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Bridges Donostia Mathematics, Music, Art, Architecture, Culture 2007 Celebrate the Tenth Annual Bridges Conference BRIDGES DONOSTIA Mathematics, Music, Art, Architecture, Culture http://www.BridgesMathArt.Org School of Architecture The University of the Basque Country San Sebastian (Donostia), Spain Conference Proceedings 2007 Reza Sarhangi and Javier Barrallo, Editors Tarquin publications Bridges Donostia Scientific Organizers Javier Barrallo Reza Sarhangi School of Architecture Department of Mathematics The University of the Basque Country Towson University San Sebastian, Spain Towson, Maryland, USA Local Organizers Angel Fernandez Luis Martin Department of Mathematics Department of Mathematics School of Architecture School of Architecture The University of the Basque Country, Spain The University of the Basque Country, Spain Santiago Sanchez Alberto Zulueta Department of Physics Department of Physics School of Architecture School of Architecture The University of the Basque Country, Spain The University of the Basque Country, Spain 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 Robert W. Fathauer Ann Burns Nat Friedman Department of Mathematics, Long Department of Mathematics and Tessellations Company Phoenix, Arizona, USA Island University, New York, USA Statistics, University at Albany 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 Advisory Nat Friedman Carlo Séquin John Sharp Department of Mathematics and Computer Science Division, London Knowledge Lab Statistics, University at Albany EECS Department, University Institute of Education New York, Albany, USA of California, Berkeley, USA University of London, UK Bridges Donostia, Mathematics, Music, Art, Architecture, Culture Conference Proceedings, 2007 Editors: Reza Sarhangi Department of Mathematics Towson University Towson, Maryland, USA Javier Barrallo School of Architecture The University of the Basque Country San Sebastian, Spain © Bridges Donostia 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 2007 Conference Proceedings remain under the control of the original authors. ISBN: 0-9665201-8-1 ISSN: 1099-6702 Printed in the UK by Print Solutions Partnership Distributed by MATHARTFUN.COM (http://mathartfun.com) and Tarquin Books (www.tarquinbooks.com) Cover design: Imaginary Garden by Anne M. Burns Logo: A (7, 2) star polygon based on the Buzjani’s approximation of heptagon by Reza Sarhangi and Robert Fathauer Cover layout: Jeffrey Rutzky CDROM: Chris K. Palmer Contents Preface xiii Modular Kirigami 1 George W. Hart Some Monohedral Tilings Derived From Regular Polygons 9 Paul Gailiunas Composite Diffusion Limited Aggregation Paintings 15 Gary R. Greenfield Symmetry and Structure in Twist-Hinged Dissections of Polygonal Rings 21 and Polygonal Anti-Rings Greg N. Frederickson Imaginary Gardens – A Model for Imitating Plant Growth 29 Anne M. Burns Allahverdi Khan Bridge (Si-O-Seh Pol) of Esfahan 37 An Example of Art and Mathematics Hourieh Mashayekh and Hayedeh Mashayekh Light, Movement and 3D – Light images Viewed as Photographs 39 Jack Tait Spiral Developable Sculptures of Ilhan Koman 47 Tevfik Akgün, Irfan Kaya, Ahmet Koman, and Ergun Akleman “Gödel, Escher, Bach”, in other Eras 53 Dirk Huylebrouck Mathematics and Symmetry: A Bridge to Understanding 59 Gail Kaplan A Proposal for the Classification of Mathematical Sculpture 67 Ricardo Zalaya Báez The Spirograph and Beyond 75 Susan McBurney Entwined Circular Rings 81 Rinus Roelofs 2D and 3D Animation Using Rotations of a Jordan Curve 91 Peter Hamburger, Edit Hepp, and Richard Wartell Mathematical Models for Binarization and Ternarization of Musical Rhythms 99 Francisco G´omez, Imad Khoury, J¨org Kienzle, Erin McLeish, Andrew Melvin, Rolando P´erez-Fern´andez, David Rappaport, and Godfried Toussaint vii Portraits of Groups II, Orientation Reversing Actions 109 Jay Zimmerman Baskets for the Mathematics Classroom 115 S. Louise Gould The Pentagram: From the Goddess to Symplectic Geometry 123 Elisa Prato The Modular Color Palette: Systems of Color Selection in the Paintings of James Mai 127 James L. Mai A “Sound” Approach to Fourier Transforms: Using Music to Teach Trigonometry 135 Bruce Kessler Magritte: Analogies in Mathematical Reasoning 143 Rozhkovskaya Natasha Zany Projects – The Art of Mixing Compass with Computer 147 Rebecca Kessler Breaking Color Symmetry 149 Carla Farsi Addled Tangles of Sanguine Language—an Eclectic Syncretic Syntactic Taxonomy 151 Benjamin