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Frameworks Counting on Frameworks AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL AMS / MAA DOLCIANI MATHEMATICAL EXPOSITIONS VOL 25 25 Counting on Frameworks Counting on Frameworks Jack Graver earned his PhD from Indiana University. After two years at Dartmouth College as a John Wesley Research instructor, he came to Syracuse University where he Mathematics to Aid the Design of Rigid Structures has been for 35 years. He has published research papers in design theory, integer and linear programming and in several different areas in graph theory. He has co-authored three books: two graduate level texts and an elementary level book on group theory Graver and graph theory for students of architecture. He has been active in the Mathematical Association of America (MMA) Seaway Section for 35 years. ln 1993 he was awarded a Certi cate of Meritorious Service by the MMA. Starting in the early 60’s, he has taught a variety of summer workshops for high school teachers; in Indiana, New York, the Virgin Jack E. Graver Islands and England. lt is an activity that he continues to nd particularly satisfying. Rigidity theory is a body of mathematics developed to aid in designing structures. Consider scaffolding that is constructed by bolting together rods and beams. The ultimate question is: “ls the scaffolding sturdy enough to hold the workers and their equipment?” There are several features of the structure that have to be considered in answering this question. The rods and beams have to be strong enough to hold the weight of the workers and their equipment, and to withstand a variety of stresses. Also important in determining the strength of the scaffolding is the way the rods and beams are put together. And, although the bolts at the joints must hold the scaffolding together, they are not expected to prevent the rods and beams from rotating about the joints. So, ultimately, the sturdiness of the scaffolding depends on the way it is braced. Just how to design properly braced scaffolding (or the basic skeleton of any structure) AMSMAA / PRESS is the problem that motivates rigidity theory. The purpose of this book is to develop a mathematical model for rigidity. 4-Color Process 192 page • 3/8” • 50lb stock • fi nish size: 6” x 9” Counting on FraIlleworks Mathematics to Aid the Design of Rigid Structures Figure 4.30 on page 171 i courtesy of Kenneth Snelson American, 1927- Free Ride Home, 1974 Aluminum and tainle teel 30' x 60' x 60' Gift of the Ralph E. Ogden Foundation, Inc. Storm King Art Center Mountainville, New York © 2001 by The Mathematical Association ofAmerica (Incorporated) Library of Congress Catalog Card Number 2001089227 ISB 0-88385-331-0 Printed in the United States ofAmerica Current Printing (la t digit): 10 9 8 7 6 5 4 3 2 1 ii 10.1090/dol/025 The Dolciani Mathematical Expositions NUMBER TWENTY-FIVE Counting on Frameworks Mathematics to Aid the Design of Rigid Structures Jack E. Graver Syracuse University Published and distributed by The Mathematical Association of America iii DOLCIANI MATHEMATICAL EXPOSITIONS Committee on Publications WILLIAM WATKINS, Chair Dolciani Mathematical Expositions Editorial Board DANIEL J. VELLEMAN, Editor EDWARD J. BARBEAU DONNA L. BEERS ROBERT BURCKEL GUILIANA DAVIDOFF SUSAN C. GELLER LESTER H. LANGE Wli,LIAM S. ZWICKER The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical A ociation ofAmerica wa e tabli hed through a generous gift to the Association from Mary P. Dolciani, Profes or of Mathematic at Hunter College of the City University of New York. In making the gift, Profe or Dolciani, her elf an exceptionally talented and successful expo itor of mathematics, had the purpose of furthering the ideal of excellence in mathematical exposition. The Association, for its part, wa delighted to accept the gracious gesture initiating the revolving fund for thi eries from one who has erved the Association with distinction, both a a member of the Committee on Publications and as a member of the Board of Governor . It was with genuine pleasure that the Board chose to name the eries in her honor. The book in the series are selected for their lucid expository style and timulating mathematical content. Typically, they contain an ample supply of exercises, many with accompanying solutions. They are intended to be sufficiently elementary for the undergraduate and even the mathematically inclined high- chool student to under tand and enjoy, but al 0 to be intere ting and sometimes challenging to the more advanced mathematician. 