One way of coloring the inside of the M-set is to work with
the the various derivatives that describe the 'flow' of
Zn. These images are all based on the computation
of the derivative of Zn
with respect to c. The expression for this is simple:
Z'n+1 = dZn+1/dc = 2 * Zn *
Z'n + 1, which must be iterated.
Note that this derivative can be though of as a vector: it has a real
component and an imaginary component. Thus, we can take its
curl or divergence as well as it's absolute value.
Because Z'n has a high 'rotation', it is handier to deal with
the normalized vector Pn = Z'n/Zn
Here we graph the phase
phi=arctan (Re Pn / Im Pn).
Note that not all buds are
alike: the interior of some buds see a full rotation of the phase by
360 degrees; others do not.
Very curiously, this seems to depend on the
factorization of the loop termination count. When the loop termination
count is prime, none of the buds rotate; when it is a product of many
small (repeated) factors, most buds rotate. For example, in this
image, most buds rotate because iteration was stopped at 1260=4*9*5*7,
whereas this image stopped at 1259 (which is a prime number) and
none of the buds rotate.
The phase of the second normalized derivative
Qn = Z''n/Zn
2 * (Zn * Z''n + Z'2n)
Modulus of second derivative.
Iterated to a high order (N=4153) before stopping.
Same image as above, but iterated to only N=19. Note the presence of
'zeros' (black dots) on the interior, which are missing from the
As above, but iterated to N=18. Note that this image contains one less
black dot on the belt.
Iterated to N=17. Notice how the images generated with prime N are
subtly different than those with composite iteration counts.
Iterated to N=16. Note that 18 is divisible by 3, while 16 is not:
thus the large bud has a green dot in the middle, but the next-smaller
one doesn't (unlike the N=18 case).
The second derivative is handy for exhibiting another phenomenon that is
hinted at elsewhere in this page, but otherwise difficult to exhibit.
This image is created quite artificially: the interior color is recorded
only if the modulus of
Qn = Z''n/Zn
is less than one for some n. In other words, we record the color only if
Qn is near a zero. Thus we see a simultaneous
depiction of the zero's of the various Qn for various n.
These seem to go to a hyperbolic-like limit-circle.
In this image, black is zero, moving
through blue, green, yellow to a red of 1.0.
Note the Moire
patterning make it difficult to discern where the actual limit is.
To eliminate this, this image was iterated only to N=50, and so we can
see the limit-circle at 50 dots.
The 'distance estimator' is the inverse infinitesimal flow of the
iteration number. It gets this name because it provides a rough
estimate of how far away an exterior point is from the boundary
of the M-Set. The smooth (real-valued) iteration count is given
by mu = n+1-log(log(|z|))/log2, as demonstrated in the
Escape Theory Room
(which is, in turn, the logarithm of the
Taking the derivative of mu w.r.t. c
we get dmu/dc = d|z|/dc / (|zl log |z| log 2).
The 'distance estimator' is one over this quantity.
This first picture shows the iteration to n=9. (We've dropped the factor
of log 2).
Iterate to n=19.
Iterate to n=49.
Iterate to n=2520=8*9*5*7 a product of small primes.
Oddly, the n=2 bulb is unlit, but n=3 and so on light up.
Iterate to n=2519 which is a prime number. Oddly, the n=2 bud lights
up, the none of the other buds do. Thus, the counting rules here seem
to be a bit different than one might expect ...
Don't confuse the absolute value of the derivative with the derivative of the
This image shows |z| / |z'|.
This image shows |z| / |z|'.
Copyright (c) 1997, 2000 Linas Vepstas
by Linas Vepstas is licensed under a
Attribution-ShareAlike 4.0 International License.