The Cubic Surfaces DVD

by David A. Madore

What is this?

The Cubic Surfaces DVD is a series of computer-generated animations (54 in total) displaying cubic surfaces and deformations thereof. It was created to help better visualize what cubic surfaces look like and what their singular points can be: it tries to be of both mathematical interest and artistic value.

The DVD ISO image is available for download from this page, and the individual animations can also be downloaded separately.

For some mathematical generalities about cubic surfaces, please consult this historical article, this MathWorld article and this Wikipedia article. Further information can be found in Yuri Manin (Юрий Иванович Манин)'s classic book, Cubic Forms: Algebra, Geometry, Arithmetic (North-Holland: 1974; second enlarged edition 1986 ISBN 0-444-87823-8). Of related interest is Arthur Cayley's historical Memoir on Cubic Surfaces (Phil. Trans. R. Soc. London, 159 (1869), 231–326) classifying all possible singular loci. The DVD author's doctoral dissertation also pertains to cubic (hyper)surfaces.

(Of course, you don't need to know any mathematics just to watch the animations!)

The DVD animations were computed using the Persistence of Vision raytracer (on a PC runing the Debian GNU/Linux operating system). Their source code is also made available for download (although it might be hard to make anything useful of).

The entire contents of the DVD and its source code and related material (including this Web page) are placed in the Public Domain by the author (David A. Madore): they can be copied or reused at will and for any purpose whatsoever. However, I would appreciate getting some recognition for my work: so, please cite me as the original author of this DVD if you like it.

The DVD

Downloading it

To save server bandwidth, the DVD ISO image for burning is made available as a bit torrent: you will need a program such as BitTorrent or Azureus to download it. Once you have BitTorrent (or some other similar program) installed, please give it the following link to download the image:

download link for the DVD (bit torrent file)

If you absolutely cannot use BitTorrent (e.g., if you are behind a fascist firewall that blocks the bit torrent protocol because it is a form of peer-to-peer), try downloading the previous link without the final .torrent extension in the address: this will fetch the image directly from the server.

The ISO image is 610MB in size.

Burning it

Once you have the ISO image downloaded (it should be called cubic-dvd.iso), you should use some DVD burning software to rip it. Make sure you burn the image as a raw ISO-9660 image, do not copy this as a file on a DVD (it is already a DVD image, not a file to be included on a DVD image!). Under Linux (or some other Unixen), for example, one might use a command line such as: nice --18 growisofs -dvd-compat -speed=4 -Z /dev/dvd=cubic-dvd.iso. Under Windows, these instructions might be of some help (though they pertain to burning CD's, I imagine that burning DVD's has much in common). Under Mac OS X, select Image then Burn Image in the Disk Copy application.

For maximal compatibility with certain standalone DVD players, it might be preferable to use a DVD−R (as opposed to DVD+R or DVD±RW) medium. This is not universally true, however (in fact, I have observed the opposite); and, of course, the medium must be appropriate for the recording drive. Since the image is small enough to fit on a CD, you can try using that medium instead of a DVD (but again, make sure to burn an ISO image, not a CD containing a single file): this might work on certain players, but it generally will not. Your mileage may vary.

Buying it

✱ If you don't feel like downloading and burning the DVD yourself, you can also buy it online for 4€, thanks to Ikarios (with whom I am not affiliated, nor do I get any profit from the sales!). Make sure to get the most recent version, though (currently, 2006-10-22). This is mostly convenient in France, although they will also ship to other countries.

Notes on the disc's content

The videos on the DVD are in the PAL format (720×576, 25 frames per second) which is common in Europe. Hopefully this should not make it difficult to view the DVD elsewhere in the World. Naturally, the image is not region-encoded, copy-protected or content-scrambled in any way.

Once you select a video from the DVD menu, it will loop forever: use the buttons on your remote control to return to the menu. To play each video for one minute, in order, highlight the title in the DVD menu (this is only valid starting from version 2006-10-22 of the disc): then you can use your remote control to switch chapters.

The version described on this page is that of 2006-10-22. A previous version was released on 2006-10-16. Version 2006-10-22 has the same videos, but differs in the following aspects: a bug was fixed which caused the highlight in the menus not to be visible on certain standalone DVD players, a title (and indication of author's name) was added in the intro sequence, and the possibility was added of viewing all videos sequentially.

I plan to make a new version of the DVD, with optional subtitles describing each animation, possibly some time in the future. For the moment, there are no comments whatsoever: please refer to the list below for a very short description of each animation.

The animations

Each of the following is a link to one of the animations of which the four thumbnail images are taken. Each is a twelve-second sequence, encoded as an MPEG2 file roughly 2MB in size (on the average), and which is meant to be played in a loop. If you don't know what program to use to view animation files, you could try, for example, VLC.

An explanation might be useful, especially for those who aren't algebraic geometers, concerning the use of the phrase deformation of foo to bar in the descriptions below: this means that most frames of the animation represent bar, and only one (or perhaps two, or some small number) represent foo. For example, deformation of an E6 singularity to a D5 singularity means that you mostly see a D5 singularity (bar) which for one moment becomes an E6 singularity (foo) and then back to D5, and not the other way around: yes, I know this is counterintuitive, but I chose to place the specialization (foo) not at the very beginning of the animation so as to make it more clearly visible. It is always exactly 3 seconds (75 frames) from the start, and it is also the third thumbnail image in the list below.

Note that no matter how the surface deforms or rotates, it is always lit in the same way: a green light source is directly above, a red one is above and slightly to the right, and a blue one is directly below. This never changes. The surface itself is colorless, only the light sources (and reflections) account for the colors. Another thing is that, to prevent it from going off to infinity, the surface is intersected with a fixed sphere (always the same), which provides for the smooth borders.

Here is the list of all (54) animations:

The source code

The source code for the Cubic Surfaces DVD is available for download (it's a standard Unix gzipped tarball, and it's meant for various flavours of Unixen). I'm also including it on the DVD itself. It might be hard to make anything useful of, however, since it is just a raw dump of my working directory and I haven't recorded every command line I typed. I'm afraid I don't have the time to explain how to create a DVD image from this source: you'll have to figure it out yourself. So, do what you will with this. Enjoy!