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Ununfoldable Polyhedra with Convex Faces
Ununfoldable Polyhedra with Convex Faces Marshall Bern¤ Erik D. Demainey David Eppsteinz Eric Kuox Andrea Mantler{ Jack Snoeyink{ k Abstract Unfolding a convex polyhedron into a simple planar polygon is a well-studied prob- lem. In this paper, we study the limits of unfoldability by studying nonconvex poly- hedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that \open" polyhedra with triangular faces may not be unfoldable no matter how they are cut. 1 Introduction A classic open question in geometry [5, 12, 20, 24] is whether every convex polyhedron can be cut along its edges and flattened into the plane without any overlap. Such a collection of cuts is called an edge cutting of the polyhedron, and the resulting simple polygon is called an edge unfolding or net. While the ¯rst explicit description of this problem is by Shephard in 1975 [24], it has been implicit since at least the time of Albrecht Durer,Ä circa 1500 [11]. It is widely conjectured that every convex polyhedron has an edge unfolding. Some recent support for this conjecture is that every triangulated convex polyhedron has a vertex unfolding, in which the cuts are along edges but the unfolding only needs to be connected at vertices [10]. On the other hand, experimental results suggest that a random edge cutting of ¤Xerox Palo Alto Research Center, 3333 Coyote Hill Rd., Palo Alto, CA 94304, USA, email: [email protected]. -
Pleat Folding, 6.849 Fall 2010
Demaine, Demaine, Lubiw Courtesy of Erik D. Demaine, Martin L. Demaine, and Anna Lubiw. Used with permission. 1999 1 Hyperbolic Paraboloid Courtesy of Jenna Fizel. Used with permission. [Albers at Bauhaus, 1927–1928] 2 Circular Variation from Bauhaus [Albers at Bauhaus, 1927–1928] 3 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 4 Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. 5 “Black Hexagon” Demaine, Demaine, Fizel 2006 Courtesy of Erik Demaine, Martin Demaine, and Jenna Fizel. Used with permission. 6 Hyparhedra: Platonic Solids [Demaine, Demaine, Lubiw 1999] 7 Courtesy of Erik Demaine, Martin Demaine, Jenna Fizel, and John Ochsendorf. Used with permission. Virtual Origami Demaine, Demaine, Fizel, Ochsendorf 2006 8 “Computational Origami” Erik & Martin Demaine MoMA, 2008– Elephant hide paper ~9”x15”x7” Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/Computational/. 9 Peel Gallery, Houston Nov. 2009 Demaine & Demaine 2009 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/Limit/. 10 “Natural Cycles” Erik & Martin Demaine JMM Exhibition of Mathematical Art, San Francisco, 2010 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/NaturalCycles/. 11 Courtesy of Erik Demaine and Martin Demaine. Used with permission. See also http://erikdemaine.org/curved/BlindGlass/. Demaine & Demaine 2010 12 Hyperbolic Paraboloid Courtesy of Jenna Fizel. Used with permission. [Demaine, Demaine, Hart, Price, Tachi 2009/2010] 13 θ = 30° n = 16 Courtesy of Erik D. -
Folding Concave Polygons Into Convex Polyhedra: the L-Shape
Rose- Hulman Undergraduate Mathematics Journal Folding concave polygons into convex polyhedra: The L-Shape Emily Dinana Alice Nadeaub Isaac Odegardc Kevin Hartshornd Volume 16, No. 1, Spring 2015 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics aUniversity of Washington Terre Haute, IN 47803 bUniversity of Minnesota c Email: [email protected] University of North Dakota d http://www.rose-hulman.edu/mathjournal Moravian College Rose-Hulman Undergraduate Mathematics Journal Volume 16, No. 1, Spring 2015 Folding concave polygons into convex polyhedra: The L-Shape Emily Dinan Alice Nadeau Issac Odegard Kevin Hartshorn Abstract. Mathematicians have long been asking the question: Can a given convex polyhedron can be unfolded into a polygon and then refolded into any other convex polyhedron? One facet of this question investigates the space of polyhedra that can be realized from folding a given polygon. While convex polygons are relatively well understood, there are still many open questions regarding the foldings of non-convex polygons. We analyze these folded realizations and their volumes derived from the polygonal family of ‘L-shapes,’ parallelograms with another parallelogram removed from a corner. We investigate questions of maximal volume, diagonalflipping, and topological connectedness and discuss the family of polyhedra that share a L-shape polygonal net. Acknowledgements: We gratefully acknowledge support from NSF grant DMS-1063070 and the 2012 Lafayette College Research Experience for Undergraduates, where the majority of this research was undertaken. We would like to thank our research advisor, Dr. Kevin Hartshorn, who helped us with his great ideas, feedback, problem solving abilities and support throughout the project. -
