Understand the Turtle Graphics description in the
L-systems tutorial at Australian National
University
and answer exercise 1 at the end about how to create a random walk
L-system.
is true, then you can conclude
that 
is true.
.
be the sequence of generations produced in the
Fibonacci L-system. Since 
is the beginning of 
, we
can consider the limit of as 
, which is an
infinite sequence 
of a's and b's. The beginning is
.
, where each
is an a or a b. From your calculation in the first part,
write down the sequence of integers n for which 
. (The
first few are 
.)
(the golden
mean). For 
(or more), compute 
, where 
means the integer part of the
real number x. Compare your results with the previous answer and
discover a pattern! 
Show that there do not exist positive integers k and m such that
(This problem asks you to prove the theorem that the Thue-Morse
sequence is cube-free; to do this you must use the production rules
and 
. As preliminary work, try to prove the
following:
find the ``global process,'' i.e. express 
as some
combination of and .
be the generations of the sequences of L's and R's
that come out of the paperfolding process described
here. So, in unfolding
the paper the L's correspond to creases pointing downward, and the
R's correspond to creases pointing upward.
from . (Hint: the new folds occur between the old ones.
Consider the left and right halves of the paper separately.
Describe how to enumerate the new folds.)
. as 
where 
corresponds to an L and 
corresponds to an R.
Find a simple formula for 
.
.
. (You need to write 
.)
be the generations of L's and R's coming out
of the ``alternating'' paperfolding sequence, where first you fold
right side over left, then you fold right side under left, then you fold
right over left, and so on, alternating the two types of folding operations.
. The figure below
gives you some help. as 
where each 
is L or R.
, n=5,6,7, write down the sequence of
vertices which are waist points. For example, for the first four
generations the sequences are as follows.
 | 1 |
 | 1-3 |
 | 1-7 |
 | 1-6,12-15 |
The vertices 7-11 bound a square in generation 4.
, the dragon curve tends to a
sequence of ``islands'' touching at waist points. What is the
relation of one island to the next?
discussed
in this section.
is a similarity by explicitly determining a dilation 
,
a rotation 
, and a translation 
such that
, where 
denotes the
transpose of the column vector 
, to deduce an equation
satisfied by A.
, we let 
denote the usual Pythagorean length
of the vector 
, we define just as before a
similarity 
to satisfy, for some c>0,
for all vectors 
.
If c=1, U is an isometry.
Let 
and 
just as in the
two-dimensional case.
where 
is an isometry such that 
,
where 
.
where A is a 
real matrix
satisfying 
, the identity matrix. (Hint:
construct some rotations 
, 
such that 
fixes (1,0,0), (0,1,0) as well as (0,0,0). Try to follow
the rest of the argument in the section on
similarities.)
. (see the
discussion around this figure.)
and the region R as described in
the section on self-similar tilings. Find the L-system in a and b that
generates this tiling. To do this, try to find the integral matrix
for the linear transformation T that has eigenvectors
and 
and eigenvalues 
.
, 
. Explain the pattern
that emerges.
for 
.
in the code-space that 
is always
equal to the same infinite Fibonacci sequence of a's and b's.
by
So 
, 
, 
, etc.
, 
.
of 
,
for 
. as .
, explain how
to find a covering of the Cantor set by disks of radius smaller than 
that are
mutually disjoint (that is, no overlap).
is exactly 0.5.
and points on the Koch curve was
described. Characterize all points on the Koch curve for which there
correspond two different sequences in the code-space.
be the similarity that takes S onto tile number i. In
the second figure, we use these linear transformations to repeatedly
subdivide the original Sphinx tile. Find the code sequences (in
1,2,3,4) that correspond to the small tiles labelled A, B and
C.
and 
be the
two affine maps whose attractor is the Cantor set. Show that
belongs to the Cantor set by explicitly determining the
code sequence in 1,2 that corresponds to x.
, where
A is the 
coefficient matrix.
, prove that
for all x satisfying 
.
(Hint: start by proving 
whenever .)
and 
.
Suppose there is a linear map L(x)= ax+b, 
, such that 
(provided 
). Find the relation
between c and 
and determine the linear map L.
and 
are linearly conjugate. for which 
has a stable cycle of period 5, 7, 6, resp. (numerically to 10
digits!) for which 
has a stable 3-cycle which begins
with 
.
