# A function with a self-similar, fractal graph
# Putting together geometric shapes (as in the Sierpinski gasket and
# vonKoch snowflake) is not the only way to construct self-similar
# fractals.  We can apply the same idea to familiar analytic functions. 
# Consider the function f(x) defined as follows:
> f:=x->sum(cos(2^k*x)/(1.5)^k,k=0..NumTerms);

                               NumTerms
                                -----        k
                                 \      cos(2  x)
                     f := x ->    )     ---------
                                 /           k
                                -----     1.5
                                k = 0

# Certainly the sum converges for all x (because we are adding terms
# whose absolute value is smaller than a geometric series of ratio 1.5),
# so f(x) is well defined, even if we take NumTerms to be infinity. 
# Unfortunately, for arbitrary values of x, the infinite sum is
# difficult to evaluate.  We'll truncate the sum after, say, 50 terms.
> NumTerms:=50:
#  This isnt a problem, since  the resulting function is off from the
# "real one" by at most
> sum(1/(1.5)^k,k=(NumTerms+1)..infinity);

                                         -8
                           .3136657091 10

# which is small enough for us to neglect here.
# Why should we expect the graph of the function to be self-similar?  
# We can think of the sum as telling us how to build the function
# iteratively, with each additional term giving us the next "level",
# just as we did for the previous fractals.  At each stage, we take the
# graph we had before, and push it around by a small copy of the cosine
# graph.  (notice that the period of cos(2^k*x) is half that of
# cos(2^(k-1)*x), so we're doubling the number of "wiggles".  Also, we
# decrease the height of the added wiggles by 2/3 each  time.  This is
# almost the same idea as what we did with the geometric objects.
# 
# Enough chatting.   Let's see what the function looks like.
> plot(f(x),x=-2*Pi..2*Pi,numpoints=1001,axes=framed);

# Just to emphasise the self-similarity, lets compare two small pieces
# of the graph.
> plot(f(x),x=-0.1..0.1,numpoints=1001,axes=framed);

> plot(f(x),x=-.00078..0.00078,numpoints=1001,axes=framed);

# Despite the fact that the second graph is over a domain about 125
# times smaller than the upper, the two graphs (after rescaling) are
# nearly identical.
# 
# Also notice that the choice of the rescaling factors were somewhat
# arbitrary (as was the use of cosine... just about any bounded function
# should work, although being periodic makes things easier).  What do
# you think would happen to the dimension of the graph if we change the
# 1.5^k to 2^k?