1. Mathematical Gems, Ros Honsberger 2. Mathematical Gems II, Ross Hon berger 3. Mathematical Morsels, Ross Honsberger 4. Mathematical Plums, Ross Honsberger (ed.) 5. Great Moments in Mathematics (Before 1650), Howard Eves 6. Maxima and Minima without Calculus, Ivan Niven 7. Great Moments in Mathematics (After 1650), Howard Eves 8. Map Coloring, Polyhedra, and the Four-Color Problem, David Barnette 9. Mathematical Gems Ill, Ross Honsberger 10. More Mathematical Morsels, Ross Honsberger 11. Old and New Unsolved Problems in Plane Geometry and Number Theory, Victor Klee and Stan Wagon 12. Problems for Mathematicians, Young and Old, Paul R. Halmos 13. Excursions in Calculus: An Interplay ofthe Continuous and the Discrete, Robert M. Young 14. The Wohascum County Problem Book, George T. Gilbert, Mark Kruse­ meyer, and Loren C. Larson 15. Lion Hunting and Other Mathematical Pursuits: A Collection of Math­ ematics, Verse, and Stories by Ralph P. Boas, Jr., edited by Gerald L. Alexanderson and Dale H. Mugler 16. Linear Algebra Problem Book, Paul R. Halmos v 17. From ErdtJs to Kiev: Problems of Olympiad Caliber, Ross Honsberger 18. Which Way Did the Bicycle Go? ... and Other Intriguing Mathematical Mysteries, Joseph D. E. Konhauser, Dan Velleman, and Stan Wagon 19. In Polya's Footsteps: Miscellaneous Problems and Essays, Ross Hons­ berger 20. Diophantus and Diophantine Equations, 1. G. Bashmakova (Updated by Joseph Silverman and translated by Abe Shenitzer) 21. Logic as Algebra, Paul Halmos and Steven Givant 22. Euler: The Master of Us All, William Dunham 23. The Beginnings and Evolution ofAlgebra, 1. G. Bashmakova and G. S. Smirnova (Translated by Abe Shenitzer) 24. Mathematical Chestnuts from Around the World, Ross Honsberger 25. Counting on Frameworks: Mathematics to Aid the Design of Rigid Structures, Jack E. Graver MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789 vi Preface The formal name for the topic discussed in this book is rigidity theory. It is a body of mathematics developed to aid in designing structures. To illustrate, consider a scaffolding that has been constructed by bolting together rods and beams. Before using it, we must ask the crucial question: Is it sturdy enough to hold the workers and their equipment? Several features of the structure have to be considered in answering this question. The rods and beams must be strong enough to hold the weight and withstand a variety of stresses. In addition, the way they are put together is important. The bolts at the joints must hold the structure together, but they are not expected to prevent the rods and beams from rotating about the joints. So the sturdiness of the structure depends on the way it has been braced. Just how to design a properly braced scaffolding (or the structural skeleton of any construction) is the problem that motivates rigidity theory. Thinking further about the scaffolding problem, we might conclude that it really has two separate parts. Designing the optimal placement of the rods, beams and braces might require mathematics that is quite different from what is needed to analyze the strength and weight-bearing properties of the materials used. In fact, we would like to separate these two aspects of structure building. Our ultimate aim is to be able to take a very rough sketch of a proposed scaffolding-a sketch without reference to scale, types of materials or exact alignments of its members-and decide whether the final structure will be rigid, assuming that the other aspects (strength of materials etc.) are properly attended to. The information depicted in this rough sketch is called the combinatorial information about the structure: the way various elements are to be combined to form the structure. The combinatorial information is in contrast to the geometrical information, the exact lengths and locations of the various elements, and to the remaining information about the materials to be used. vii viii Counting on Frameworks The purpo e of this book is to develop a mathematical model for rigidity. We will actually develop three di tinct models. The tructure we wi h to study will be represented by a framework: a configuration of traight line segment joined together at their endpoints. We will a sume that the joints are flexible and will explore whether a given configuration can be deformed without changing the lengths of its segments. For instance, a triangle is a simple rigid framework in the plane, while the square is not rigid-it deform into a rhombu . The fir t model we will develop for rigidity i the degrees offreedom model. It i very intuitive and very ea y to use with mall frameworks. But at the outset thi model lack a rigorou foundation, and in some cases it actually fails to give a correct prediction. The next model we will build is based on sy terns of quadratic equations. Each egment of the framework yields the quadratic equation stating that the distance between the joints at the egment' two ends mu t equal the length of the egment itself. It is rather ea y to ee that this model will give the correct answer to our question about the pre ence or ab ence of rigidity, in all cases.
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