From: Chris Hillman Newsgroups: sci.math Subject: Re: DART & KITE (Game) Date: Thu, 22 Jan 1998 01:11:38 -0800 On Thu, 22 Jan 1998, Bill Taylor wrote: > Chris Hillman writes: > > |> Grunbaum and Shephard coined the term "aperiodic" prototile set for a set > |> of such decorated tiles which can only tile the plane aperiodically. > |> Curiously, all known aperiodic planar prototile sets have at least two > |> tiles, although no-one has proven (so far) that this is the smallest > |> possible number. (In three dimensions and higher, there are aperiodic > |> prototile sets with only one tile.) > > Is it also the case, as I seem to recall, that no set of CONVEX tiles is > known that only tile aperiodically? If you allow decorated tiles, then the Penrose rhombs are an aperiodic set (in the sense of Grunbaum and Shephard) of convex prototiles. If you mean undecorated tiles, then yes, I believe you are correct. However, there is a right triangle with a geometric subdivision into rescaled copies of itself, which yields an aperiodic "substitution tiling" with exceptionally interesting properties. See Radin, "The pinwheel tilings of the plane", Annals of Math. 139(1994), 661-702, also available over the web at Radin's website, http://www.ma.utexas.edu/users/radin/ Another simple rule is the "domino tiling" with substitution rule _ __ _ ______ | |__| | <--- | | |_|__|_| |______| Such substitution rules are closely analogous to the construction of of aperiodic binary sequences like the substitution rules 01 <-- 0, 0 <--- 1 The construction of aperiodic tilings by substitution rules and by matching rules turn out to be closely related: Chaim Goodman-Strauss has shown that any tiling which can produced by substitution rules can also be defined by matching rules; see http://www.geom.umn.edu/~strauss/papers/sub.html These things are also related to the continued fraction algorithm; for instance the substitution rules given above yield the sequence of partial quotients for the Golden Mean. Another interesting construction is due to Petra Gummelt, who noticed that you can build Penrose tilings using marked decagons; you place the decagons so that they -overlap- in a way obeying the markings. This isn't so much a tiling as a covering of the plane, but as an idealized model of physical quasicrystals, such coverings may be more realistic; see http://dept.physics.upenn.edu/~www/astro-cosmo/walker/walker.html > If there are any, what is the smallest such set? If you allow decorations, it is possible that the smallest aperiodic prototile sets (for tilings, not coverings) in the plane contain two tiles. Recently Penrose thought he had found a set with one tile (if I understood him correctly) but it turns out that his construction actually needs additional "key tiles" to work. Gummelt's construction shows that for coverings, there are aperiodic "prototile" sets with a single convex (decorated) tile. > And what is the status of convex aperiodic tiling in 3 dimensions? The situation in higher dimensions is totally different. See Radin & Conway, "Quaquaversal tilings and rotations", to appear in Inventiones Mathematicae; a preprint is available at Radin's website. (Danzer says that an unknown German amateur had a similar construction a decade earlier!) > Anyone have any details on these matters? Radin, Conway, Penrose, Steinhardt, and Danzer, would between them probably know the current situation. Penrose and Danzer are hard to reach by email but Radin & Steinhardt have web pages and Conway has an email address listed in the AMS Directory. For general information on tilings and dynamical systems, see my page http://www.math.washington.edu/~hillman/research.html and the references thereupon. Chris Hillman