This book uses geometry in its broad physical sense: the organization of position, boundary, curvature, scale, topology, and relation within a system. Geometry is not treated as a diagram imposed after an event. It is the set of spatial and relational conditions under which the event can occur.
A medium is anything capable of carrying, sustaining, transforming, or refusing a form. Water is a medium for waves and vortices. A membrane is a medium for curvature, tension, charge, and transport. A resonant cavity is a medium for selected modes. Living tissue is a medium whose material, electrical, chemical, and mechanical relations continuously alter one another. The term does not imply that all media are made of the same substance. It identifies the role each plays: each supplies a field of possibilities with real limits.
Form means an organized relation that persists long enough to do work. It may be a standing wave, a fold, a spiral, a branch, a periodic sequence, a closed boundary, or a topology that preserves information through transformation. A form is not opposed to matter. It is matter held in a particular relation, or a relation transmitted through changing matter. Function follows because different relations permit different operations.
Three distinctions govern the argument.
First, recurrence is not identity. A spiral in a shell, a spiral chemical wave, and a spiral traveling across cortex do not share one local mechanism. They belong to a shape family because different media admit related geometric organizations. The recurring form is the question that crosses the disciplines; each discipline remains responsible for its own mechanism.
Second, transmission is not necessarily transport. A wave may travel while its constituent parcels do not travel with it. A melody may survive changes of key, instrument, and physical carrier. The persistent element is the relation capable of being formed again. This distinction provides the bridge from nonliving pattern to biological inheritance.
Third, selection begins before natural selection. Physical and chemical systems already select among possible forms through boundary conditions, energy gradients, stability ranges, and phase transitions. Biology adds a decisive capacity: a bounded form can preserve a local history, divide, and carry an organized relation into another generation. Evolution then searches the possibilities admitted by living form.
The inquiry moves through four levels. It begins with the admission rule, which states the relation between a medium and the forms it can hold. It then gathers a cross-disciplinary shape family: light, electromagnetic fields, resonance, membranes, water, music, the nautilus, Fibonacci structures and anyons, the possible Riemann architecture, and the folded cortex. It next examines the biological threshold at which a form becomes capable of lineage. Finally, it asks how evolution operates when geometry supplies capacities, variation explores them, and selection retains what a living system can continue to carry.
The book therefore does not propose one mechanism operating unchanged from light to life. It proposes one order of inquiry. Begin with the ingredients, but do not stop there. Identify the boundary, the permitted transformations, the stable relations, and the forms the system refuses. Then ask what changes when one of those forms can remember its own organization and hand it forward.