such device, manufactured by Burleigh Instruments under the trademark INCHWORM™, is illustrated in Figure 4.1.

FIGURE 4.1 INCHWORM™ clamps C1, C2, stretcher S, and armature A.

This actuator, consisting of three sleeves/tubes made from piezoelectric material mounted on a frame and enclosing a linear armature, works on the physical principle that the piezoelectric material deforms under electrical stimulus (the outer sleeves independently clamp down, and the middle sleeve stretches in length). Running the actuator through a succession of clamp-stretch-unclamp-unstretch cycles, one generates incremental motion of the armature in a specified direction. It is possible to make linear movements as small as 4 nanometers. Other actuators based on piezoelectric effects are increasingly finding their way into consumer products, including ultrasonic motors for autofocusing in cameras based on surface wave excitation (see Ueha and Tomikawa, 1993, for detailed discussions of these devices). A common design principle in these devices is a type of rectification of small cyclical motions to produce gross motions.

Turning to the natural world, much attention has been devoted to the systematic understanding of how various microscopic organisms move in fluids under various conditions. Since movement is essential to reaching food particles, efficiency considerations have also been of interest (see Childress, 1981, for related discussion). Apparently, the paramecium gets around in a fluid under conditions of a very low Reynolds number through a process of cyclical change in its boundary contour (or more precisely, the envelope determined by the oscillating cilia that make up the contour). (See Figure 4.2.) In the work of Shapere and Wilczek (1989) this has been shown, under appropriate fluid mechanical assumptions, via the mathematics of gauge theory (which has played an important role elsewhere in modem physics and geometry over the last three decades). Here again a type of rectification is at work.

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