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Gray Is 25C25m25y25k gray is 25c25m25y25k Bridges Leeuwarden Mathematics, Music, Art, Architecture, Culture 2008 Celebrating the 110th Birth Year of M.C. Escher During the Eleventh Annual Bridges Conference BRIDGES LEEUWARDEN CHN University Netherlands Leeuwarden, the Netherlands Municipality of Leeuwarden The Netherlands The Foundation Eindhoven University of Passages University of Technology Twente Mathematical Connections in Art, Music, and Science Proceedings 2008 Reza Sarhangi and Carlo Séquin, Chief Editors Tarquin Publications Bridges Leeuwarden, Mathematics, Music, Art, Architecture, Culture Conference Proceedings 2008 Editors: Reza Sarhangi Department of Mathematics Towson University, Towson, Maryland, USA Carlo H. Séquin EECS Computer Science Division University of California, Berkeley, USA © Bridges Leeuwarden Conference. ( http://www.bridgesmathart.org ). All rights reserved. General permission is granted to the public for non-commercial reproduction, in limited quantities, of individual articles, provided authorization is obtained from individual authors and a complete reference is given for the source. All copyrights and responsibilities for individual articles in the 2008 Conference Proceedings remain under the control of the original authors. ISBN: 9780966520194 ISSN: 1099-6702 Printed in the UK by InType Libra Distributed by MATHARTFUN.COM (http://mathartfun.com) and Tarquin Books (www.tarquinbooks.com) Front cover: Day and Night by M.C. Escher, 1938, © The M.C. Escher Foundation. Used with permission. All the Escher images used in the 2008 Bridges Proceedings are published with the kind permission of the M.C. Escher Foundation – Baarn – The Netherlands. Back cover: Trefoil Knot by Karel Vreeburg; No. 1, Type 1, 1 ,1, 1 (a) by David Bailey; Brainstrain 1, Atom and Eve 2 by Dick Termes; Polyhedra de los Leones by Briony Thomas; Echinodermania 2 by George Hart. Cover design by Jeffrey Rutzky. Bridges Leeuwarden Scientific Organizers Reza Sarhangi Rinus Roelofs Department of Mathematics The Foundation Passages Towson University Hengelo, the Netherlands Towson, Maryland, USA Arno Pronk Zsófia Ruttkay Department of Architecture, Department of Computer Science Building and Planning University of Twente Eindhoven University of the Netherlands Technology, the Netherlands Local Organizer Cor Wetting The City of Leeuwarden The Foundation Passages Leeuwarden, the Netherlands Bridges for Teachers, Teachers for Bridges Mara Alagic Paul Gailiunas Department of Curriculum and Instruction Newcastle, England Wichita State University Wichita, Kansas, USA Bridges Visual Art Exhibit Ann Burns Nat Friedman Robert W. Fathauer Department of Mathematics, Long Department of Mathematics and Tessellations Company Island University, New York, USA Statistics, University at Albany Phoenix, Arizona, USA New York, Albany, USA Conference Website and Electronic Correspondence George W. Hart Craig Kaplan Department of Computer Science David R. Cheriton School of Computer Stony Brook University, New York, USA Science, University of Waterloo, Canada Conference Board of Advisors Craig Kaplan George W. Hart Carlo H. Séquin David R. Cheriton School of Department of Computer EECS, Computer Science Computer Science, University Science University of California, of Waterloo, Canada Stony Brook University Berkeley, USA New York, USA Chief Editors: Reza Sarhangi Carlo H. Séquin Department of Mathematics EECS Computer Science Division Towson University, Towson, Maryland, USA University of California, Berkeley, USA Program Committee Mara Alagic Javier Barrallo Ann Burns Department of Curriculum School of Architecture Department of Mathematics, and Instruction The University of the Basque Long Island University, New Wichita State University Country San Sebastian, Spain York, USA Wichita, Kansas, USA Paul Gailiunas Francisco Gómez George W. Hart Newcastle, England Escuela U. de Informática Department of Computer Science U. Politécnica de Madrid, Spain Stony Brook University New York, USA Craig Kaplan Douglas McKenna Robert V. Moody Cheriton School of Computer Mathemaesthetics Inc. Mathematics Department Science, University of Waterloo, Boulder, Colorado, USA University of Victoria Canada British Columbia, Canada Rinus Roelofs Reza Sarhangi Carlo H. Séquin, (Chair) The Foundation Passages Department of Mathematics Computer Science Division Hengelo, the Netherlands Towson University University of California Towson, Maryland, USA Berkeley, USA John Sharp Godfried T. Toussaint London Knowledge Lab John M Sullivan School of Computer Science Institute of Education Technische Universität Berlin Schulich School of Music University of London, UK Berlin, Germany McGill University, Canada Contents Preface Lessons in Duality and Symmetry from M.C. Escher 1 Doris