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Bridges 2021 Mathematics・Art・Music・Architecture・Culture Conference Proceedings Bridges 2021 Mathematics・Art・Music・Architecture・Culture Conference Proceedings Editors Program Chair David Swart Miovision, Inc. Waterloo, Ontario, Canada Short Papers Chair Workshop Papers Chair Frank Farris Eve Torrence Santa Clara University Mathematics Department Santa Clara, California, USA Randolph-Macon College Ashland, Virginia, USA Production Chair Craig S. Kaplan Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, Canada Bridges 2021 Conference Proceedings (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 2021 Conference Proceedings remain under the control of the original authors. ISBN: 978-1-938664-39-7 ISSN: 1099-6702 Published by Tessellations Publishing, Phoenix, Arizona, USA (© 2021 Tessellations) Distributed by MathArtFun.com (mathartfun.com). Cover design: Uyen Nguyen, New York City, New York, USA. Image credits: David Swart; H. A. Verrill; Anduriel Widmark; Jin Yamauchi and Chamberlain Fong; Robert Bosch and Zejian Huang; Loe Feijs; Carlo H. Séquin; Roger Antonsen and Laura Taalman; Stefan Pautze; Chirag Mehta; Ulrich Reitenbuch, Martin Skrodzki, and Konrad Polthier; Frank A. Farris; John Berglund and Craig S. Kaplan. Bridges Board of Directors Susan Goldstine George Hart Department of Mathematics and Computer Wiarton, Ontario, Canada Science St. Mary’s College of Maryland St. Mary’s City, Maryland, USA Craig S. Kaplan Carlo H. Séquin Cheriton School of Computer Science Computer Science Division University of Waterloo University of California, Berkeley Waterloo, Ontario, Canada Berkeley, California, USA Sujan Shrestha Eve Torrence Science, Information Arts & Technologies Department of Mathematics University of Baltimore Randolph-Macon College Baltimore, Maryland, USA Ashland, Virginia, USA Area Coordinators Steve Abbott Robert Fathauer Department of Mathematics Tessellations Company Middlebury College Phoenix, Arizona, USA Vermont, USA Art Exhibition Theater Event Kristóf Fenyvesi Sarah Glaz Finnish Institute for Educational Research Department of Mathematics University of Jyväskylä The University of Connecticut Jyväskylä, Finland Storrs, Connecticut, USA Family Day Poetry Reading Tiffany Inglis Nathan Selikoff D2L Digital Awakening Studios Waterloo, Ontario, Canada Orlando, Florida, USA Technical Support Technical Support Bruce Torrence Bianca Violet Randolph-Macon College IMAGINARY Ashland, Virginia, USA Berlin, Germany Art Exhibition Short Film Festival Proceedings Program Committee Steve Abbott Kazushi Ahara Abdalla G. M. Ahmed Middlebury College Meiji University Khartoum, Sudan Vermont, USA Tokyo, Japan Marco Aldi Roger Antonsen António Araújo Virginia Commonwealth University of Oslo Universidade Aberta University Oslo, Norway Lisbon, Portugal Richmond, Virginia, USA Ellie Baker Robert Bosch Christopher Brownell Cambridge, Massachusetts, USA Oberlin College Fresno Pacific University Ohio, USA California, USA Doug Burkholder Stephen M. Campbell Andrea Capozucca Lenoir-Rhyne University The Puzzle Factory Ltd University of Camerino Hickory, North Carolina, USA Lancashire, United Kingdom Camerino, Italy Christopher Carlson Kelly Delp Neil Dodgson Wolfram Research Cornell University Victoria University of Wellington Champaign, Illinois, USA Ithaca, New York, USA New Zealand Doug Dunham Frank Farris Robert Fathauer University of Minnesota Santa Clara University Tessellations Company Duluth, USA California, USA Phoenix, Arizona, USA Loe Feijs Kristóf Fenyvesi Chamberlain Fong Eindhoven University of University of Jyväskylä San Francisco, California, USA Technology Finland The Netherlands Paul Gailiunas Susan Gerofsky Sarah Glaz Newcastle, England University of British Columbia The University of Connecticut Vancouver, Canada Storrs, Connecticut, USA Susan Goldstine Chaim Goodman-Strauss Louise Gould St. Mary’s College of Maryland University of Arkansas Central Connecticut State St. Mary’s City, Maryland, USA Fayetteville, Arkansas, USA University New Britain, Connecticut, USA Emily Grosholz Richard Hammack Edmund Harriss Pennsylvania State University Virginia Commonwealth University of Arkansas University Park, Pennsylvania, University Fayetteville, Arkansas, USA USA Richmond, Virginia, USA George Hart Andrea Hawksley Judy Holdener Wiarton, Ontario, Canada San Francisco, California, USA Kenyon College Ohio, USA Guy Inchbald Tiffany Inglis Veronika Irvine Worcestershire, England D2L tesselace.com Waterloo, Ontario, Canada Sudbury, Ontario, Canada Craig S. Kaplan Karl Kattchee Toni Kotnik University of Waterloo University of Wisconsin Aalto University Waterloo, Ontario, Canada La Crosse, USA Helsinki, Finland Alice