VI Preface to the Fifth Edition It is now almost twenty years that the idea to this project was born dur- ing some leisurely discussions at the Mathematisches Forschungsinstitut in Oberwolfach with the incomparable Paul Erdos.At that time we could not possibly imagine the wonderful and lasting response our book about The Book would have,with all the warm letters,interesting comments and suggestions,new editions,and as of now thirteen translations.It is no exaggeration to say that it has become a part of our lives. In addition to numerous improvements and smaller changes,many of them suggested by our readers,the present fifth edition contains four new chap- ters,which present an extraordinary proof for the classical spectral theorem from linear algebra,the impossibility of the Borromean rings as a highlight from geometry,the finite version of Kakeya's problem,and an inspired proof for Minc's permanent conjecture. We thank everyone who helped and encouraged us over all these years.For the second edition this included Stephan Brandt,Christian Elsholtz,Jurgen Elstrodt,Daniel Grieser,Roger Heath-Brown,Lee L.Keener,Christian Leboeuf,Hanfried Lenz,Nicolas Puech,John Scholes,Bernulf WeiBbach, and many others.The third edition benefitted especially from input by David Bevan,Anders Bjorner,Dietrich Braess,John Cosgrave,Hubert Kalf,Gunter Pickert,Alistair Sinclair,and Herb Wilf.For the fourth edi- tion,we were particularly indebted to Oliver Deiser,Anton Dochtermann, Michael Harbeck,Stefan Hougardy,Hendrik W.Lenstra,Guinter Rote, Moritz W.Schmitt,and Carsten Schultz for their contributions.For the present edition,we gratefully acknowledge ideas and suggestions by Ian Agol,France Dacar,Christopher Deninger,Michael D.Hirschhorn,Franz Lemmermeyer,Raimund Seidel,Tord Sjodin,and John M.Sullivan,as well as help from Marie-Sophie Litz,Miriam Schloter,and Jan Schneider. Moreover,we thank Ruth Allewelt at Springer in Heidelberg and Christoph Eyrich,Torsten Heldmann,and Elke Pose in Berlin for their continuing sup- port throughout these years.And finally,this book would certainly not look the same without the original design suggested by Karl-Friedrich Koch,and the superb new drawings provided for each edition by Karl H.Hofmann. Berlin,June 2014 Martin Aigner.Giinter M.ZieglerVI Preface to the Fifth Edition It is now almost twenty years that the idea to this project was born dur￾ing some leisurely discussions at the Mathematisches Forschungsinstitut in Oberwolfach with the incomparable Paul Erdos. At that time we could ˝ not possibly imagine the wonderful and lasting response our book about The Book would have, with all the warm letters, interesting comments and suggestions, new editions, and as of now thirteen translations. It is no exaggeration to say that it has become a part of our lives. In addition to numerous improvements and smaller changes, many of them suggested by our readers, the present fifth edition contains four new chap￾ters, which present an extraordinary proof for the classical spectral theorem from linear algebra, the impossibility of the Borromean rings as a highlight from geometry, the finite version of Kakeya’s problem, and an inspired proof for Minc’s permanent conjecture. We thank everyone who helped and encouraged us over all these years. For the second edition this included Stephan Brandt, Christian Elsholtz, Jürgen Elstrodt, Daniel Grieser, Roger Heath-Brown, Lee L. Keener, Christian Lebœuf, Hanfried Lenz, Nicolas Puech, John Scholes, Bernulf Weißbach, and many others. The third edition benefitted especially from input by David Bevan, Anders Björner, Dietrich Braess, John Cosgrave, Hubert Kalf, Günter Pickert, Alistair Sinclair, and Herb Wilf. For the fourth edi￾tion, we were particularly indebted to Oliver Deiser, Anton Dochtermann, Michael Harbeck, Stefan Hougardy, Hendrik W. Lenstra, Günter Rote, Moritz W. Schmitt, and Carsten Schultz for their contributions. For the present edition, we gratefully acknowledge ideas and suggestions by Ian Agol, France Dacar, Christopher Deninger, Michael D. Hirschhorn, Franz Lemmermeyer, Raimund Seidel, Tord Sjödin, and John M. Sullivan, as well as help from Marie-Sophie Litz, Miriam Schlöter, and Jan Schneider. Moreover, we thank Ruth Allewelt at Springer in Heidelberg and Christoph Eyrich, Torsten Heldmann, and Elke Pose in Berlin for their continuing sup￾port throughout these years. And finally, this book would certainly not look the same without the original design suggested by Karl-Friedrich Koch, and the superb new drawings provided for each edition by Karl H. Hofmann. Berlin, June 2014 Martin Aigner · Günter M. Ziegler