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This page features still images of the Burning Ship fractal.

For a deep zoom video, visit this animation page.

Introduction

The "Burning Ship" fractal is a spectacularly beautiful variation on the Mandelbrot set invented by Michael Michelitsch and Otto E. Rössler way back in 1992, consisting of a small modification to the Mandelbrot set's formula (see below).

A seemingly small change in the formula makes a striking difference in the appearance of the fractal!! Images of this thing are incredibly beautiful and radically different from any "analytic" fractals (my term). The non-analytic part is the absolute value, which takes away a lot of the curviness and turns it into angles and lines.

This fractal is absolutely seductive. Every time I zoom into a new area, I want to render it at some insane resolution with 25X noise reduction. It just compels enormity of implementation, it is so beautiful.

Technical Details

The formula for the Burning Ship fractal is very similar to the formula for the Mandelbrot set. Instead of iterating

z = z² + c

which expands to

z = [Re(z) + i Im(z)]² + c

which generates the Mandelbrot set, the Burning Ship fractal is obtained by iterating

z = [|Re(z)| + i |Im(z)|]² + c

The difference, in case it's not clear from the 1-pixel wide lines in the above equation, is the absolute values of the real and imaginary components are used in the Burning Ship fractal.

A small detail to note: Traditionally, the Burning Ship fractal is drawn with at least the y-axis inverted (i.e. negative numbers at the top), so that it looks like a ship. Often, the x-axis is reversed as well, so the ship appears to be heading to the left rather than to the right. All images here have only the y-axis reversed.

Images

Click on any image to download a high-quality version.

Overview

Here is a link to a 1024x768 overview of the Burning Ship fractal. This is centered at (-0.5,0.5) and has a size of 3 x 4 on the complex number plane.

To the left we can see a little tail reminiscent of the western antenna structure in the Mandelbrot set. Magnifying that 10X shows there is an entire armada of little mini-ships.

Magnifying still more, zooming into the largest ship structure, we find one of the most famous structures in this fractal, the one that inspired the name. This image size is 0.095, about 32X the size of the overview image.

Different Colorings

The same fractal data can of course be colorized differently to give a different visual feel.

Here is a nighttime burning ship.

And here is another rendering with eerie blue-green colorizing (more creepy looking on magnified images -- see below)

Burning Ship Lady

The ship has at least one passenger, a lady waving her arms, probably calling for help. She is standing on the forward deck where the lacy structures take a deep dip down. This image has a size of 0.04 and was made with 3x3 oversampling for noise reduction.

Embedded Mini-Ships

Just as other Mandelbrot-like fractals have infinitely many embedded mini-sets, the Burning Ship fractal has embedded mini-ships, and not just those on the waterline behind the main ship. Zooming into the sails of the ships reveals incredibly intricate structures filled with mini-ships. The image on the left is a 1024x768 extreme deep-zoom to 2.34e-100. The mini-ship on the right is at a size of 3.6e-40 and 640x480 resolution.

Some of the mini-ships look like aircraft carriers. This 1280x720 image has a size of 1.37e-37 and was drawn with 3x3 stochastic supersampling.

Some of the little embedded mini-sets look very little like the main ship. This is a 1024x768 image at 1.52e-16.

Other Ships in the Armada - And the Water Reflections

Magnifying different areas of the armada area at the left can reveal some spectacular ships. The image below has a size of 9.1e-3 and is 1920x1080.

This is magnified about 1 million times relative to the main fractal, with a size of 4e-6. It was drawn at 1024x768 resolution with 3x3 noise-reduction oversampling.

The next image is a deep-zoom to 3.1e-13 and was drawn with 3x3 oversampling. The "reflection" of many of the ships are similar to how the Mandelbrot set's mini-brots look in the Western Antenna region.

Here's a microscopically tiny ship (you can't see it, but you know it has to be there) in one of the creepy sheets draped off the port side of one of the ships in the armada. The reflection in the water is even more strikingly similar to how the Mandelbrot set's Western Antenna looks.

Magnifying the Sails

The sails contain extraordinary structures that look nothing like the ship itself. Here are some examples. Many more images are in the galleries below. What the Mandelbrot set makes of curlicues and spirals, the Burning Ship fractal does with angles, lines, and boxes.

Note the bottom right image, which is a magnification of the bottom center image: There's a little squadron of mini-ships in there, with little mini-squadrons ad infinitum.

2.56e-4 640x480			3e-5 1024x768 5x5 NR-OS			3.3e-15 1280x1024			1.3e-10 1280x1024
1.58e-9 1600x800				7.9e-12 1600x800				7.9e-13 with 3x3 NR-OS 1600x800

Higher Powers

The function that is iterated to generate this fractal can be generalized to any arbitrary power:

z = [|Re(z)| + i |Im(z)|]^N + c

As with the Mandelbrot set, this creates shapes that resemble the original shape but have increasingly many attached arms or buds. Below is a table showing the first few of these.

With N=2 we get the Burning Ship. The N=3 version has been called the "Bird of Prey."

Integeral Powers of the Burning Ship Function

N=2	N=3	N=4	N=5	N=6
N=7	N=8	N=9	N=10	N=11

The odd powers are symmetric around an axes that are oriented at +/- 45 degrees. This is probably due to the symmetry of the absolute value functions. This property has been mentioned at this site (external link) back in 1999.

These images are all centered at (0,0) in the complex number plane, and all have a size of 5 horizontally and 4 vertically. This size is a bit too big and results in a lot of blank space in the images, but the N=2 fractal is offset from the origin, so the larger size is needed to have all the images drawn with a consistent location and size so they can be compared accurately.

Cubic Burning Ship Images

Here are a few images of the Cubic Burning Ship. The "bug" images are named because they look like actual bugs, not because of a software error! These are the first deep zoom images ever published of the cubic Burning Ship fractal. The left image size is 1.0e-19, and the right image is 1.0e-23.

Update 19 Jun 2011: A high-def video based on these images was published.

Additional Image Galleries

Below are a few galleries of more-or-less related images of the Burning Ship fractal.

Super High Resolution 3200x1600 Images (Uploaded to Wikimedia)

Overview	The Armada at the left 10x10 oversampling
Closeup view of the Armada 10x10 oversampling	Large image at 3e-9 size 10x10 stochastic supersampling

Series 1

8.14e-8 3x3 oversampling

Series 2

3.2e-11	3.3e-12	6.6e-10	2.6e-33	3.55e-48
2.27e-50 3x3 stochastic supersampling	7.18e-51	5.7e-72	4.0e-11	4.0e-12
2.2e-29	7.0e-33	1.92e-42 6x6 oversampling	8.5e-45 5x5 oversampling	7.5e-52 3x3 oversampling

Series 3

2.6e-13	4.2e-14	1.67e-14	4.5e-16
1.05e-13 3x3 NR-OS		1.4e-13 3x3 NR-OS

Series 4

2.8e-28	8.9e-29	8.9e-30	2.8e-30	8.9e-31
2.8e-31	1.9e-32	3.8e-33	5.0e-24	1.5e-16 Same as above

Here are a couple of deep-zoom images of the foamy fiery area above the deck of the ship.

Series 5

4.34e-39

1.37e-39

Finally this. I lost track of where it came from but I can't just throw it away. It's size is 1.87e-15 and it was drawn with 3x3 oversampling.