Some substitution
tilings (or self-similar tilings) from two
colors of squares and the Fourier transforms of their matrices.
A substitution
tiling is generated by first defining a substitution rule on the tiles.
The substitution rule cuts up the white tile into an n
by n array of black and white tiles (based
on the whimsy of the person defining the substitution), and does the same
for the black tile (keeping the same
n).
These rules are fixed and then applied repeatedly to generate the substitution
tilings shown below. In the first tiling shown (the nonbijective
one), the substitution rule was to cut up the white tile into a 3 by 3
array, all white except a black one in the middle. The substitution
rule on the black one was to cut it up into a 3 by three array of all white
tiles. In the second example the substitution rule on the white was
the same as before---all white with a black in the middle. The substitution
on the black tile is the reverse---all
black with a white in the middle. That is what makes it bijective---the
fact that the substitution on the black is the opposite of the substitution
on the white tile. Check it out and see if you can see that.
One can see tilings out of black and
white square tiles as big matrices with a 1 in the (j,k)th entry if there
is a black tile with its lower left-hand corner at the point (j,k), and
a 0 if the tile there is white. OK, so I don't quite match the normal
system of counting rows and columns, but I'm pretty sure that you get the
idea. Anyway, you can run Matlab's fft2 program on the matrix and
then plot the intensities to ``see" the Fourier transform of the matrix,
and that is what I've done here. The Fourier transform of a
tiling is related to the dynamical spectrum of that tiling, and as it happens
it is possible to show (mathematically) that the dynamical spectrum of
bijective substitution tilings is a lot more complicated than that for
nonbijective substitutions. The examples below just begin to scratch
the surface of the order and disorder possible in the dynamical spectra
of substitution tilings. I haven't even included any tilings with
more than two tile colors, or nonsquare tiles... Also, you wouldn't believe
how much subtle changes in the substitution affects the Fourier plot.
I would like to thank Billie J. Rinaldi
for creating the Matlab program which generates these, and many other fabulous
tiling pictures. If you'd like to get your hands on the code, we
can send it to you. Email napriebe@vassar.edu
or rinalb@rpi.edu. The program
is really quite user-friendly, but there are many little things which aren't
totally intuitive (or explained). Be
warned: Your results may vary. Obviously,
we'll happily answer any questions.
Extreme order:
nonbijective 3 by 3 substitution (all square symmetries) with the Fourier
plot of its matrix.
Here's a 3 by 3 bijective substitution
that seems to be very ordered (but not nearly as ordered as the first one):
A 5 by 5 bijective substitution with
all of the square symmetries and its Fourier plot:
A 4 by 4
bijective substitution (rotational symmetry only) and its Fourier plot:
Finally, here's a 5 by 5 substitution
which is just a blobby mess, and so is its Fourier plot:
It is really fun to experiment with this program.