ERIK D DEMAINE and MARTIN L DEMAINE CV
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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. -
Algorithmic Folding Complexity∗
Algorithmic Folding Complexity∗ Jean Cardinal1, Erik D. Demaine2, Martin L. Demaine2, Shinji Imahori3, Tsuyoshi Ito4, Masashi Kiyomi5, Stefan Langerman1, Ryuhei Uehara5, Takeaki Uno6 1 D´epartement d'Informatique, Universit´eLibre de Bruxelles, CP 212, B-1050 Brussels, Belgium. e-mail: fjcardin,[email protected] 2 Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technol- ogy, Cambridge, MA 02139, USA. e-mail: fedemaine,[email protected] 3 Department of Computational Science and Engineering, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan. e-mail: [email protected] 4 Institute for Quantum Computing and School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. e-mail: [email protected] 5 School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, Japan. e-mail: fmkiyomi,[email protected] 6 National Institute of Informatics, Chiyoda-ku, Tokyo 101-8430, Japan. e-mail: [email protected] Abstract. How do we most quickly fold a paper strip (modeled as a line) to obtain a desired mountain-valley pattern of equidistant creases (viewed as a binary string)? Define the folding complexity of a mountain-valley string as the minimum number of simple folds required to construct it. We first show that the folding complexity of a length-n uniform string (all mountains or all valleys), and hence of a length-n pleat (alternating mountain/valley), is O(lg2 n). We also show that a lower bound of the complexity of the problems is Ω(lg2 n= lg lg n). Next we show that almost all mountain-valley patterns require Ω(n= lg n) folds, which means that the uniform and pleat foldings are relatively easy problems. -
Stem+Visual Art
STEM+VISUAL ART A Curricular Resource for K-12 Idaho Teachers a r t Drew Williams, M.A., Art Education Boise State University + Table of Contents Introduction 1 Philosophy 2 Suggestions 2 Lesson Plan Design 3 Tips for Teaching Art 4 Artist Catalogue 5 Suggestions for Classroom Use 9 Lesson Plans: K-3 10 Lesson Plans: 4-6 20 Lesson Plans: 6-9 31 Lesson Plans: 9-12 42 Sample Images 52 Resources 54 References 55 STEM+VISUAL ART A Curricular Resource for K-12 Idaho Teachers + Introduction: Finding a Place for Art in Education Art has always been an integral part of students’ educational experiences. How many can remember their first experiences as a child manipulating crayons, markers and paintbrushes to express themselves without fear of judgement or criticism? Yet, art is more than a fond childhood memory. Art is creativity, an outlet of ideas, and a powerful tool to express the deepest thoughts and dreams of an individual. Art knows no language or boundary. Art is always innovative, as each image bears the unique identity of the artist who created it. Unfortunately as many art educators know all too well, in schools art is the typically among the first subjects on the chopping block during budget shortfalls or the last to be mentioned in a conversation about which subjects students should be learning. Art is marginalized, pushed to the side and counted as an “if-we-have-time” subject. You may draw…if we have time after our math lesson. We will have time art in our class…after we have prepared for the ISAT tests. -
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. -
A Brief Look at Mathematics and Theology Philip J
Humanistic Mathematics Network Journal Issue 27 Article 14 Winter 1-1-2004 A Brief Look at Mathematics and Theology Philip J. Davis Brown University Follow this and additional works at: http://scholarship.claremont.edu/hmnj Part of the Logic and Foundations of Mathematics Commons, Mathematics Commons, and the Religious Thought, Theology and Philosophy of Religion Commons Recommended Citation Davis, Philip J. (2004) "A Brief Look at Mathematics and Theology," Humanistic Mathematics Network Journal: Iss. 27, Article 14. Available at: http://scholarship.claremont.edu/hmnj/vol1/iss27/14 This Article is brought to you for free and open access by the Journals at Claremont at Scholarship @ Claremont. It has been accepted for inclusion in Humanistic Mathematics Network Journal by an authorized administrator of Scholarship @ Claremont. For more information, please contact [email protected]. 1 A Brief Look at Mathematics and Theology Philip J. Davis "Such a really remarkable discovery. I wanted your opinion on it. You know the formula m over naught equals infinity, m being any positive number? [m/0 = ]. Well, why not reduce the equation to a simpler form by multiplying both sides by naught? In which case you have m equals infinity times naught [m = x 0]. That is to say, a positive number is the product of zero and infinity. Doesn't that demonstrate the creation of the Universe by an infinite power out of nothing? Doesn't it?" Aldous Huxley, Point Counter Point, (1928), Chapter XI. I Introduction We are living in a mathematical age. Our lives, from the personal to the communal, from the communal to the international, from the biological and physical to the economic and even to the ethical, are increasingly mathematicized. -
