Abstract:

Mathematics students often joke about the word ‘trivial’. The word in its serious usage means that a given object in a specific setting does not contain interesting information. Incidence geometry is simple – its definition is even trivial. But does that mean the subject itself is easy and ‘trivial’? Not necessarily.

Blog Post:

Mathematics students often joke about the word ‘trivial’. The word in its serious usage means that a given object in a specific setting does not contain interesting information. For instance, in the context of the field of mathematics known as ‘group theory’, which is concerned with transformations like the multiplication and division of numbers and symmetries of shapes, a group of transformations is called ‘trivial’ if it has only one element – the do-nothing element. The trivial group has no structure apart from the fact that it does nothing, which is not very interesting, even to a mathematician.

This handy little word makes it easy to avoid writing out all the steps required in solving a problem, since at least some of them, and if we’re lucky, maybe even all of them, are ‘trivial’. It also makes it easy to make fun of your maths friends by calling whatever they are working on ‘trivial’, and therefore beneath you.

This brings me to incidence geometry, a beautiful and simple, which is not to say trivial, field of mathematics.

What is incidence geometry? That’s easy – even trivial. By ‘geometry’ I mean space, and by ‘incidence’ I mean that in space we consider points and lines, and we say a point is ‘incident’ with a line if the point lies on that line. In mathematical terminology we say that incidence is a ‘relation’ (think ‘relationship’) between points and lines.

That’s it.

We don’t have rulers and lengths. We don’t have compasses and angles. We don’t have equations describing curves. We don’t even have to define the dimension of our space.

We have points and lines, and points on lines, in space, and that is all.

Sounds kind of… trivial, no?

No! No, it is not trivial…

When I have shared with family that I am doing a Pure Mathematics Honours in geometry, I have more than once heard the reply, “Geometry! I thought you finished all that in high school!” I know they don’t know the word ‘trivial’, or how it’s used in mathematics, because if they did they would surely be crying “Trivial! Trivial!”

So, why is incidence geometry non-trivial?

Allow me to give you an example.

In case you don’t know, projective geometry is defined by three incidence axioms. It is known that for all primes p and natural numbers n, there is a projective plane of prime power ‘order’ N = p^n (this is the number of points on each line). However, it is unknown whether there are projective planes of non-prime power order, except that if one exists, its order N will have remainder 1 or 2 when divided by 4. It was shown via exhaustive computation that there are no projective planes of order 10, but the order 12 case and beyond remain completely open.

The take-away: geometry even in its simplest form is far from finished, and incidence geometry is far from trivial.

Jolyon Joyce
The University of Western Australia