We have covered the process for drilling square holes (with rounded corners) using the Watts drill before, but in the intervening time some cool videos demonstrating the process have surfaced on YouTube, e.g. the one I’ve embedded, above, from user jacquesmaurel.
Turns out it’s also possible to drill hexagonal hole using a very similar tool based on the Reuleaux pentagon. The video immediately above, again from jacquesmaurel, shows a tool he describes as a “Vika attachment,” mounted in a lathe, boring an hexagonal hole in a piece of stock. The video below, part of the Wolfram Demonstrations Project, illustrates the process.