Major Vincent J. Matsko Elisabetta Matsumoto Edmonton, Alberta, Canada St. Petersburg, Florida, USA Georgia Institute of Technology Atlanta, Georgia, USA Dan May Douglas McKenna Kerry Mitchell Black Hills State University Mathemæsthetics, Inc. Phoenix, Arizona, USA Spearfish, South Dakota, USA Boulder, Colorado, USA Mike Naylor Doug Norton Kirsi Peltonen Matematikkbølgen Villanova University Aalto University Math Creativity and Competency Pennsylvania, USA Helsinki, Finland Center Vanvikan, Norway Ulrich Reitebuch Rinus Roelofs Radmila Sazdanovic Freie Universität Berlin Hengelo, The Netherlands North Carolina State University Germany Raleigh, North Carolina, USA Karl Schaffer Katherine Seaton Henry Segerman De Anza College and La Trobe University Oklahoma State University MoveSpeakSpin Melbourne, Australia Stillwater, Oklahoma, USA Scotts Valley, California Carlo H. Séquin Sujan Shrestha Donald Spector University of California, Berkeley University of Baltimore Hobart & William Smith Colleges USA Maryland, USA Geneva, New York, USA Catherina Steyn David Swart Laura Taalman Nelson Mandela University Waterloo, Ontario, Canada James Madison University Port Elizabeth, South Africa Harrisonburg, Virginia, USA Tara Taylor Briony Thomas Bruce Torrence St. Francis Xavier University University of Leeds Randolph-Macon College Antigonish, Nova Scotia, Canada England Ashland, Virginia, USA Eve Torrence Eva Ulbrich Walt van Ballegooijen Randolph-Macon College Johannes Kepler University Wijk en Aalburg, The Netherlands Ashland, Virginia, USA Linz, Austria Tom Verhoeff Charles Wampler Phil Webster Eindhoven University of General Motors Research and Phil Webster Design Technology Development Chandler, Arizona, USA The Netherlands Warren, Michigan, USA Jiangmei Wu Carolyn Yackel Indiana University Bloomington Mercer University Indiana, USA Macon, Georgia, USA Art Exhibition and Catalog Program Committee Robert Fathauer Bruce Torrence Tessellations Company Randolph-Macon College Phoenix, Arizona, USA Ashland, Virginia, USA Co-curator Co-curator Katie McCallum Conan Chadbourne University of Brighton San Antonio, Texas, USA Brighton, England, UK Catalog design Jury member Nathan Selikoff Taneli Luotoniemi Digital Awakening Studios Aalto Math&Arts Orlando, Florida, USA Aalto University Technical Support Espoo, Finland Local coordinator Short Film Festival Program Committee Susan Gerofsky Henry Segerman University of British Columbia Oklahoma State University Vancouver, Canada Stillwater, Oklahoma, USA Bianca Violet IMAGINARY Berlin, Germany Chair Contents Preface ............................................................................... xiv Regular Papers Creativity and Rigor: A Bead Crochet Mathematics Course ..................................1 Eve Torrence The Joy of Polar Zonohedra ..............................................................7 George Hart Structured Knight’s Tours ...............................................................15 Robert Bosch and Zejian Huang Animated Map Colourings of Hinged Squares .............................................23 Craig S. Kaplan Beyond the Great 96 ....................................................................31 John Berglund and Craig S. Kaplan Mathematics in the Poetry of Sefer Yetzirah ...............................................39 Sarah Glaz The Flat Klein Bottle Rendered in Curved-Crease Origami .................................47 Stepan Paul Logarithmic Spiral Tilings of Triangles ...................................................55 Robert Fathauer Polyhedral-Edge Knots .................................................................63 Carlo H. Séquin Wallpaper Patterns from Looping Strands: The Layer Groups ...............................71 Frank Farris Dancing Topologically ..................................................................79 Karl Schaffer Categorizing Celtic Knot Designs ........................................................87 Roger Antonsen and Laura Taalman viii Approximating Logarithmic Spirals by Quarter Circles .....................................95 Ulrich Reitebuch, Martin Skrodzki, and Konrad Polthier Invisible Forces: Baskets without Corners ...............................................103 Paul Gailiunas Pretty 3D Polygons: Exploration and Proofs .............................................111 Melissa van Veenendaal and Tom Verhoeff Conjunction-forms: Three-Circle Combinations ..........................................119 James Mai The Short Tiles Category ...............................................................127 Lars Eriksson Quadrilateral Spiral Tilings and Escheresque Art ........................................135
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