITECTURE AND CIVIL ENGINEERING MODALITIES OF TENANTS PARTICIPATION IN THE REVITALIZATION Vol. 16, No 2, 2018 OF OPEN SPACES IN COMPLEXES WITH HIGH-RISE HOUSING ......................203 Biljana Arandelovic THE NEUKOELLN PHENOMENON: THE RECENT MOVE OF AN ART SCENE IN BERLIN ...........................................213 2, 2018 Velimir Stojanović o THE NATURE, QUANTITY AND QUALITY OF URBAN SEGMENTS.................223 Maja Petrović, Radomir Mijailović, Branko Malešević, Đorđe Đorđević, Radovan Štulić Vol. 16, N Vol. THE USE OF WEBER’S FOCAL-DIRECTORIAL PLANE CURVES AS APPROXIMATION OF TOP VIEW CONTOUR CURVES AT ARCHITECTURAL BUILDINGS OBJECTS ........................................................237 Ana Momčilović-Petronijević, Predrag Petronijević, Mihailo Mitković DEGRADATION OF ARCHEOLOGICAL SITES – CASE STUDY CARIČIN GRAD .................................................................................247 Vesna Tomić, Aleksandra Đukić -
Plato, and the Fractal Geometry of the Universe
Plato, and the Fractal Geometry of the Universe Student Name: Nico Heidari Tari Student Number: 430060 Supervisor: Harrie de Swart Erasmus School of Philosophy Erasmus University Rotterdam Bachelor’s Thesis June, 2018 1 Table of contents Introduction ................................................................................................................................ 3 Section I: the Fibonacci Sequence ............................................................................................. 6 Section II: Definition of a Fractal ............................................................................................ 18 Section III: Fractals in the World ............................................................................................. 24 Section IV: Fractals in the Universe......................................................................................... 33 Section V: Fractal Cosmology ................................................................................................. 40 Section VI: Plato and Spinoza .................................................................................................. 56 Conclusion ................................................................................................................................ 62 List of References ..................................................................................................................... 66 2 Introduction Ever since the dawn of humanity, people have wondered about the nature of the universe. This was -
Helix, and Its Implementation in Architecture
Appendix 1 to Guidelines for Preparation, Defence and Storage of Final Projects KAUNAS UNIVERSITY OF TECHNOLOGY CIVIL ENGINEERING AND ARCHITECTURE FACULTY Emin Vahid CONTEMPORARY TRENDS OF ARCHITECTURAL GEOMETRY. HELIX AND ITS IMPLEMENTATION IN ARCHITECTURE. Master’s Degree Final Project Supervisor Lect. Arch. Vytautas Baltus KAUNAS, 2017 KAUNAS UNIVERSITY OF TECHNOLOGY CIVIL ENGINEERING AND ARCHITECTURE FACULTY TITLE OF FINAL PROJECT Master’s Degree Final Project Contemporary Trends of Architectural Geometry. Helix and its implementation in architecture (code 621K10001) Supervisor (signature) Lect. Arch. Vytautas Baltus (date) Reviewer (signature) Assoc. prof. dr. Gintaris Cinelis (date) Project made by (signature) Emin Vahid (date) KAUNAS, 2017 ~ 1 ~ Appendix 4 to Guidelines for Preparation, Defence and Storage of Final Projects KAUNAS UNIVERSITY OF TECHNOLOGY Civil Engineering and Architecture (Faculty) Emin Vahid (Student's name, surname) Master degree Architecture 6211PX026 (Title and code of study programme) "Contemporary trends of architectural geometry. Helix and its implementation in architecture" DECLARATION OF ACADEMIC INTEGRITY 22 05 2017 Kaunas I confirm that the final project of mine, Emin Vahid, on the subject “Contemporary trends of architectural geometry. Helix and its implementation in architecture.” is written completely by myself; all the provided data and research results are correct and have been obtained honestly. None of the parts of this thesis have been plagiarized from any printed, Internet-based or otherwise recorded sources. All direct and indirect quotations from external resources are indicated in the list of references. No monetary funds (unless required by law) have been paid to anyone for any contribution to this thesis. I fully and completely understand that any discovery of any manifestations/case/facts of dishonesty inevitably results in me incurring a penalty according to the procedure(s) effective at Kaunas University of Technology. -