Wells Structure and Form in the Design Curriculum 161 M.A. Hann and B.G. Thomas Electrostatic Patterns in the Interior of a Circular Region 169 N.G. Nicolis Ricochet Compositions 177 I.A. de Kok, T. Lucassen, and Zs. Ruttkay Golden Fractal Trees 181 T. D. Taylor Anticlastic Form – Manifesting Fields of Tension 189 Benjamin Storch Patterned Polyhedra: Tiling the Platonic Solids 195 B.G. Thomas and M.A. Hann Frieze Patterns of the Alhambra 203 B. Lynn Bodner Modeling D-Forms 209 Özgür Gönen, Ergun Akleman, and Vinod Srinivasan viii A Simple Procedure to Generate Curves and Surfaces 217 Alan Sutcliffe The 7 Curve, Carpets, Quilts, and Other Asymmetric, Square-Filling, 225 Threaded Tile Designs Douglas McKenna Geometric Constructions and their Arts in Historical Perspective 233 Reza Sarhangi The Ideal Vacuum: Visual Metaphors for Algebraic Concepts 241 Jessica K. Sklar Inout Sculptures 247 Yutu Liu, Hernan Molina, and Ergun Akleman Painting by the Numbers: A Porter Postscript 253 Chris Bartlett Poverty and Polyphony: A Connection between Economics and Music 259 Rachel W. Hall and Dmitri Tymoczko From Modeling Foliage with L-systems to Digital Art 269 Glyn M. Rimmington and Mara Alagic Does it Look Square? Hexagonal Bipyramids, Triangular Antiprismoids, and their Fractals 277 Hideki Tsuiki Shape, Time and Chemistry: Some Platonic Meditations 287 Farzad Mahootian Shiva: Two Views of Burnside’s Lemma at Work 289 James Mai and Daylene Zielinski Modeling High Genus Sculptures Using Multi-Connected Handles and Holes 297 Vinod Srinivasan, Hernan Molina, and Ergun Akleman Revisiting the Geometry of the Sala de Dos Hermanas 303 Ann Robertson Ancient Harmonic Law 311 Jay Kappraff The Effect of Human Experience on Formal Word Meaning 313 Russell Jay Hendel Transgenic Visual-and-Sound Compositions 315 Artemis Moroni, Rafael Bocaletto Maiolla, and Jônatas Manzolli Geometry and New Urban Order 323 Cristina Argumedo, Mª Francisca Blanco, Dora Giordano, and Miriam Pisonero ix The Power and Potential of Art in Literature to Teach Mathematics 331 William P. Bintz and Sara Delano Moore Planar Symmetry with Turtles 333 James Dean Palmer Fractal Knots Created by Iterative Substitution 335 Robert W. Fathauer Sculptures which Stellarize Non-Planar Hexagons 343 Douglas G. Burkholder Edge-Constrained Tile Mosaics 351 Robert Bosch Hyperbolic Semi-Regular Tilings and their Symmetry Properties 361 Ma. Louise Antonette N. De Las Peñas, Glenn R. Laigo, and Eden Delight B. Provido Fractal Art: Closer to Heaven? 369 Modern Mathematics, the art of Nature, and the nature of Art Charalampos Saitis Images of the Ammann-Beenker Tiling 377 Edmund Harriss Symmetric Embedding of Locally Regular Hyperbolic Tilings 379 Carlo H. Séquin When is a picture not a picture? What is really in a Random Tandem? 389 Simon Bexfield A “Circle Limit III” Calculation 395 Douglas Dunham Designing a Modern Tower in a Mathematically–Based World 403 Zafer Sagdic and Barbaros Sagdic Amazing Labyrinths 405 Samuel Verbiese The Automorphism of Amalgamation Polytopes and Tessellation 413 Lin Hsin Hsin Geometrical Transformation: A Method for the Creation of Form in Contemporary Architecture 415 Ülkü İnceköse x Bridges for Teachers, Teachers for Bridges Understanding Math via Arts, Creating Arts via Math 423 Mara Alagic and Paul Gailiunas Imaginative Quilted Geometric Assemblages 425 Elaine Krajenke Ellison The Geometry of Asian Trousers 427 Penelope Woolfitt Math/Art Projects 431 Ann Hanson Building Models to Transition from Dimension to Dimension 433 Robert McDermott Exploring Cubes Woven on the Skew 441 Felicity Wood Using Art to Teach Maths, Using Maths to Create Art 445 Julie Dobson and Jenny Gage From Folding and Cutting to Geometry and Algorithms: Integrating Islamic Art into the Mathematics Curriculum 453 Carol Bier Zome Workshop 459 Paul Hildebrandt Index 465 xi xii Preface Celebrate with us this tenth year of Bridges! The conference began in 1998 and has evolved into the premier annual event for the interdisciplinary study of mathematics and the arts. You are holding the tenth in the series of annual proceedings volumes recording the diversity, richness, and depth of ideas presented by conference participants. For many artists, mathematicians, computer scientists, and educators, the Bridges Conference is an essential annual excursion to meet with like-minded multidisciplinarians and recharge one’s creative batteries. The talks, workshops,
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