The simplest dissection polynomial for the 6th unit cubic Pisot number
corresponds to a 3 element tile, so we can deduce that there are no 2 element
self similar tiles associated with this number. Restricting consideration to
tiles with all elements directly similar to the tile, we can arbitarily
restrict the IFSs for 3 element tiles to { p → ±ap;
p → ±ap + 1; p → ±a3p +
v2 } (or { p → ±ap - 1;
p → ±ap + 1; p → ±a3p +
v2 }). In principle we can perform an
exhaustive search by iterating the value of v2,
and subsequently closing in from values visually identifiable as close to a
tile. However in practice a step size sufficiently small to be certain to find
the tiles requires too many candidates to be examined for this to be
computationally practicable. It also produces an impracticably high number of
candidates for visual inspection. The process would also not necessarily find
cryptic tiles (tiles like the Levy curve, where the area of the plane wholly
covered by the tile is small compared to the outline of the tile). However for
3 element tiles a heuristic search with v2 =
P(a), that is v2 is a polynomial in
a, is practicable.
The problem is to select a practical but effective range of polynomials. What values of the exponent should be used? How many non-zero coefficients should be allowed? What values should be allowed for the coefficients?
Investigation of other Perron number tilings led to the hypothesis is that
the maximum exponent is not large compared to the maximum exponent of the
contraction elements of the transforms. Values beween -2 and 6 were surveyed.
An initial survey using ±an and ±1 +
±an turned out to be incomplete (applied to the two
alternative sets of IFSs didn't find the same set of tiles). A second survey
was performed using aj + ak + al.
This also turned out to be incomplete. At this point a survey using
(aj + ak + al)/(1-a3)
was performed. (Vectors of the form a/(1-a3) turn up in
tilings using these tiles.) More tiles were found, but using a table of values
of polynomials in a it was found that the corresponding vectors
were equivalent to sums of powers divided by 2 or 4, supporting the conjecture
that the coefficients are ratios of small numbers. (It was already known from
the study of rep-tiles that non-integer coefficients occurred.)
The resulting final survey produced 24 values of
v2 which produce tiles, composed of 8 versions
of the symmetric tile, and 8 pairs of rotationally
equivalent tiles, composed of two different dissections of the windowed tile, two demi-symmetric tiles, two external tiles and two complex
teragons, giving 24 IFSs producing 8 distinct tiles. This is pleasingly
symmetric (even if the symmetry is broken in the case of the redundancy of the
windowed tile) and gives some hope that the search found all tiles. Overlooking
cryptic tiles cannot however be excluded.
© 2016 Stewart R. Hinsley