Ptolemy constructed planetary orbits from compound circular motions.
Consider a point p1 moving in a circular path of radius R at a constant
speed of 1 rad/sec around a point O. A second point, p2, revolves
around p1 at a radius of R/2 and a speed of 2 rad/sec (relative to
a frame of reference rigidly connected to p1). Continuing to add
epicycles in this way, each with half the radius and twice the angular
velocity of the preceeding one, we arrive at an interesting fractal
orbit, which can be expressed in complex form as
/ exp(2it) / exp(4it) / exp(8it) /
f(t) = exp(it) ( 1 + -------- ( 1 + -------- ( 1 + -------- ( 1 + ...
\ 2 \ 2 \ 2 \
A plot of this orbit is shown below:
QUESTION: What is the smallest eliptical envelope of this orbit, i.e.,
what is the elipse with the smallest area that contains this
Incidentally, there is another interesting object that might be called
"Ptolemy's Leaf" (because of it's resemblence to an oak leaf) formed by
plotting Re(f(t))cos(t) vs Re(f(t))sin(t). This is illustrated in
the plot below:
A somewhat similar construction begins with an arbitrary triangle
with edge lengths 1,a,b where a and b are both less than 1. Then
we construct similar triangles on each of the legs, and then
similar triangles on each of those legs, and so on. To illustrate,
here is a picture based on a triangle with sides proportional to
1, a=0.4, b=0.77629...
This shows the intermediate stages as it approaches the ultimate
fractally looping curve. The perimeter length of the original ramp
is just a+b, and the next stage has segments of length a^2, ab, ab,
and b^2, so it's length is (a+b)^2. And so on. Obviously at the
nth stage the path contains exactly C(n,k) segments of length
a^(n-k) b^k, and the total length is (a+b)^n, going to infinity as
n increases, and the path approaches the true fractal curve with
loops on all scales. This is quite similar to Koch's "snowflake"
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