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.
You'll need to know a little bit of measure theory in order to
fully
understand my comments about the pictures, but if you don't, just enjoy
looking! I have written a semi-expository paper discussing the
spectrum of this type of tiling from a dynamical viewpoint (you will
need background in measure theory and dynamical systems to appreciate
it). If you want to download it, click here.
Extreme
order:
nonbijective 3 by 3 substitution (all square symmetries) with the
Fourier
plot of its matrix. This tiling has a "purely discrete
spectrum": its spectral measure gives nonzero values to points
only. No additional weight is carried by open
sets. The symmetries of the plot are a side effect of the
tiling's symmetry.
Here's a 3 by
3 bijective
substitution
that seems to be very ordered (but not nearly as ordered as the first
one). It has a "mixed spectrum": there is a piece
that is discrete, but there is another piece that is continuous with
respect to Lebesgue measure. The singular continuous
part is a measure of the level of disorder that has been introduced.
A 5 by 5
bijective substitution
with
all of the square symmetries and its Fourier plot.
This one, like the previous one and the next two, have a mixed
spectrum, but the continuous part is still singular
continuous. So the pictures are all different but the
spectrum is essentially the same from a measure-theoretic perspective.
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:
In order to create substitution tilings with an absolutely
continuous portion to their spectrum, I believe it is necessary to use
more than two colors. I have published an class of
substitutions that have an absolutely continuous component to their
spectrum which you can download here.
The example I'm putting here has eight tile colors in its original
substitution, instead of two. The image you see is the
result of identifying four of them to the white tile and four to the
black, which separates out the absolutely continuous piece of the
spectrum from the discrete part. Notice the radical
difference between the appearance of the absolutely continuous spectrum
from the singular continuous! It represents a higher level
of "disorder", at least as perceivable from the Fourier plot.
I would
like to thank Billie J.
Rinaldi
for creating the original Matlab code that generates these, and many
other fabulous
tiling pictures. If you'd like to get your hands on the code,
you can download it from my home page
or I'll send it to you: just email nafrank@vassar.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,
I'll happily answer any questions.