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N PS Form 10-900 OMB No. 1024-0018 United States Department of the Interior RECEIVED 2280 National Park Service National Register of Historic Places Registration Form N-'~lG!SlERTH!ST6RIC PLACES This form is for use in nominating or requesting determinations for individual properties and districts. See instructions in How to Complete the National Register of Historic Places Registration Form (National Register Bulletin 16A). Complete each item by marking V in the appropriate box or by entering the information requested. If an item does not apply to the property being nominated, enter "N/A" for "not applicable." For functions, architectural classification, materials, and areas of significance, enter only categories and subcategories from the instructions. Place additional entries and narrative items on continuation sheets (NPS Form 10-900a). Use a typewriter, word processor, or computer, to complete all items. 1. Name of Property Catalina American Baptist Church historic name other name/site number Catalina Baptist Church 2. Location street & number: 1900 North Country Ciub Road _not for publication city/town: Tucson __ vicinity state: Arizona code: AZ county: Pima code: 019 zip code: 85716 3. State/Federal Agency Certification As the designated authority under the National Historic Preservation Act, as amended, I hereby certify that this^STnomination D request for determination of eligibility meets the documentation standards for registering properties in the National Register of Historic Places and meets the procedural and professional requirements set forth in 36 CFR Part 60. In my opinion, the propertybLmeets Q does not meet the National Register criteria. I recommend that this property be considered significant D nationally D statewidtjxcf locally. -
Luce Chapel.Pdf
1 Luce Chapel is a renowned architecture in Taiwan. With its outstanding achievements, it certainly stands outin the modern architectural movement of Preface post-war Taiwan. In October 2014, Luce Chapel was chosen to be one of the ten global classic modern architectures, and the first project within Asian architecture,which received the first “Keeping It Modern” (hereafter abbreviated as KIM) Grant from the Getty Foundation. The Grant acknowledgesthese 20th century modern architectures as milestones of human civilization. With high experimental mentalities, groups of architects and engineersof the previous centuryboldlytried out exploratory materials and cutting edge construction techniques, and built innovative architectures thathave stimulated changes in their surrounding environments, histories, local culture, and forever transformed the philosophical approaches of architecture. However, the Getty Foundation also regards these architectures to be under various degrees of risks. Being fifty to sixty or even older, many of these innovative materials and techniques boldly used at the time oftheirsconstruction were not, and still have not been scientifically tested and analyzed to this very day. Furthermore, the productions of many of these materials have been discontinued due to low adoptions in the market, making conservations even more difficult. Therefore, the Getty Foundation KIM grants promote the sustainable conservation of modern architecture. This focus has also been the core value of the Luce Chapel conservation project. Built in 1963, Luce Chapel has stood on the campus of Tunghai University for over 50 years. This building was designed and built to function as a church building, and has maintained its religious purpose over the years.However, as the number of faculty and students continues to grow,the space demand for community engagements and ceremonial activities of colleges and departments on campus have also increased extensively. -
Design of Hyperboloid Structures
MOJ Civil Engineering Research Article Open Access Design of hyperboloid structures Abstract Volume 3 Issue 6 - 2017 In this paper, we present the design of hyperboloid structures and techniques for hyperboloid fitting which are based on minimizing the sum of the squares of the geometric distances Sebahattin Bektas Faculty of Engineering, Ondokuz Mayis University, Turkey between the noisy data and the hyperboloid. The literature often uses “orthogonal fitting” in place of “best fitting”. For many different purposes, the best-fit hyperboloid fitting to Correspondence: Sebahattin Bektas, Ondokuz Mayis a set of points is required. Algebraic fitting methods solve the linear least squares (LS) University, Faculty of Engineering, Geomatics Engineering, 55139 problem, and are relatively straightforward and fast. Fitting orthogonal hyperboloid is Samsun, Turkey, Email a difficult issue. Usually, it is impossible to reach a solution with classic LS algorithms. Because they are often faced with the problem of convergence. Therefore, it is necessary to Received: October 11, 2017 | Published: December 27, 2017 use special algorithms e.g. nonlinear least square algorithms. We propose to use geometric fitting as opposed to algebraic fitting. Hyperboloid has a complex geometry as well as hyperboloid structures have always been interested. The two main reasons, apart from aesthetic considerations, are strength and efficiency. Keywords: hyperboloid structure, design of hyperboloid, orthogonal fitting, algebraic fitting, nonlinear least square problem Introduction When designing these cooling towers, engineers are faced with two problems: The industrial manufacturing is making more widespread use of three-dimensional object recognition techniques. The shapes i. The structure must be able to withstand high winds and of many 3D objects can be described using some basic surface ii. -
