...Let's start with the book's title, which is connected to the problem of how to determine whether a roundish object is exactly round, to a certain tolerance. This turns out to be much trickier than one would expect. For a start, there are the curves of constant width (such as the Reuleaux triangle, which is made by drawing three 60-degree arcs of a circle centered at the vertices of an equilateral triangle). Because one can make such curves with many bumps, a device that just checks several diameters for equality can be fooled. The authors describe various ways in which one might try to confirm roundness, but they all have drawbacks, and when it comes to the definitive answer, Bryant and Sangwin admit that it takes very complicated machinery to perform a proper check (basically by rotating the given object around an axis)...
)