Projective and polar spaces

I have produced a new edition of my lecture notes on Projective and Polar Spaces and put them with my lecture note collection.

I did this because it seems that people still find some use for these notes. According to Google Scholar, they are my 16th most cited publication, with 113 citations. The first edition was hard copy in the Queen Mary Maths Notes series in 1990; I had just taken over the editorship of the series from Karl Gruenberg. It was probably a bad time to do that; we sold copies by email (though we had to ask our customers to send a cheque), but didn’t realise how the web would become the location of choice for lecture notes.

In 2000, I produced a second edition which I did put on the web. Back then, downloading a big file was a tedious procedure, so I made each chapter a separate PDF file. Now things have changed, and so I have re-integrated them. Apart from that, the changes are very small, detailed below. I regard this as, in some sense, a historical document, and I am too busy at the moment to produce a proper revision.

In 1990, I had only recently moved to Queen Mary, and even more recently started using TeX. By choice, in those days, I used plain TeX, since I wasn’t very fond of the default LaTeX style and wanted to make my own decisions about how the notes looked. (I still don’t like default LaTeX very much, but this is what journals want these days.) So this was my first attempt at a large-scale TeX document.

Of course, there was a difficulty: a book like this is full of diagrams; not only geometric diagrams like Desargues’ Theorem but also Buekenhout diagrams, which sometimes had to be embedded within the text. LaTeX had a picture environment, which would do the job; how could I do it in plain TeX?

Someone pointed out to me that the code for the LaTeX picture environment is almost completely self-contained and can be detached from the rest of LaTeX. I did so. If I recall correctly, there was only one incompatibility that I had to deal with. Plain TeX used the command \line to refer to a line of text, and complained about the double use. So I simply replaced \line by \Line everywhere in the LaTeX code.

One interesting problem I had to face using this was the restriction on slopes of lines (which were produced by typesetting small line segments from special fonts). A line, other than a horizontal or vertical line, had to have slope which was a rational number with numerator and denominator coprime and at most 6 in absolute value. I needed to draw things like the Desargues configuration so as to appear fairly “generic”, without obvious symmetries or coincidence of slopes. This took a bit of work, but I think I managed fairly well.

However, one thing defeated me. I wanted to show the diagram showing how Desargues’ Theorem follows from Pappus’ Theorem, which involves adding some extra lines to the Desargues configuration so that the three invocations of Pappus are visible. The slopes of the extra lines I needed did not fit the restrictions! In the hard-copy first edition, I drew them in by hand; in the second edition, I threw in the towel, and put a statement in the preface apologising that several lines were missing. Now the LaTeX curves package will draw a straight line through any two points, so in the new edition I have been able to include them. I have also tried to make the diagram easier to follow by use of colour: the line whose existence we are trying to prove is red, projections of existing lines are blue, and other construction lines are green.

Incidentally, there is a problem related to the LaTeX picture environment, to which I don’t know the answer:

What is the maximum number of points in the plane with the properties that no three are collinear, and the line joining any two can be drawn in the LaTeX picture environment (that is, its slope is one of the allowed values)?

In the second edition I referred to some SOCRATES lecture notes containing related material. There was a European network run by Frank De Clerck in Ghent, which ran summer schools on finite geometry. The notes I referred to appear to have disappeared, I am not sure where. But I think that one set was my own notes on “Finite geometry and coding theory”, covering various topics related to quadratic forms over GF(2) including Reed-Muller, Kerdock and Preparata codes and an introduction to quantum codes. These may also have some historical value, so I have posted them here as well.

Enjoy!