Forms of [engineered] artifacts

Much as in biological evolution, once a particular engineering construct has been found to work it normally continues to be used. Examples with characteristic forms include (in rough order of their earliest known use): arrowheads, boomerangs, saws, boxes, stairs, fishhooks, wheels, arches, forks, balls, kites, lenses, springs, catenaries, cogs, screws, chains, trusses, cams, linkages, propellers, clocksprings, parabolic reflectors, airfoils, corrugation, zippers, and geodesic domes. It is notable that not even nested shapes are common, though they appear in cross-sections of rope (see page 874), as well as in address decoder trees on chips—and have recently been used in broadband antennas. (Some self-similarity is also present in standard log-periodic antennas.) When several distinct components are involved, more complicated structures are not uncommon—as in escapements, and many bearings and joints. More complex shapes for single elements sometimes arise when an analog of area maximization is desired—as with tire treads or fins in devices such as heat exchangers. Quadratic residue sequences Mod[Range[n]², n] (see page 1081) are used to give profiles for acoustic diffusers that operate uniformly over a range of frequencies. Musical instruments can have fairly complicated shapes maintained for historical reasons to considerable precision. Some knots can also be thought of as objects with complex forms. It is notable that elaborate types of mechanical motion (and sometimes surprising phenomena in general) are often first implemented in toys. Examples are early mechanical automata and model airplanes, and modern executive toys claiming to illustrate chaos theory through linkages, magnets or fluid systems. Complex trajectories (compare page 972) have sometimes been proposed or used for spacecraft. (See also notes on ornamental art on page 872.)