Annual Report 2010–11
ANNUAL REPORT 2010–11 ANNUAL REPORT 2010–11 The National Gallery of Australia is a Commonwealth (cover) authority established under the National Gallery Act 1975. Thapich Gloria Fletcher Dhaynagwidh (Thaynakwith) people The vision of the National Gallery of Australia is the Eran 2010 cultural enrichment of all Australians through access aluminium to their national art gallery, the quality of the national 270 cm (diam) collection, the exceptional displays, exhibitions and National Gallery of Australia, Canberra programs, and the professionalism of Gallery staff. acquired through the Founding Donors 2010 Fund, 2010 Photograph: John Gollings The Gallery’s governing body, the Council of the National Gallery of Australia, has expertise in arts administration, (back cover) corporate governance, administration and financial and Hans Heysen business management. Morning light 1913 oil on canvas In 2010–11, the National Gallery of Australia received 118.6 x 102 cm an appropriation from the Australian Government National Gallery of Australia, Canberra totalling $50.373 million (including an equity injection purchased with funds from the Ruth Robertson Bequest Fund, 2011 of $15.775 million for development of the national in memory of Edwin Clive and Leila Jeanne Robertson collection and $2 million for the Stage 1 South Entrance and Australian Indigenous Galleries project), raised $27.421 million, and employed 262 full‑time equivalent staff. © National Gallery of Australia 2011 ISSN 1323 5192 All rights reserved. No part of this publication can be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording or any information storage and retrieval system, without permission in writing from the publisher. -
A Salamander Sculpture Barn Raising
BRIDGES Mathematical Connections in Art, Music, and Science A Salamander Sculpture Barn Raising George W. Hart Computer Science Department. Stony Brook University Stony Brook, NY 11794-4400, USA E-mrul:[email protected] Abstract Salamanders is a thirty-piece wooden sculpture that was group assembled by thirty volunteers in an exciting sculpture "barn raising" when I was artist-in-residence at M.I.T. in OctoberlNovember 2003. It is composed of laser-cut salamander-shaped components which lie in the planes of a rhombic triacontahedron and were mathematically designed to weave through each other and exactly fit together on the outside. 1. Introduction M.e. Escher playfully incorporated chameleons and other reptiles or amphibians in his two-dimensional geometric artwork [1]. In homage to his creative spirit, I designed my sculpture Salamanders to feature flat salamanders which interweave in three dimensions. Figure 1 shows it hanging temporarily inside a window overlooking the construction of Frank O. Gehry's new Stata Center at M.I.T. [2], where the sculpture will eventually reside. Figure 1: Salamanders 54 2004 Bridges Proceedings My ultimate concept, if funding can be found, is for a large metal double sphere as shown in Figure 2. The inner and outer spheres are each made of thirty identical two-headed salamander shapes. Each part is parallel to an identical part similarly oriented in the other sphere. I find it visually interesting to show that the same salamander parts can be joined in these two contrasting arrangements---one very open and one very interlocked. It is a puzzle with two very different solutions. -
Art Studio Classes Adult • Teens • Children Spring 2015All Events Are FREE Assand Open to the Calendar Public Unless Otherwise Noted
art studio classes adult • teens • children spring 2015All events Are FREE Assand open to the calendar public unless otherwise noted. welcome feb When was the last time you lost yourself eXHIBITIONS | Feb 12 – Apr 26 in a creative activity, one that ignited OPENING RECEPTION | thurs, Feb 12 | 6:00 – 8:00 pm The Crochet Coral Reef Project your passion and curiosity, where you russell hill rogers Gallery II | navarro This exhibition by the Institute for Figuring, found yourself completely absorbed? celebrates the intersection of higher geometry and feminine handicraft, and a testimony to the disappearing wonders of the marine world. You can go back to that time of inspiration and unleash the artist inside of you Confections and Fictions once again. What’s stopping you? The