Introductory remarks -- A self-referential boundary -- Expanding forms: the geometrical representation -- The geometrical expansions as a relational network -- The geometrical representation and its computability -- Conclusion -- References.

This book offers a thought-provoking exploration of dynamic geometry and its connections to self-reference and theoretical biology. The authors explore how a self-referential boundary can be translated into remarkable relations between expanding geometrical forms, with a particular focus on triangles and circles. The essence of this work lies in revealing not only how these forms expand and interact with others but also how their interactions lead to closed loops of definitions between processes, where triangles and circles reciprocally define one another. These unique geometrical relations offer fresh perspectives on the interaction and emergence of forms. Through the introduction of time and a fixed velocity of expansions, a rich tapestry of encounters and coalescences unfolds, pushing beyond the boundaries of traditional insights on context dependence and state transitions of systems. These captivating movements elude prediction other than by numerical approximation within unpredictable durations. Unlike cellular automata, they defy stepwise progression on a predefined grid, presenting themselves as unprogrammable construction processes that leave readers in awe of their unexpected elegance. This book is essential reading for researchers and students in theoretical biology seeking to deepen their understanding of the intersections of geometry and systems theory and seeking to gain new insights into the processes that underlie the origination of complexity. "What is unique to the authors' attempt is to shed a new light on extending the notion of cohesive interaction so as to make it applicable even to biology at large without offending the established physics so far. To the best of my knowledge, their work has been the first attempt of this kind in explicating the intricate relationship between geometric topology of the network and the realizable temporal cohesion to be observed widely in biology." (Professor Koichiro Matsuno, 1st foreword to this book) "I am delighted that the authors use Robert Rosen's (M,R)-systems — impredicative networks that are inherently geometrical — to illustrate (see Chapter 4 of this book) their self-referential systems of geometrical expansions." (dr. Aloisius Louie, 2nd foreword to this book).