Christopher J. Kimmer, Ph.D.
IU Southeast Informatics
iSci
the informatics of scientific computing
Pruning Vertex Sets
The too short separation vectors along 3-fold directions are
the projections of the vectors
{111000}
, which are integral and
odd-summed;
two vertices joined by these separation vectors will have
coordinate sums of
opposite parity. The 5-fold too-short vectors are projections of
{
½
}
which
implies that two vertices separated by these vectors will have either all
integral or all half-integral coordinates with coordinate sums
of the same parity.
These observations can be immediately exploited to
create different inflation rules by choosing to keep the member of
pairs of vertices separated by 3-fold directions based on parity,
and keeping members of pairs separated by too-short
5-fold directions by their integrality or lack thereof. These choices are
independent, so four rules are possible. An independent set of
options when using cosets
to determine inflation rules is provided by considering the tiling
before shortest separations have been generated, i.e.
immediately after a pruning operation before
has been applied.
The short 3-fold edges contain vertices that have the same parity but
different integrality, so the integrality of the
original
vertices
can be used to specify a different inflation rule.
Integrality can be chosen, say, and the
vertex closest to the vertex with integral coordinates in 6D will be
kept. For the short 5-fold edges,
the endpoints have different parity and integrality. Choosing based on
parity is equivalent to the rules already discussed for five-fold edges, but
different vertices will be eliminated by choosing based on
integrality. Four new rules can be made in this way,
and together with the first set of 4 rules, there are
4
x
4 = 16
rules
that can be specified at each stage. By making a definite choice from
these 16 possibilities
at every vertex that is pruned
,
an infinity of distinct inflation
rules can be specified. It should be
mentioned that words like ``parity'' and ``integrality'' used above
are fleeting; these labels can change based on the inflation convention
used, i.e. based on the length scale used to originally define
the tiling. What is
inflation-independent is that 3- and 5-fold edges each have two distinct
ways of making distinct choices based on the cosets of the separation
vectors so that there are 16 rules to choose from at each pruning operation.
The symmetry of tilings chosen in this way can be determined by changing the coset of the starting vertex. Changing a point's coset means that its integrality, parity, or both, will change, so changing the coset of the starting point used to generate the tiling will result in a different tiling in physical space. In other words, there is always more than one kind of acceptance domain for tilings generated by these rules, and the tilings themselves must be SCI or FCI. Generically, if a series of choices for the pruning operation has visited all 16 possibilities, there will be a different AD for each coset, and an FCI tiling results. This result can be tightened, though, for the AD of half-integral odd points is always different from the AD for integral even points using these rules. This fundamental distinction between the two types of ADs is a consequence of always having to eliminate too-short 5-fold edges based on integrality, for opposite members of pairs of too-close vertices will be selected depending on whether the starting point for growing the tiling is integral or half-integral. Moreover, any time the pruning choice is based on the cosets of the too-short 3-fold edge, distinctions between even and odd-summed starting points will be introduced, and there will be 4 ADs and an FCI tiling. Whenever the choice is made based on the cosets of the short 3-fold edge, the integral lattice points will have the same AD and the tiling will be an SCI tiling with a basis of two different ADs. These few rules have generated an infinite number of SCI and FCI icosahedral tilings.
Another way of choosing between too-close point pairs is to make the choice based on knowledge of the point's perp space coordinates. One good way to choose the points is to always choose the with the smallest displacement in perp space from x 0 . Since the perp-space displacement from x 0 is considered, this method is independent of the coset of x 0 , and tilings of this form will have only 1 AD and be BCI. The displacement that is furthest away can also always be chosen, and this yields a different AD decorating the BCC lattice. By choosing based on displacement in perp space at each pruning stage, a countable infinity of BCI tilings can be generated. Separate choices for eliminating one of the 3-fold and 5-fold too-close pairs can be made at each stage, of course.
A final method for generating tilings is to take a coin, flip it, and make a random choice of points for each too-close pair. This method is capable of breaking the icosahedral symmetry, so care must be taken to make choices only for each symmetry-inequivalent pair. In general, the symmetric method leads to an FCI tiling.
The advantage of the 16 coset-based choices is that their simplest implementations can be used to construct inflation matrices. The inflations of tiles listed in Appendix A can be used formulate inflation matrix entries that distinguish not only between which tiles are in a particular subdivision of a τ2 -inflated tile, but also which coset they belong to. For instance, when using inflation rules based on inegrality, an O 1 half-pyramid can be deemed integral if its first vertex (as listed in Table 3.1 ) is integral, and half-integral otherwise. There will be two distinct kinds of O 2 tile--each with a particular decomposition after inflation, and the inflation matrix will in general be larger than the 4 x 4 matrix expected if there are only four distinct kinds of tiles. This example is presented in more detail in Appendix A for the intrepid reader. The advantage of the perp-space based rules is that they lead to only one AD, so there are fewer objects to keep track of when specifying the tiling. Also, since the quasiperiodic surface of Chapter 2 is BCI, the BCI inflations rules are the ones that would be directly relevant to studying the relationship, if any, between these tilings and the surface.
Chris Kimmer 2011-06-01