Les Houches School, October 11-15, 1999

Nature's design magic is found in its perfect ratios. This illustrated drawing book focuses on four natural designs--Star, Branch, Spiral, and Fan--and why they work.

While the natural world is often described as organic, it is in fact structured to the very molecule, replete with patterned order that can be decoded with basic mathematical algorithms and principles. In a nautilus shell one can see logarithmic spirals, and the Golden Ratio can be seen in the seed head of the sunflower plant. These patterns and shapes have inspired artists, writers, designers, and musicians for thousands of years. "Patterns in Nature: Why the Natural World Looks the Way It Does" illuminates the amazing diversity of pattern in the natural world and takes readers on a visual tour of some of the world s most incredible natural wonders. Featuring awe-inspiring galleries of nature s most ingenious designs, "Patterns in Nature" is a synergy of art and science that will fascinate artists, nature lovers, and mathematicians alike."

Written in a style that is accessible to a wide audience, The Fractal Geometry of Nature inspired popular interest in this emerging field. Mandelbrot's unique style, and rich illustrations will inspire readers of all backgrounds.

Branching morphogenesis, the creation of branched structures in the body, is a key feature of animal and plant development. This book brings together, for the first time, expert researchers working on a variety of branching systems to present a state-of-the-art view of the mechanisms that control branching morphogenesis. Systems considered range from single cells, to blood vessel and drainage duct systems to entire body plans, and approaches range from observation through experiment to detailed biophysical modelling. The result is an integrated overview of branching.

This book introduces biological examples of Branching Processes from molecular and cellular biology as well as from the fields of human evolution and medicine and discusses them in the context of the relevant mathematics. It provides a useful introduction to how the modeling can be done and for what types of problems branching processes can be used.

Branching processes form one of the classical fields of applied probability and are still an active area of research. The field has by now grown so large and diverse that a complete and unified treat ment is hardly possible anymore, let alone in one volume. So, our aim here has been to single out some of the more recent developments and to present them with sufficient background material to obtain a largely self-contained treatment intended to supplement previous mo nographs rather than to overlap them. The body of the text is divided into four parts, each of its own flavor. Part A is a short introduction, stressing examples and applications. In Part B we give a self-contained and up-to-date pre sentation of the classical limit theory of simple branching processes, viz. the Gal ton-Watson ( Bienayme-G-W) process and i ts continuous time analogue. Part C deals with the limit theory of Il!arkov branching processes with a general set of types under conditions tailored to (multigroup) branching diffusions on bounded domains, a setting which also covers the ordinary multitype case. Whereas the point of view in Parts A and B is quite pedagogical, the aim of Part C is to treat a large subfield to the highest degree of generality and completeness possi"ble. Thus the exposition there is at times quite technical.

Sunday Times Bestseller‘A paradigm-smashing chronicle of joyous entanglement’ Charles Foster Waterstones Non-Fiction Book of the Month (September) Are trees social beings? How do trees live? Do they feel pain or have awareness of their surroundings?

The purpose of this book is to give a unified treatment of the limit theory of branching processes. Since the publication of the important book of T E. Harris (Theory of Branching Processes, Springer, 1963) the subject has developed and matured significantly. Many of the classical limit laws are now known in their sharpest form, and there are new proofs that give insight into the results. Our work deals primarily with this decade, and thus has very little overlap with that of Harris. Only enough material is repeated to make the treatment essentially self-contained. For example, certain foundational questions on the construction of processes, to which we have nothing new to add, are not developed. There is a natural classification of branching processes according to their criticality condition, their time parameter, the single or multi-type particle cases, the Markovian or non-Markovian character of the pro cess, etc. We have tried to avoid the rather uneconomical and un enlightening approach of treating these categories independently, and by a series of similar but increasingly complicated techniques. The basic Galton-Watson process is developed in great detail in Chapters I and II.

Leading paleontologist J. David Archibald explores the rich history of visual metaphors for biological order from ancient times to the present and their influence on humans' perception of their place in nature, offering uncommon insight into how we went from standing on the top rung of the biological ladder to embodying just one tiny twig on the tree of life. He begins with the ancient but still misguided use of ladders to show biological order, moving then to the use of trees to represent seasonal life cycles and genealogies by the Romans. The early Christian Church then appropriated trees to represent biblical genealogies. The late eighteenth century saw the tree reclaimed to visualize relationships in the natural world, sometimes with a creationist view, but in other instances suggesting evolution. Charles Darwin's On the Origin of Species (1859) exorcised the exclusively creationist view of the "tree of life," and his ideas sparked an explosion of trees, mostly by younger acolytes in Europe. Although Darwin's influence waned in the early twentieth century, by midcentury his ideas held sway once again in time for another and even greater explosion of tree building, generated by the development of new theories on how to assemble trees, the birth of powerful computing, and the emergence of molecular technology. Throughout Archibald's far-reaching study, and with the use of many figures, the evolution of "tree of life" iconography becomes entwined with our changing perception of the world and ourselves.