GEOMETRIC FOLDING ALGORITHMS I
P1: FYX/FYX P2: FYX 0521857570pre CUNY758/Demaine 0 521 81095 7 February 25, 2007 7:5 GEOMETRIC FOLDING ALGORITHMS Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500s but have only recently been studied in the mathemat- ical literature. Over the past decade, there has been a surge of interest in these problems, with applications ranging from robotics to protein folding. With an emphasis on algorithmic or computational aspects, this comprehensive treatment of the geometry of folding and unfolding presents hundreds of results and more than 60 unsolved “open prob- lems” to spur further research. The authors cover one-dimensional (1D) objects (linkages), 2D objects (paper), and 3D objects (polyhedra). Among the results in Part I is that there is a planar linkage that can trace out any algebraic curve, even “sign your name.” Part II features the “fold-and-cut” algorithm, establishing that any straight-line drawing on paper can be folded so that the com- plete drawing can be cut out with one straight scissors cut. In Part III, readers will see that the “Latin cross” unfolding of a cube can be refolded to 23 different convex polyhedra. Aimed primarily at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from high school students to researchers. Erik D. Demaine is the Esther and Harold E. Edgerton Professor of Elec- trical Engineering and Computer Science at the Massachusetts Institute of Technology, where he joined the faculty in 2001. He is the recipient of several awards, including a MacArthur Fellowship, a Sloan Fellowship, the Harold E. -
Professor, EECS Massachusetts Institute of Technology [email protected] Artist-In-Residence, EECS Massach
ERIK D. DEMAINE MARTIN L. DEMAINE Professor, EECS Artist-in-Residence, EECS Massachusetts Institute of Technology Massachusetts Institute of Technology [email protected] [email protected] http://erikdemaine.org/ http://martindemaine.org/ BIOGRAPHY Erik Demaine and Martin Demaine are a father-son math-art team. Martin started the first private hot glass studio in Canada and has been called the father of Canadian glass. Since 2005, Martin Demaine has been an Artist-in-Residence at the Massachusetts Institute of Technology. Erik is also at the Massachusetts Institute of Technology, as a Professor in computer science. He received a MacArthur Fellowship in 2003. In these capacities, Erik and Martin work together in paper, glass, and other material. They use their ex- ploration in sculpture to help visualize and understand unsolved problems in science, and their scientific abilities to inspire new art forms. Their artistic work includes curved origami sculptures in the permanent collections of the Museum of Modern Art (MoMA) in New York, and the Renwick Gallery in the Smith- sonian. Their scientific work includes over 60 published joint papers, including several about combining mathematics and art. They recently won a Guggenheim Fellowship (2013) for exploring folding of other materials, such as hot glass. SOLO/JOINT EXHIBITIONS work by Demaine & Demaine 2014 Duncan McClellan Gallery, Florida (joint with Peter Houk) 2013 Edwards Art Gallery, New Hampshire (joint with Shandra McLane) 2013 Dwight School, New York City (solo show) 2013 The Art Museum, -
Bridges Stockholm 2018 Mathematics | Art | Music | Architecture | Education | Culture 2018 Conference Proceedings Editors
Bridges Stockholm 2018 Mathematics | Art | Music | Architecture | Education | Culture 2018 Conference Proceedings Editors Program Chairs Eve Torrence Bruce Torrence Department of Mathematics Department of Mathematics Randolph-Macon College Randolph-Macon College Ashland, Virginia, USA Ashland, Virginia, USA Short Papers Chair Workshop Papers Chair Carlo H. Séquin Kristóf Fenyvesi Computer Science Division Department of Music, Art and Culture Studies University of California University of Jyväskylä Berkeley, USA Jyväskylä, Finland Production Chair Craig S. Kaplan Cheriton School of Computer Science University of Waterloo Waterloo, Ontario, Canada Bridges Stockholm 2018 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 2018 Conference Proceedings remain under the control of the original authors. ISBN: 978-1-938664-27-4 ISSN: 1099-6702 Published by Tessellations Publishing, Phoenix, Arizona, USA (© 2018 Tessellations) Distributed by MathArtFun.com (mathartfun.com). Cover design: Margaret Kepner, Washington, DC, USA Bridges Organization Board of Directors Kristóf Fenyvesi George W. Hart Department of Music, Art and Culture Studies Stony Brook University University of Jyväskylä, Finland New York, USA Craig S. Kaplan Carlo H. -
