You can read about the Mandelbox and its construction on its wikipedia page.
My implementation was made shortly after discovering OpenGL and is perhaps one of the worst ever, but I still find images interesting. This is the beauty of graphics programming - it's wrong in so many ways, but still looks good! If this was a device driver, you would find it in Windows.
If you are interested in generating high quality images, have a look at mandelbulber.
The images are made from point clouds of raycasted points. The rays are projected from the view port in a spherical pattern, instead of through a rectangle representing the screen, as one would expect. The idea was to display all visible areas on the mandelbox from the current location. This way the user could look around without having to re-render the scene, which takes a long time. This spherical method was eventually abandoned in later versions.
The rays where not evenly distributed over the projection sphere resulting in rings that can be seen by looking closely at some of these images. Another problem was the lack of depth perception, I think the light was initially positioned at the same point as the viewer. After rendering you had to navigate away from the render point to see the resulting point cloud in detail. This is the reason for the random white space in these images.
Here is every screenshot I have. Initially the entire box was rendered as a massive point cloud. At this point the algorithm was also wrong but still produces a boxlike object. Then came the proper raycasting method in an attempt to reduce the points and increase the resolution, and eventually the correct mandelbox algorithm.