Geometry Under Compression–Decompression: A Soft-Matter Atlas

This document is a set of notes developed while exploring how common shapes in soft and biological materials arise from basic mechanical conditions. The central idea is to look at geometry through a simple compression–decompression rule, used here as an alternative way of organizing well-known physical behaviors.
The work goes through twenty-five families of shapes—such as spheres, tubes, sheets, folds, vesicles, foams, porous structures, and flow-induced forms—and describes the physical “settings” that make each one appear.
The goal is not to introduce a new physical theory, but to provide a clear and accessible summary of how different geometries can emerge from changes in tension, curvature, pressure, anisotropy, adhesion, and flow.
The content is based on established equations from soft-matter physics and differential geometry, presented in a unified and easy-to-follow format. This document is meant as a reference or starting point for readers interested in how shape and mechanics connect in a broad range of systems.