Guilloché is a decorative technique in which a very precise, intricate, and repetitive pattern is mechanically engraved into an underlying material (usually metal) via Engine Turning. It is a form of Fixed Tool Turning.
There is an online blog about engine turning and Guilloché engraving machines, their restoration, and use. This site also includes a nice Engine Turning/Guilloché Guide & Glossary.
There is also a web site titled, Guilloché Toy, where you can play around with the various parameters to see the resultant pattern produced. An example is shown to the right.
If you are interested in pursuing this segment of ornamental turning, Calina C. Shevlin wrote a great book. The book came out in 2017, and is titled, Guilloché: A History & Practical Manual.
Calina finished her MFA in the USA, then moved to Switzerland to work in the watch industry making watch faces.
The Santa Fe Symposium is an annual conference for jewelry makers, and often has topics around ornamental turning and especially guilloché. These include a bit of history and are great reads. Some of note include:
The patterns can be in a radial manner using a rose engine lathe. Below is a YouTube video showing a coin (a flat, round piece of metal) with guilloché patterns being applied in a radial pattern :
There are also some other really nice videos by Roger Smith (rwsmithwatches) showing similar work.
Alternatively, the pattern can be in a linear manner using a straight line engine. The YouTube video shows a watch face with a basket weave guilloché pattern being applied in a linear pattern :
Some brands of straight line engines include :
The Vimeo video below shows both types of machines in use. I find that watching this video of such a master craftsman at work is mesmerizing.
This is a nice video showing a 1914 Duget "tour a Guilloché" (rose engine) machine and an 1891 "ligne droite" (straight line) from Duget machine (an old French engine maker).