Canton Tower MARQUESA FIGUEROA, ERIC HENDERSON,, ANDREW ILGES, ASHLEY JOHNSTON, MIRAY OKTEM Overview
Canton Tower MARQUESA FIGUEROA, ERIC HENDERSON,, ANDREW ILGES, ASHLEY JOHNSTON, MIRAY OKTEM Overview Location : Guangdong, China Height : 600 Meters Project Area : 114,000 sqm Project year : 2010 Architect : Mark Hemel and Barbara Kuit are the founding partners of Information Based Architecture. Engineer : ARUP Premise . International competition held in 2004 . Program included : the tower, park at its base and the master-plan which includes Plaza, pagoda-park, retail facilities, offices, television centre and hotel . Competition ran from April to August 2004 with 13 different proposals . February 2005 finally choose Information Based Architecture’s Design Design Concept . Wanted to create a ‘female’ tower, being dynamic, transparent, curvy, gracious and sexy . Free-form tower with a rich and human-like identity nicknaming it 'the supermodel’ . The non-symmetrical form portraying the building 'in movement' represents Guangzhou as a vibrant and exciting city Design Concept . Tightest waist as possible . Twisted waist shape inspired by the east ward turning of Pearl River . Eccentric core placement aligns the mast with the new central axis of the city . Creates dynamic views of the north and south banks of the river Building Layout Five functional areas . Section A : Two floors lobby, exhibition, public service, connections to underground . Section B : 4d cinema . Section C : leisure places observation hall snack shop Hollow space with spiral staircase . Section D : tea rooms, resting places . Section E : restaurants and damping floor open terrace observation square Free fall ride on antenna Four hollow vertical parts 37 occupied functional floors Innovative Features . Triangular glazed panels . Vertical transportation system . Damping control system . Structural health monitor system- five key cross sections monitored in real time . -
Arxiv:1601.04786V1 [Math.MG] 19 Jan 2016 Much of the Notation Will Look Similar, but Mean Very Different Things
HAUSDORFF DIMENSION OF GENERALIZED FIBONACCI WORD FRACTALS TYLER HOFFMAN AND BENJAMIN STEINHURST Abstract. Fibonacci word fractals are a class of fractals that have been stud- ied recently, though the word they are generated from is more widely studied in combinatorics. The Fibonacci word can be used to draw a curve which pos- sesses self-similarities determined by the recursive structure of the word. The Hausdorff dimension of the scaling limit of the finite Fibonacci word curves is computed for i-Fibonacci curves and any drawing angle between 0 and π=2. 1. Introduction Motivated by the work of Monnerot-Dumaine [7], we analyze a generalized class of Fibonacci fractals for their Hausdorff dimension. The two parameters we consider are the drawing angle as introduced by Monnerot-Dumaine and the use of the i−Fibonacci words as introduced by Ram´ırez et al [10, 11, 12]. The necessity of this analysis is that the dimensions estimated in the previous works are for the self-similarity dimension despite occasional references to Hausdorff dimension. The equality of these two dimensions often holds but is not automatic. Equality holds in the familiar case when there exists an iterated function system satisfying the open set condition whose attractor is the fractal in question. Many of the already mentioned papers and those of Blondin-Mass´e[2] also include a study of the Fibonacci snowflake fractal which we do not consider in the present paper. The basic construction of the Fibonacci fractals is based on three operations: one is a recursive construction of a string of symbols, the second is the processing of a finite string into a curve, and the third is taking a scaling limit of the finite curves to a fractal limit. -
The Mote in God's
The Mote In God’s Eye Larry Niven Pocket Books ISBN: 0-6717-4192-6 Contents Prologue PART ONE - A.D. 3017 - THE CRAZY EDDIE PROBE 1 Command 2 The Passengers 3 Dinner Party 4 Priority OC 5 The Face of God 6 The Light Sail 7 The Crazy Eddie Probe 8 The Alien 9 His Highness Has Decided 10 The Planet Killer 11 The Church of Him 12 Descent into Hell PART TWO - THE CRAZY EDDIE POINT 13 Look Around You 14 The Engineer 15 Work 16 Idiot Savant 17 Mr. Crawford’s Eviction 18 The Stone Beehive 19 Channel Two’s Popularity 20 Night Watch 21 The Ambassadors 22 Word Games 23 Eliza Crossing the Ice 24 Brownies 25 The Captain’s Motie PART THREE - MEET CRAZY EDDIE 26 Mote Prime 27 The Guided Tour 28 Kaffee Klatsch 29 Watchmakers 30 Nightmare 31 Defeat 32 Lenin 33 Planetfall 34 Trespassers 35 Run Rabbit Run 36 Judgment 37 History Lesson 38 Final Solution PART FOUR - CRAZY EDDIE’S ANSWER 39 Departure 40 Farewell 41 Gift Ship 42 A Bag of Broken Glass 43 Trader’s Lament 44 Council of War 45 The Crazy Eddie Jump 46 Personal and Urgent 47 Homeward Bound 48 Civilian 49 Parades 50 The Art of Negotiation 51After the Ball Is Over 52 Options 53 The Djinn 54 Out of the Bottle 55 Renner’s Hole Card 56 Last Hope 57 All the Skills of Treason Prologue “Throughout the past thousand years of history it has been traditional to regard the Alderson Drive as an unmixed blessing.