This page describes a method for making lines in the Mandelbrot set. The lines are made of repeating ovals and can be made as long as you desire by repeating a set of simple steps. Note that longer lines are at deeper zooms and will take longer to calculate. For some example images see: Shut Eye

First, a little background. I was on the lookout for a method to create lines after a post on the FractInt Mail List that had a link to an image of a long line of side by side X's. There was no parameter file and the hint on how the image was created was vague. I eventually stumbled on the solution and now the hint makes sense. To make it easier to figure out, I decided to make this web page with annotated pictures and a parameter file at the end. If you see something that is confusing or incorrect please let me know.

The first thing you need is a Mandelbrot set fractal with a minibrot visible. Any minibrot should work as long as it is deep enough to have visible arms that double. A typical minibrot is shown in the image below left. Pick a dead end arm like the one shown by the arrow below left. The image below right is zoomed in to better show the details of the dead end arm. Find a point with two way symmetry as shown by the arrow below right. The two way symmetry reveals that there will be a much smaller minibrot at that location if you were to zoom in far enough.

The magnification of the lower left image is 1.9e8 and the lower right is 7.7e8.

lm8ed.jpg lm9ed.jpg

Zoom in on the intersection of the four arms shown by the arrow in the image above right until you see an oval as shown in the image below left. This oval will be the tool used to make the line.

Note that your starting fractal may already have ovals. The kind of oval you need is shown in the simplified image below right. The starting oval has some stuff that is a smaller oval, a center, and a mirror of the smaller oval. If your fractal already has an oval of this type you can start with your oval and skip the above steps.

The magnification of the image below is 3.2e12.


Zoom in on the point at the tip of the arrow in the image above left. Keep zooming down the center until you come to the oval below. The magnification of the image below is 2.6e15.


Now you can see the pattern. To lengthen the line, you zoom into the oval on the extreme right side. Zoom in on the point at the tip of the arrow in the image above. Keep zooming down the center until you come to the oval below. The magnification of the image below is 5.2e19.


In the above image, the oval you want to zoom into is barely visible at the far right. I didn't add an arrow to the image this time. The next few operations are shown below. The magnifications of the images below are 6.6e26, 1.3e38, and 3.3e55 respectively.




The parameter file for the bottom image is below. You can get all the other images by zooming out to the magnifications noted in the text. The rotation and position of the images were changed to better fit the web page.

I noticed there seems to be a pattern to the increase of the magnifications. You can predict that the next line will be at a magnification of about e81. The one after that should be at about e120. Hint: Take the difference of the last two exponents, multiply by 1.5, and add that to the last exponent. For example: 55 - 38 = 17, 17 x 1.5 ~ 26, 55 + 26 = 81

Line_Method { ; Exported from Fracton.
 reset=2004 type=mandel passes=1 float=y
 params=0/0/0/0/0/0/0/0/0/0 maxiter=8000 inside=0


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