High-Precision Deep Zoom


Buffalo Deep 1

The first deep-zoom animation of the Buffalo Fractal


The Buffalo fractal is an obscure fractal related to the Burning Ship fractal. The formula that is iterated to make this fractal uses absolute values of the complex number components like the Burning Ship does, but the Buffalo formula is a little different. The details of the math are below, if you're interested.

As far as HPDZ is aware, this is the first and only deep-zoom animation into the Buffalo fractal, and the only animation of any kind into the Buffalo fractal.


This is an odd fractal -- at least this part of it. Unlike the Burning Ship fractals (the second-order and third-order ones that I've animated so far) this one seems to be a "dust" off at the western end. By that I mean that I have been unable to find any mini-Buffalos anywhere over there. The visual centers of all the structures, where you normally find mini-sets, just keep going with no mini-sets at all. You can see several points in the video where it looks like it's zooming into nothingness, but a tiny dot appears, then a whole new world of new shapes and structures emerges.

Download Options

In order to conserve limited server hosting space, the larger versions of this video are only available as MP4 files. A small WMV file is provided for IE users who would like an on-demand viewing experience, but the best way to view this animation is to download the big HD MP4 file.

Trouble Downloading?
Video download options
MP4 Files (QuickTime player)
HD Quality Download1158 MB1600x90010 Mbps
DVD Quality (sort of)243 MB800x4502 Mbps FASTSTART
Fast On-Demand Viewing28 MB256x144250 Kbps FastStart
WMV Files (Windows Media Player)
Fast On-Demand Viewing36 MB320x180500 Kbps

Video Capture Images

Click on a thumbnail to go to my new simple image gallery viewer, where you can download the full-size images. This gallery has 5 pages, and you can use the Next and Prev buttons below to move to different pages.

Production Details

Vital Statistics
Date Generated:30 Nov - 6 Dec 2011
Initial Image Size:10
Final Image Size:1e-227
Length:15:00 of fractal, 2:36 total
Rendering Time:151.1 hours
Rendering System:Core i7 980X 4.0GHz overclocked
Method:Buffalo fractal, exponential smoothing, interpolation from 870 primary images, 20% overlap, weighted-average pixel interpolation
Noise Reduction:None
Audio:"Corridors of Confusion" by Technetium


The 27,000 frames of this animation were generated by interpolation from 870 primary images that were directly calculated from the fractal data. The primary images are 3597x2027 pixels. The primary images were blended together with 20% overlap on the leading and trailing ends of each primary frame. The weighted-average type of pixel interpolation was used. Exponential smoothing was used to achieve smooth gradients in the images.

The primary images add up to 23.6 GB, and the interpolated video frames take up 144 GB! This was so much data that on the first attempt to render this, the target hard disk actually ran out of space.


The music is "Corridors of Confusion" by Technetium. I never thought I'd make a video long enough to be able to use this piece, but its 15:06 length works perfectly here.

Fractal Formula

The formula is easiest to understand if it's written in two steps:

(1) Let z' = |Re(zn)| + i |Im(zn)|
(2) zn+1 = z'2 - z' + c

In other words, first take the absolute values of the real and imaginary components of zn, then apply the formula on the second line to get the next iteraton, zn+1.

Zoom depth

This video is an extraordinarly deep zoom -- an overal factor of 10228!!! There are a few other fractal videos out there that zoom this deeply, but very few. This is also an extremely long video -- 15 minutes, with 27,000 video frames. With such a huge zooming ratio, it has to be long so the zooming doesn't go too fast.


The only reference to this fractal anywhere on the internet is a small site called theory.org. I have no idea who invented this or who named it.

Gallery Viewer and Thumbnail Pages

Previously, video captures have been put in table format with links to individual files. For the first time, I've made an interactive image gallery for the video capture files, since there are so many of them (76). The table below is filled in by a JavaScript script, and if you click on one of the thumbnails, you'll be taken to a gallery viewer that's a simple PHP script to scroll through the images. From there, full-size images can be downloaded as desired.

This is nice for such a large number of thumbnails, and it's a easier for visitors to browse the images -- you don't have to click on a thumbnail, then click "Back" etc. It also will save time in the future since the code to generate the thumbnail table is now all JavaScript.

Rendering Times

For those who are interested, click on this link for a graph of the rendering time. This shows the time for each primary image, plus the time to interpolate the video frames from each primary image, about 31 frames per primary.

Buffalo Deep 1 Render Time Graph

The big jump in the per-frame render time at about frame 1100 is due to the switch from native double-precision arithmetic to software-based high-precision math, about a 10X slowdown. As each increasing level of precision is reached, you can see a little bump in the per-frame time. When the code gets beyond the optimized, loop-unrolled high-precision math and switches to a loop-based general purpose set of functions, around frame 16500, you can see another jump in the per-frame rendering times.

Primary Data Images

As with several previous porjects, I have included a sampling of the large primary images that are used for the interpolation. Ths file has 87 of the 870 enormous uncompressed BMP files. The file was made with WinZip using the maximum compression, so it may not be compatible with all unzip utilities.

The coloring of these images is different from the video coloring. This is explained in more detail on the Burning Ship video page.