Fields Institute Communications

VOLUME 70

The Fields Institute for Research in Mathematical Sciences

Fields Institute Editorial Board:

Carl R. Riehm, Managing Editor

Walter Craig, Director of the Institute

Matheus Grasselli, Deputy Director of the Institute

James G. Arthur, University of

Kenneth R. Davidson,

Lisa Jeffrey,

Barbara Lee Keyfitz, Ohio State University

Thomas S. Salisbury,

Noriko Yui, Queen’s University

The Fields Institute is a centre for research in the mathematical sciences, located in Toronto, Canada. The Institutes mission is to advance global mathematical activity in the areas of research, education and innovation. The Fields Institute is supported by the Ontario Ministry of Training, Colleges and Universities, the Natural Sciences and Engineering Research Council of Canada, and seven Principal Sponsoring Universities in Ontario (Carleton, McMaster, Ottawa, Queen’s, Toronto, Waterloo, Western and York), as well as by a growing list of Affiliate Universities in Canada, the U.S. and Europe, and several commercial and industrial partners.

For further volumes: http://www.springer.com/series/10503

Robert Connelly • Asia Ivic´ Weiss • Walter Whiteley Editors

Rigidity and Symmetry

The Fields Institute for Research 123 in the Mathematical Sciences Editors Robert Connelly Asia IvicWeiss´ Department of Mathematics Department of Mathematics and Statistics Cornell University York University Ithaca, NY, USA Toronto, ON, Canada

Walter Whiteley Department of Mathematics and Statistics York University Toronto, ON, Canada

ISSN 1069-5265 ISSN 2194-1564 (electronic) ISBN 978-1-4939-0780-9 ISBN 978-1-4939-0781-6 (eBook) DOI 10.1007/978-1-4939-0781-6 Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2014938202

Mathematics Subject Classification (2010): 05C, 05E, 20B, 20D, 20E, 51E, 51M, 52B, 52C

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Cover illustration: Drawing of J.C. Fields by Keith Yeomans

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Springer is part of Springer Science+Business Media (www.springer.com) Preface

The thematic program on Discrete Geometry and Applications took place at the Fields Institute for Research in Mathematical Sciences in Toronto between July 1 and December 31, 2011. The papers included in this book are based on some research conducted during the semester and on some of the lectures there, in particular those related to the part of the program under the heading “Rigidity and Symmetry”. This includes the study of the theory of rigidity as applied to discrete objects such as bar and joint frameworks, tensegrities, body and bar frameworks especially including such symmetric objects, periodic frameworks, and the combinatorics when the objects are symmetric. When the configuration of points that define the object is generic, the rigidity properties reduce to combinatorial properties usually of some underlying graph. When the object is symmetric, it automatically becomes non-generic, but nevertheless it is possible to consider the case when the configuration is generic modulo the symmetry group. This leads to a lot of interesting and intricate theory. It is useful to keep in mind that there are two approaches to a symmetric rigid object. Incidental rigidity is when the object is rigid and symmetric, but it is not constrained to stay rigid under a flex. Forced rigidity is when the object is rigid and symmetric, and the symmetry is part of constraints. Both situations occur here. Another part concerns symmetry as applied to abstract as well as geometric objects. Central to this theme are polytopes, the generalizations of polygons and polyhedra to higher rank (the abstract analogue of dimension). Several articles are devoted to regular maps on surfaces, which are just polyhedra in a general sense. These usually permit operations – replacing faces by different edge-circuits – that change their combinatorial type, an important idea relating different maps. Such operations can be applied in higher rank as well. Regular and chiral polytopes (the latter roughly speaking half-regular) often correspond to interesting groups, particularly simple ones; such connexions are explored in several papers. Variants of regularity, further weakening the condition, also lead to interesting questions. Closely related to polytopes are graphs and complexes; these are the subject of

v vi Preface other articles. More metrical in scope are papers on volume in non-euclidean spaces, symmetric configurations in the plane, and a concept of rigidity of polytopes that provides a bridge to the previous part.

Ithaca, NY, USA Robert Connelly Toronto, ON, Canada Asia IvicWeiss´ Walter Whiteley Contents

Volumes of Polytopes in Spaces of Constant Curvature ...... 1 Nikolay Abrosimov and Alexander Mednykh Cubic Cayley Graphs and Snarks ...... 27 Ademir Hujdurovic,´ Klavdija Kutnar, and Dragan Marušicˇ Local, Dimensional and Universal Rigidities: A Unified Gram Matrix Approach ...... 41 A.Y. Alfakih Geometric Constructions for Symmetric 6-Configurations ...... 61 Leah Wrenn Berman On External Symmetry Groups of Regular Maps ...... 87 Marston D.E. Conder, Young Soo Kwon, and Jozef Širánˇ Variance Groups and the Structure of Mixed Polytopes...... 97 Gabe Cunningham Mobility in Symmetry-Regular Bar-and-Joint Frameworks ...... 117 P.W. Fowler, S.D. Guest, and B. Schulze Generic Global Rigidity in Complex and Pseudo-Euclidean Spaces ...... 131 Steven J. Gortler and Dylan P. Thurston Chiral Polytopes and Suzuki Simple Groups ...... 155 Isabel Hubard and Dimitri Leemans Globally Linked Pairs of Vertices in Rigid Frameworks ...... 177 Bill Jackson, Tibor Jordán, and Zoltán Szabadka Beauville Surfaces and Groups: A Survey ...... 205 Gareth A. Jones Generic Rigidity with Forced Symmetry and Sparse Colored Graphs .... 227 Justin Malestein and Louis Theran

vii viii Contents

Rigidity of Regular Polytopes...... 253 Peter McMullen Hereditary Polytopes ...... 279 Mark Mixer, Egon Schulte, and Asia IvicWeiss´ One Brick at a Time: A Survey of Inductive Constructions in Rigidity Theory...... 303 A. Nixon and E. Ross Polygonal Complexes and Graphs for Crystallographic Groups ...... 325 Daniel Pellicer and Egon Schulte Two Notes on Maps and Surface Symmetry ...... 345 Thomas W. Tucker Buildings and s-Transitive Graphs ...... 357 Richard M. Weiss