This exhibit is a spin-off of NON-ROUND WHEELS. Basically it is just another way to look at non-circular shapes that have a constant width. There are four shapes made in white plastic: one is a circle; one is basically triangular with sharp corners; one is basically triangular but with rounded corners; and one is irregularly shaped. There is also a pattern board with file openings cut into it: there is a matching opening for each of the four shapes and one square.
Currently the pattern board is made of wood. The holes have been cut very close to the actual size of the four shapes so the exhibit is more effective. But wood changes size slightly with changes in humidity - when the humidity is higher there is more moisture in the air. Some of this moisture is absorbed by the wood, causing it to swell. When the wood swells the holes are slightly smaller, and sometimes the shapes are then a tight fit.
Put any of the four shapes in the square. No matter how the shape is turned, it is always touching all four sides of the square, demonstrating that each of the shapes has a constant width.
Put the triangular shape with sharp corners in the square cutout. While watching one of the corners of the shape, rotate it in the square. Notice that the corner traces out the perimeter of the square. What would happen if the corners were in fact cutters? As you rotated the shape you would begin cutting a square hole in the table top.
A company in Pennsylvania makes equipment based upon this action that can drill, squares, hexagons and octagons in wood and metal.