Schattschneider The Sculpture Manifold: A Band from a Surface, a Surface from a Band 9 Bernd Krauskopf, Hinke M. Osinga, and Benjamin Storch Color Mixing using Colliding Particles 15 Gary R. Greenfield Hankin’s ‘Polygons in Contact’ Grid Method for Recreating a Decagonal Star Polygon Design 21 B. Lynn Bodner Connected Holes 29 Rinus Roelofs Metamorphosis in Escher’s Art 39 Craig S. Kaplan The Catenary: Art, Architecture, History, and Mathematics 47 Gail Kaplan Mathematical Beauty in Architecture 55 Huib Koman, Stephan Luijks, and Arno Pronk The Borromean Rings: A Video about the New IMU Logo 63 Charles Gunn and John M. Sullivan Systematic Approaches to Color Interaction: Limited Palettes for Simultaneous Contrast Effects 71 James Mai Building a Möbius Bracelet Using Safety Pins: A Problem of Modular Arithmetic and Staggered Positions 79 Eva Knoll The Brachistochrone Problem between Euclidean and Hyperbolic 87 Robert Smits Sculpting from the Inside Out “540 Hidden Split Torus” 93 Karel Vreeburg Patterning by Projection:Tiling the Dodecahedron and other Solids 101 B.G. Thomas and M.A. Hann Pull-up Patterned Polyhedra: Platonic Solids for the Classroom 109 E.B. Meenan and B.G. Thomas From Sierpinski Triangle to Fractal Flowers 117 Anne M. Burns Designing Symmetric Peano Curve Tiling Patterns with Escher-esque Foreground/Background Ambiguity 123 Douglas McKenna Doyle Spiral Circle Packings Animated 131 Alan Sutcliffe Intricate Isohedral Tilings of 3D Euclidean Space 139 Carlo H. Séquin The Mathematics of the Channel Anamorphosis 149 James L. Hunt and John Sharp A Sangaku Revived 155 Zsófia Ruttkay Inpainting of Ancient Austrian Frescoes 163 Wolfgang Baatz, Massimo Fornasier, Peter A. Markowich, and Carola-Bibiane Schönlieb A Computer Aided Geometric Model of a Ten-Plane Polyhedral Transformation 171 Robert McDermott and Will Hawkins Neo-Riemannian Geometry 179 Rachel W. Hall A Comparative Phylogenetic-Tree Analysis of African Timelines and North Indian Talas 187 Eric Thul and Godfried T. Toussaint The Maths of Churches, Mosques, Synagogues and Temples 195 Jenny Gage Interactive 3D Simulation of Escher-like Impossible Worlds 201 E.M. Orbons and Zs. Ruttkay Procedural Generation of Sculptural Forms 209 George W. Hart Counting the Number of Site Swap Juggling Patterns with Respect to Particular Ceilings 219 Carl Bracken The Mathematics of Mitering and its Artful Application 225 Tom Verhoeff and Koos Verhoeff Connecting the Dots: The Ins and Outs of TSP Art 235 Robert Bosch Manipulating Images with Fractal Julia Sets 243 Stanley Spencer Making Patterns on the Surfaces of Swing-Hinged Dissections 251 Reza Sarhangi Edge-Based Intersected Polyhedral Paper Sculptures Constructed by Interlocking Slitted Planar Pieces 259 Jace Miller and Ergun Akleman Diffusion Processes and Light Installations: Mathematics, Visualisation, and Perception 265 Mike Kostner, Arjan Kuijper, and Franz Schubert Interactive Demonstrations with Mathematica 6 273 Sándor Kabai A Closer Look at Jamnitzer's Polyhedra 281 Albert van den Broeke and Zsófia Ruttkay A Fractal Crystal Comprised of Cubes and Some Related Fractal Arrangements of other Platonic Solids 289 Robert W. Fathauer, Hank Kaczmarski, and Nicholas Duchnowski Medieval Islamic Architecture, Quasicrystals, and Penrose and Girih Tiles: Questions from the Classroom 297 Raymond Tennant Recursion, Symmetry & Tessellation with Software based Video Feedback Systems 305 Paul Prudence Coxeter Groups in Colored Tilings and Patterns 311 Glenn R. Laigo, Ia Kristine D. Puzon, and Ma. Louise Antonette N. De Las Peñas Porter’s Golden Section, Experimentally 319 Chris Bartlett and D Huylebrouck Perspective Drawings of Lattices 327 Chaim Goodman-Strauss and Hop David Some Interesting Observations Regarding the Spidrons 333 Dániel Erdély and Walt van Ballegooijen A “Circle Limit III” Backbone Arc Formula 341 Douglas Dunham and Luns Tee Connecting Mathematics and the Arts through the Magic of Escher for Elementary School Students 347 Marilyn Sue Ford and Sarae Gold Module-Based Sculptural Constructions 353 Roland de Jong Orlando Painting the Total Picture 363 Dick Termes Cosmati Pavements: The Art of Geometry 369 Tristram de Piro Teaching Group Theory Using Portraits of Groups 377 Jay Zimmerman Four Bar Linkages 381 Paul Gailiunas The Tangramoid: Recent Developments 385 Samuel Verbiese TENSEGRITIES: Design, Analysis and Constructing 389 Jan W. Marcus Pattern by Design – Not by Chance 393 M.A. Hann and B.G. Thomas Using Binary Numbers in Music 397 Vi Hart Image Stitching: From Mathematics to Arts 401 Franz Schubert, Arjan Kuijper, and Mike Kostner Can Geometry Create Art? 405 András Bezdek Masterpiece: Interactive Evolution in Visual Domain 409 Artemis Moroni, Rafael Bocaletto Maiolla, and Jônatas Manzolli The Art of Equations
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