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The Geometry of Biological Time

Arthur T. Winfree
Springer Verlag
Publication Date: 
Number of Pages: 
Interdisciplinary Applied Mathematics 12
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Jan E. Holly
, on

The new edition of The Geometry of Biological Time is a fascinating update of the delightful original. This new edition contains thoughtful commentary on new developments in the field, adding a historical and sociological dimension to the original book's elegant and unifying treatment of biological problems involving processes that repeat themselves regularly, i.e. involve "rhythmic return through a cycle of change."

The book is an enjoyable page-turner, even for those readers with only a passing interest in biology, and demonstrates well the synergistic effect between biology and mathematics. On the one hand, biology benefits from the unifying effect of mathematics; this book contains diverse biological examples in chapters such as "The Firefly Machine", "Energy Metabolism in Cells", and "The Female Cycle" (just to name a few) for which a common mathematical language has been developed. On the other hand, mathematicians are often amazed by the purely mathematical ideas that arise in biology, some charmingly cute and some intriguingly complicated. This book begins with the basic idea of a circle/ring and mathematically develops the concept of biological time through product spaces, projections, phases (and phase singularities), dynamics, and other mathematical notions. Particularly enticing are figures conceptualizing biological time, e.g. of the "fruitfly clock's time crystal" and stereographic views of the "timing of Kalanchoë flower opening after a light pulse." Though the book is intended primarily for research students, undergraduates could use the book for independent study or an undergraduate research project.

The term "page-turner" may seem unusual in mathematics; however, it is appropriate here. Not only is the reader continually tantalized by the figures appearing on the ensuing pages, but the new commentary lends a mystery-novel feeling to the book. This second edition was created by inserting new text boxes into the original, mostly intact, edition. This style leads to a fascinating historical picture. For example, the text, "(...In the latter cases the periodicity approximation gets worse closer to the pivot. I wish here to sweep such matters under the rug (in 1978))" is followed by a new text box that begins, "The bulge under the rug grew and grew...", continuing with a description of developments over the last two decades. The plot continues even now, with descriptions of the last twenty years often followed in the book by descriptions of current puzzles. In summary, the original book is good and the second edition is even better; the historical commentary is fascinating, and there are also a few reorganized and new chapters presenting additonal biological examples


Jan Holly ( is Assistant Professor of Mathematics at Colby College, where she specializes in applied mathematics. She has done research at Los Alamos National Laboratory, the Robert S. Dow Neurological Sciences Institute in Portland, Oregon, and the Center for Computational Biology at Montana State University.


 1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ("Sodium Geometrate") * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects