The Smallest Perfect Universe and Its Friends

The story of the movies "PG(3,2): The Smallest Projective Space" and "Growing Generalized Quadrangles"

by Burkard Polster

When I first came to Australia in 1995 I got very interested in finite geometries. In particular, I spent a lot of time collecting and inventing symmetric real-world models of finite geometries that capture much of their beauty, construction principles, particularities, substructures and interconnections. The book A Geometrical Picture Book was one of the outcomes of this interest. It was published in 1998 by Springer Verlag.

At around the same time I started producing little animated movies featuring the smallest projective space. Eventually, I combined these clips into a 30 minutes long documentary. Producing movies like this was not a straightforward matter back then, at least not on a regular home computer. Uncompressed, at 15 frames per second, a frame size of 320 x 240 pixels and using millions of colours, my masterpiece ended up being 2 Gigabytes in size. In fact, this did not leave room for much else on my laptop's hard drive. The relatively small frame size and frame rate were dictated by my poor computer having to render every single frame from scratch in the raytracing program POV-ray, which even with these modest settings took forever. Eventually, it was done and I was fairly happy with the end result.

The main problem with the choices I had made turned out to be that it was basically impossible to compress the movie and still get something watchable. Even now just unleashing any of the standard compression software packages on this movie does not produce acceptable results. This meant that every time I wanted to show the movie I had to first clear the hard drive of my laptop and then upload the uncompressed movie from CDs. This also meant that for a long time it was basically impossible to give the movie to other people.

Eventually, I forgot all about the movie, and it was only in 2010 that I watched it again, probably for the first time in at least 8 years. At the same time I figured out that a compact version of the movie can be produced by (screen)capturing the movie while it is playing on my computer.

The 3d model of the smallest projective space that the movie (literally) revolves around was first shown to me by Alan Offer, who had built a version from pipe cleaners. The strange Darth Vader voiceover is actually my voice in action with its pitch electronically lowered. Oh well, it seemed like a good idea at the time... . The amazing soundtrack was produced by Stuart Johnson.

Anyway, here is the movie, split up into two parts to keep YouTube happy. The movie is fairly comprehensive in what it covers, and to be able to appreciate absolutely everything you have to know a fair bit about finite geometries. Still, even if you've never heard of finite geometries, you should be able to get something out of it. In this case, you may want to first read my article YEA WHY TRY HER RAW WET HAT as an introduction. Anyway, please enjoy the movie for what it is.

The small generalized quadrangles, the stars of the short movie Growing Generalized Quadrangles which you can also find below are even more esoteric than the smallest projective space. Having said this, it is very easy to follow what the movie is trying to show, namely how some pretty models of these geometries can be constructed from scratch. Again, enjoy this one for what it is.

PG(3,2): The Smallest Projective Space

Part 1

Part 2

Growing Generalized Quadrangles