Southwest School of Art is waiting! detail. The Crochet Coral Reef Project, mixed media russell hill rogers Gallery i | navarro Jesse Amado, Andrew Havenhand, and The Southwest School of Art has been offering an extensive list of adult art Kelly O’Connor exhibit new provocative and classes for adults since 1970. enticing works created with unusual materials. • We offer classes in papermaking, ceramics, drawing, printmaking, painting, dec DEBBIE McMAhon Fiber Art syMposiuM REGISTRATION | 2015 spring term sat, Feb 14 fiber art, jewelry, metals, photography, sculpture, digital and integrated media MeMBERS’ priority the sAn Antonio sAtellite reeF | and more! • Online | tues, dec 2 | 9:00am crochet workshop | sat, Feb 14 | 9:30am – 12:30pm • Member Open | thurs, dec 4 russell hill rogers lecture hall | navarro • Never taken an art class? No art experience necessary! We offer a wealth of OPEN REGISTRATION Part of the worldwide Crochet Coral Reef Project, beginning classes, as you’ll see on page 2. -
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. -
'Saratoga Springs Satellite Reef' Expands with In- Person and Online
For immediate release ‘Saratoga Springs Satellite Reef’ expands with in- person and online crochet programs SARATOGA SPRINGS, NY (July 22, 2021) — The Frances Young Tang Teaching Museum and Art Gallery at Skidmore College, which reopened to the public on July 10, is expanding its programming for the Saratoga Springs Satellite Reef to include in-person and online events. Scores of crafters and the craft-curious have already met online to crochet corals, and now participants will be able to meet in person. The first in-person event is a craft circle at the Museum on Saturday, July 31, from 2 to 4 p.m. The Saratoga Springs Satellite Reef is part of the worldwide Crochet Coral Reef project by Christine and Margaret Wertheim and the Institute For Figuring. The Saratoga Springs Satellite Reef, which will be on view at the Tang in the exhibition Radical Fiber starting January 29, 2022, will be made up of creations crocheted by members of the public, either working on their Irish Satellite Reef at Science Gallery (Dublin), own or through the Tang’s public programs. photo © Institute For Figuring For the Saratoga Springs Satellite Reef, anyone can participate by crocheting hyperbolic structures, referring to an area of geometry whose crocheted models mimic coral-like ruffles. The project draws on the long historical connections, especially in the United States, between fiber practice and community building, and will connect hobby crafters, art professionals, and novice crocheters from Skidmore, Saratoga Springs, and global communities. Download the PDF brochure How to Crochet Hyperbolic Corals, which includes patterns for a variety of corals. -
Arts of Living on a Damaged Planet Anna Lowenhaupt Tsing, Nils Bubandt, Elaine Gan, Heather Anne Swanson
Arts of Living on a Damaged Planet Anna Lowenhaupt Tsing, Nils Bubandt, Elaine Gan, Heather Anne Swanson Published by University of Minnesota Press Tsing, Lowenhaupt & Bubandt, Nils & Gan, Elaine & Swanson, Anne. Arts of Living on a Damaged Planet: Ghosts and Monsters of the Anthropocene. Minneapolis: University of Minnesota Press, 2017. Project MUSE., https://muse.jhu.edu/. For additional information about this book https://muse.jhu.edu/book/52400 Access provided by University of Washington @ Seattle (8 Jan 2019 07:42 GMT) 2 SYMBIOGENESIS, SYMPOIESIS, AND ART SCIENCE ACTIVISMS FOR STAYING WITH THE TROUBLE Donna Haraway Symbiogenesis Shoshanah Dubiner’s vivid painting called Endosymbiosis (Figure M2.1) hangs in the hallway joining the Departments of Geosciences and Biol- ogy at UMass Amherst, near the Life and Earth Café, a spatial clue to how critters become with each other.1 Irresistible attraction toward enfolding each other is the vital motor of living and dying on earth. Critters interpenetrate one another, loop around and through one another, eat each another, get indigestion, and partially digest and partially assimilate one another, and thereby establish sympoietic arrangements that are otherwise known as cells, organisms, and eco- logical assemblages. Sym-poiesis is a simple word; it means “making-with.” Nothing makes itself; nothing is really auto-poietic or self-organizing. In the words of the Iñupiat computer “world game,” earthlings are Never Alone.2 That is the radical implication of sympoiesis. Sympoiesis is a word proper to complex, dynamic, responsive, situated, historical systems. It is a word for worlding. M25 M26 ● Donna Haraway Another word for these sympoietic entities is holobionts, or, ety- mologically, “entire beings” or “safe and sound beings.”3 That is decid- edly not the same thing as One and Individual.