The shapes of cells, skeletons, and shells can be explained by forces and geometry rather than heredity alone, and a coordinate grid deformed in a few directions maps one fish species onto another (Thompson, 1917). Parametric modelling software now performs that transformation routinely. Animal markings arise when two chemicals, an activator and an inhibitor, diffuse at different rates and react (Turing, 1952). The resulting equations produce spots, stripes, and labyrinths from a uniform field, and a change in one diffusion rate moves the result from leopard to zebra. Confirmed in living tissue only decades later, in zebrafish stripes and the ridges of the mammalian palate, the equations are still the most common method of pattern generation in creative computation.

L-systems are string-rewriting grammars that model branching in plants and algae (Lindenmayer, 1968). A rule as simple as replacing every F with F[+F]F[-F]F, applied repeatedly and drawn as turtle graphics, yields a shrub. Extended with parameters, randomness, and sensitivity to light and neighbouring branches, the grammars reproduce the architecture of specific species (Prusinkiewicz and Lindenmayer, 1990), and they still underlie most procedural vegetation in games and film.

The second lineage uses physical material to do the calculation. A hanging chain settles into a catenary, and inverted it becomes a pure compression arch; weighted chain models found the vaults of the Colònia Güell crypt, and soap films, suspended nets, and wet wool were systematized as form-finding instruments at the Institute for Lightweight Structures in Stuttgart from 1964. The Munich Olympic roofs (1972) and the Mannheim Multihalle (1975), a timber gridshell spanning 60 metres, were derived from hanging models. In each case the geometry was found by letting a model settle under gravity rather than drawn in advance.

Differential growth models a surface that expands faster than the space available to it, so it buckles and folds as coral, lettuce, and cortex do. Each vertex of a mesh pushes away from its neighbours while the surface resists stretching and bending, and the folds are where the two demands collide. Floraform (2015) grew jewellery this way, and Cellular Forms (2014) simulated cell division driven by nutrient flow to produce sculptures with millions of cells. Reaction-diffusion run on a mesh does related work for surface relief and perforation.

Three local rules, separation, alignment, and cohesion, are enough to produce flocking (Reynolds, 1986). Swarm-based design lets agents deposit material along their paths, so that structure is the accumulated trace of interaction rather than a drawn object; the practices Kokkugia and Biothing built a body of work on it. Space colonization grows branches toward scattered attractor points and removes each point once reached (Runions et al., 2005), and it remains the standard method for generating leaf venation and tree crowns.

Topology optimization removes material wherever stress is low until only a load path remains (Bendsøe and Kikuchi, 1988). Dynamic relaxation, as in the Kangaroo plug-in for Grasshopper, reproduces hanging models numerically, letting a mesh settle under gravity or pressure until its forces balance. Both are now the core of commercial generative design software, which is why bone-like brackets appear so often in examples of the method.