Reenigne blog

A good way to speed up a Mandelbrot set plotter is to eliminate the main cardioid and the largest circle. It turns out that there are simple equations for these, which can be found if you know that the cardioid consists of all the points which converge to a single point and the largest circle consists of all the points which converge to a cycle of period 2.

First the main cardioid. For each c, we can find the fixed point of iteration such that . There are two solutions, . Then we can look at what happens when we iterate a nearby point . If then the fixed point is stable. It turns out that this only happens for the solution, so points in the cardioid are , with equality for the boundary.

Having to compute a complex square root is a bit ugly, though - can we do better? It turns out that we can. It's a fiddly calculation but if you multiply out all the square roots and simplify, you can get the formula .

For the period-2 circle, we solve and eliminate the period-1 solutions to get . In this case, both solutions are equally valid, since the cycle consists of both. Picking one and doing the same derivative analysis (this time applying two iterations) gives us the circle with center at -1 and radius 1/4.

The video I made for Saturday's post made me wonder what the picture would look like if you didn't have a branch cut at all - if you make the logarithm function generate all of its multiple values (or equivalently, choose a sheet of the Riemann surface at random). Computers can't count up to infinity so the random method must be used. We can't make all the sheets equally likely as their number is unbounded so we need to pick some distribution. I chose a method that's rather simple to implement - I just roll a (virtual) dice until a 1 or a 6 comes up, and subtract the number of 2s and 3s from the number of 4s and 5s (a 1D random walk). That makes positive and negative numbers equally likely, but makes small (positive or negative) numbers more likely than large ones.

Here is the result:

The infinite tower is transformed into a fractal fern-like structure. I was surprised at how different this looked from the second image on this post - while the non-principal branches are not visited very often, they cover a lot of space so make a lot of difference to the image.

The second image in this post was quite popular, so I decided to make a movie out of it.

There aren't really many parameters in the equations used to make this image, though - (the functions are just negative, exponential and logarithm). The colouring is arbitrary but modifying it wouldn't make a very interesting movie. The zoom factor is fairly arbitrary as well but zooming very much in or out would make the image take much longer to render (besides which, zooming movies have been done to death). About the only other arbitrary thing about the image is the choice of branch cut for the logarithm.

This describes a way of adjusting the branch cut in a continuous way, and this is the movie I made by adjusting theta from -4.5 to +4.5 radians:

High quality 640x480 11Mb DivX avi version.

This took about three days to render. I could have done it faster, but my laptop gets uncomfortably warm when both its cores are fully utilized. Also, a problem with Vista networking was causing the other machine I was using to keep getting disconnected from my laptop every few frames.

A commenter wondered what the images on Saturday's post would look like using a sixth root of unity instead of a fourth. I am happy to oblige. With ω=e^πi/3, this is the image with the operations z+ω, ωz, ω^z:

And this is the image with the operations z+ω, ωz, z^ω:

More variations:

Red: z->z+i
Green: z->iz
Blue: z->i^z:

Same, but with blue: z->zⁱ:

A movie can be thought of as a three-dimensional array of pixels - horizontal, vertical and time. If you have such a three-dimensional array of pixels, there are three different ways of turning it into a movie - three different dimensions that you can pick as "time". For movies of real things this probably isn't a very interesting thing to do, but for movies of mathematical objects like the one I made yesterday, there may be mathematical insights to be gained from "reslicing" a movie.

So, here is yesterday's movie with the x and time coordinates swapped:

Higher resolution version (8Mb 640x480 DivX).

And here it is with the y and time coordinates swapped (and also rotated to landscape format):

Higher resolution version (8Mb 640x480 DivX).

Higher quality 10Mb 640x480 DivX version here.

This is a generalization of the second picture from yesterday's post, varying the coefficient of i from e^-4.5 to e^4.5. It also demonstrates how this picture is related to the third picture from Monday's post.

1.35 trillion points were calculated to make this movie, taking 4 CPUs with 6 cores most of a day (it was going to take nearly 4 days, but I decided to use most of the other computers in the house as a render farm).

Generated by a program similar to the last two pictures on Monday's post, but the functions are , and . Don't plot a point if the last operation was , and colour points according to the operation used 5 iterations ago.

Similar to the third image on Monday's post, but when we multiply by we also multiply by the golden ratio = 1.618... Most the rectangles you see in this image are golden rectangles, which are supposedly the most aesthetically pleasing.