From: Chris Hillman <hillman@math.washington.edu>
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 <hillman@math.washington.edu> 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