G U I D E D B Y I N V O I C
G U I D E D B Y I N V O I C E S 558 WEST 21ST STREET NEW YORK, NY 10011 917.226.3851 ERIK D. DEMAINE Born in Canada on February 28, 1981 PERMANENT COLLECTIONS AND INSTALLATIONS 1. “Green Waterfall” (paper sculpture, joint work with Martin Demaine), Fuller Craft Museum, Brock- ton, Massachusetts, acquired 2012. 2. “Natural Cycles”, “Hugging Circles”, “Green Balance” (3 paper sculptures, joint work with Martin Demaine), Renwick Gallery, Smithsonian American Art Museum, Washington, DC, acquired 2011. 3. “Floating glass ceiling” (waterjet-cut glass, joint work with Martin Demaine and Jo Ann Fleischhauer), part of Leonardo Dialogo, an installation at UT Health Science Center, MD Anderson Cancer Center, Houston, Texas, installed 2010. 4. “Computational Origami” (3 paper sculptures, joint work with Martin Demaine), Museum of Modern Art (MoMA), New York, acquired 2008. 5. “Hyparhedra” (paper sculptures, joint work with Martin Demaine), Southwestern College, Winfield, Kansas, acquired 1999. 6. “Topological puzzles” (wire puzzles, joint work with Martin Demaine), Elliott Avedon Museum and Archive of Games, University of Waterloo, Ontario, Canada, acquired 1987. EXHIBITIONS July 2012-Feb Renwick Gallery, “40 under 40: Craft paper sculptures 2013 Smithsonian American Art Futures” (invitational), (joint work with Museum, Washington DC curated by Nicholas R. Bell Martin L. Demaine) July-August 2012Simons Center for “Ocean Beasts” (joint paper sculptures Geometry and Physics Art exhibition with Theo (joint work with Gallery, Stony Brook, NY Jansen), curated by Flo Martin L. Demaine) Tarjan June-July 2012 College of Fine Arts “Bridges 2012 Exhibition of “Gentle Gallery, Towson University, Mathematical Art” (juried) Earthquake” (joint Towson, MD work with Martin L. -
Shaping Space Exploring Polyhedra in Nature, Art, and the Geometrical Imagination
Shaping Space Exploring Polyhedra in Nature, Art, and the Geometrical Imagination Marjorie Senechal Editor Shaping Space Exploring Polyhedra in Nature, Art, and the Geometrical Imagination with George Fleck and Stan Sherer 123 Editor Marjorie Senechal Department of Mathematics and Statistics Smith College Northampton, MA, USA ISBN 978-0-387-92713-8 ISBN 978-0-387-92714-5 (eBook) DOI 10.1007/978-0-387-92714-5 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2013932331 Mathematics Subject Classification: 51-01, 51-02, 51A25, 51M20, 00A06, 00A69, 01-01 © Marjorie Senechal 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publishers location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. -
Is Origami the Future of Tech? by Drake Bennett on May 03, 2012
Is Origami the Future of Tech? By Drake Bennett on May 03, 2012 In 1996 a young mathematician and computer scientist named Erik Demaine became fascinated by a magic trick that Harry Houdini used to do before he made his name as an escape artist. The magician would fold a piece of paper flat a few times, make one straight cut with a pair of scissors, and then unfold the paper to reveal a five-pointed star. Other magicians built on Houdini’s fold-and-cut method over the years, creating more intricate shapes: a single letter, for example, or a chain of stars. It’s an odd subject of study for a computer science professor, but Demaine had an unorthodox background. When he was hired by the Massachusetts Institute of Technology in 2001, he was, at 20, the youngest professor in the university’s history. Pale, thin, and soft-spoken, with a pickpocket’s long fingers and a fox- colored ponytail, Demaine was born in Halifax, Nova Scotia, and raised by his father, Martin, a renowned glass blower. Erik Demaine outside his MIT office Erik and Martin Demaine's shared studio at MIT When Demaine was six, he and his father started a puzzle company. When he was seven, his father took him out of school and they spent four years traveling the U.S., choosing their destinations together. They spent a few months in Florida, Demaine recalls, “because it was so flat and good for riding bicycles.” Martin home-schooled his son while they moved from place to place. -
The Origami Geometer 1999
