Mark
Dow
Geek
art
Simple
recursive systems
and fractal patterns
2x2
2-symbol
L-system Fourier transforms
2x2
2-symbol
L-system Fourier transform
thumbail chart
DFT
amplitudes (value) and phase (color)
2x2
2-symbol
L-system Fourier transform tiny thumbail chart
DFT
amplitudes (value) and phase (color)
About
Fourier transforms and their graphical representation
[To Do]
The discrete Fourier transform of an
image is always
a full description of the image. Instead of pixel values the Fourier
components are amplitudes and phases of single spatial (sine wave)
frequencies. But does it make sense to describe non-periodic
sequences in terms of Fourier transform components? Fourier components
are periodic sequences; how can sums of periodic sequences describe a
non-periodic sequence?
Remember that a Fourier transform
component
is an integral (sum) over a periodic extension of a finite sequence.
The finite sequence is not periodic. But the Fourier components
represent both the non-periodic aspects of the finite sequence and the
periodic extension. These aspects of the patterns can be distinguished
-- the two patterns are seperable. For example, as the length of the
sequence is extended toward infinity, the non-periodic pattern
maintains its form while the periodic pattern is reduced. [To Do:
Illustrate.] I guess it's not too surprising that the pattern of
Fourier components are also non-periodic sequences.
Some of the sequences (rows and/or
columns)
considered here are periodic. Some are almost periodic -- a few point
changes to the sequence will make them periodic. Others are highly
redundant. The redundant sequences have repetition of elements at many
length scales, but the repetitions are at fixed separations. Another
way of describing redundancy is "periodicity at non-periodic phases".
This is reflected in the non-periodic regularity of the Fourier
transform amplitude and phase diagrams.
Do
Fourier transforms of non-periodic patterns have any relevance? Why
would you want to know about periodic representations of non-periodic
patterns?
Yes they do. Many of
these systems have periodic and non-periodic parts. Furthermore the
non-periodic parts are almost periodic! The Fourier transform does a
nice job of highlighting the small non-periodic parts of almost
periodic systems. [To Do: Explain how phase and ampitude information
are complementary.]
Notes
[To Do: Examples of the following.]
A first rule that is all black, such
that the second
rule is never reached, is not included as it is not a two symbol system.
One pattern is included twice, one the
inverse
(symbol interchange) of the other. While I don't count this as a unique
pattern, the system occurs is properly located in two spots.
The 0 -> [0,0,0,0], 1 ->
[1,1,1,1] system
never returns to symbol zero. The pattern is equivalent to its fixed
point after the first generation. But it is a proper unique system if
the evolution is considered.
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