Research of Penrose tilings needed!

*To*: mathgroup at smc.vnet.net*Subject*: [mg20969] Research of Penrose tilings needed!*From*: "Kai G. Gauer" <gauer at sk.sympatico.ca>*Date*: Wed, 1 Dec 1999 01:50:36 -0500 (EST)*Sender*: owner-wri-mathgroup at wolfram.com

I had a quick couple of questions concerning Penrose tilings and I wasn't sure whether or not any of you could help me out. ( I ) When Martin Gardner brought out the definition of a township (I assume he meant a neighbourhood-like structure), he stated that for any radius, p, of a township boundary that we could find another township just like our township nearby (namely, 2p away). How is the term radius of a township defined? (or is it the minimal number of continuous steps needed to get from one unit of tiling to some other unit of tiling)? ( II ) Martin Gardner also mentioned that in Jan 1977 Scientific American's math column, that there can exist holes in an infinite tiling that might intend to span all of the plane, but can't (because something bad, such as Batman or an asterix can appear if we use it as a hub of a wheel). Conway had apparently conjectured that any possible hole, regardless of shape/size, is equivalent to a decapod hole. Is this still a conjecture? If yes, in what ways has is been strengthened, or does there exist a counterexample that I haven't seen yet? Thanks for any help you may be able to provide; I've got a presentation in Tiling theory due on Wednesday. Also, do you know of any way of contacting guys such as Penrose, Conway, Stewart, Gardner, or others currently involved in Tiling and Lattice theory? I can't seem to find any of their email addresses. P. S. Do you do any Mathematica package work with flowsnake tiling constructions (also in a S A Martin Garner column, probably in the eighties?), infinite/sporadic families of finite simple groups, Parquet deformations, the transaction article in Feb 1996, tiling a convex region with equilateral triangles (of which two or more triangles have coprime lengthed sides...I'd reeally like to see other results that S A was unable to publish, but claimed that they had a winner for) and/or psychological illusions such as those articles found in Nov, 1998. I'll reference you to the Class I'm taking...it's called 424 (actually... Math 424, Applied Abstract Algebra).We are to pick a topic that we can cover in 20 mins tha t has a direct relation to Applied Abstract Algebra (the book published by Lidl and Pilz 2nd ed, 1998)... (no prior (third year or higher) algebra/group theory classes were needed, and I don't remember too much about the seventeen symmetry groups or their notations for various types of rotaions, reflection actions, etc for 2 or 3 space (and how Penrose tilings may not be as nice as the periodic tilings)). However, I may need to bring a litle of this in when I present my 10 minutes on Penrose tilings. I'm covering the other 10 minutes with a short example on the banker's problem since it is easy to present. And yes, I'll be using a computer overhead, and maybe a little bit of graphics code from one or two of your examples.

**Re: Convex Hull**

**Re: What kind of math problem is this?**

**Re: Convex Hull**

**Re: What kind of math problem is this?**