Guilloche patterns --- slightly off group topic

*To*: mathgroup at smc.vnet.net*Subject*: [mg39436] Guilloche patterns --- slightly off group topic*From*: "DIAMOND Mark R." <dot at dot.dot>*Date*: Sat, 15 Feb 2003 01:42:04 -0500 (EST)*Organization*: The University of Western Australia*Sender*: owner-wri-mathgroup at wolfram.com

I hope there is no objection to my reposting this (originally alt.math.recreational) here. I wondered if anyone might be able to shed some light on the construction of Guilloche patterns. I have had a look at the entry on Mathworld http://mathworld.wolfram.com/GuillochePattern.html and also seen the Mathematica notebook, but neither really give much information about the construction. The notebook, for example, produces a series of interesting guilloche patterns, but gives no information about the multitude of parameters in the equations. Additional questions I have are (1) are all the standard mathematical constructions of guilloche patterns based on sine waves, and if not, what other forms are there? (2) was there an explicit mathematical basis for the construction of traditional guilloche patterns? (2a) if so, what was it? (2b) if not, are there implicit rules that can be easily infered from the traditional designs? Last of all, can anyone suggest a reference (a paper book would be nice!) that discusses traditional guilloche designs. I am really interested in discovering more. I have, of course, looked at the other sites referenced in the Mathworld entry. Cheers, -- Mark R. Diamond