COMMENT BOOKS & ARTS Curved-crease origami sculptures can self-fold into intricate patterns. they looked beautiful. We’re still pursuing Q & A Erik Demaine unsolved mathematical problems too. David Huffman, who invented the compression algorithms used for mobile phones, left many beautiful origami sculptures when he died in The origami geometer 1999. He was getting to the point at which he E. DEMAINE/M. DEMAINE Computer scientist Erik Demaine uses origami to advance computational geometry and create could conceive a 3D form, then reverse engi- art. His paper sculptures, made with his father, artist Martin Demaine, are now on show at the neer the steps to fold it with curved creases. Japanese American National Museum in Los Angeles, California; from August, the exhibition We hope to have him as a posthumous co- will tour the United States. He explains the challenges of folding together mathematics and art. author on some papers. How do you actually fold the paper? How did you come to making one straight cut, then unfolding it? For prototype models we use a robotically do most of your work Houdini made a five-pointed star in this way controlled laser to score the paper, but we with your father? in the 1920s. It took us years to prove that you prefer a simple compass-like device. The NICK HIGGINS When I was five, I can make any straight-edged shape; we have final folding we always do by hand. At the helped him to design designed swans, butterflies and the MIT logo. first show we contributed to, we made three wire puzzles for toy Another result was an algorithm for folding sculptures by folding two circular pieces of shops across Canada. -
Arxiv:Cs.CG/9908003 V2 27 Aug 2001 [email protected] [email protected] Email: Etcs[0.O H Te Ad Xeietlrslssugg Results Experimental Hand, Other the on [10]
Ununfoldable Polyhedra with Convex Faces Marshall Bern∗ Erik D. Demaine† David Eppstein‡ Eric Kuo§ Andrea Mantler¶ Jack Snoeyink¶ k Abstract Unfolding a convex polyhedron into a simple planar polygon is a well-studied prob- lem. In this paper, we study the limits of unfoldability by studying nonconvex poly- hedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut. 1 Introduction A classic open question in geometry [5, 12, 20, 24] is whether every convex polyhedron can be cut along its edges and flattened into the plane without any overlap. Such a collection of cuts is called an edge cutting of the polyhedron, and the resulting simple polygon is called an edge unfolding or net. While the first explicit description of this problem is by Shephard in 1975 [24], it has been implicit since at least the time of Albrecht D¨urer, circa 1500 [11]. It is widely conjectured that every convex polyhedron has an edge unfolding. Some recent support for this conjecture is that every triangulated convex polyhedron has a vertex arXiv:cs.CG/9908003 v2 27 Aug 2001 unfolding, in which the cuts are along edges but the unfolding only needs to be connected at vertices [10]. -
ERIK D DEMAINE and MARTIN L DEMAINE CV
G U I D E D B Y I N V O I C E S 558 WEST 21ST STREET NEW YORK, NY 10011 917.226.3851 ERIK D. DEMAINE and MARTIN L. DEMAINE Permanent Collections 2012 Green Waterfall, Fuller Craft Museum, Brockton, Massachusetts 2011 Natural Cycles, Renwick Gallery, Smithsonian American Art Museum, Washington, DC 2010 Floating Glass Ceiling joint work with Jo Ann Fleischhaur, UT Health Science Center, MD Anderson Cancer Center, Houston Texas 2008 Computational Origami, Museum of Modern Art, New York, NY 1999 Hyparhedra, Southwestern College, Winfield, Kansas 1987 Topological Puzzles, Elliot Avedon Museum and Archive of Games, University of Waterloo, Ontario, Canada Solo Exhibitions 2012 Curved Crease Sculptures, curated by Chris Byrne, Guided by Invoices, New York, NY Group Exhibitions 2012 40 Under 40: Craft Futures, Renwick Gallery, Smithsonian American Art Museum, Washington DC Ocean Beasts, curated by Flo Tarjan, Simons Center for Geometry and Physics Art Gallery, Stony Brook, NY Unfolding Patterns, curated by Ombretta Agrò Andruff, Dorsky Gallery Curatorial Programs, Long Island City, NY Origami: Art + Mathematics, curated by Linda Pope, Eloise Pickard Smith Gallery, University of California, Santa Cruz ¡AHA! Art Exhibition with a Mathematical Twist, Central Library, Atlanta, GA Mens et Manus, Fuller Craft Museum, Broctkon MA Math + Art Collaborative curated by Steven Broad, Moreau Center for the Arts, Saint Mary’s College, Notre Dame, IN 2011 Exhibition of Mathematical Art, curated by Robert Fathauer, Joint Mathematics Meetings of the American Mathematical Society and Mathematical Association of America, Boston, MA 2009 Paper: Torn Twisted & Cut, curated by Steven Hempel, Peel Gallery, Houston, TX Paper, Transient Matter, Lasting Impressions, curated by Isabel De Craene, Art Cézar, Rotselaar, Belgium 2008 Rough Cut: Design Takes a Sharp Edge, curated by Paola Antonelli, Museum of Modern Art, New York, NY Design and the Elastic Mind, curated by Paola Antonelli, Museum of Modern Art, New York, NY ERIK D.