There are, for possibly obvious reasons, not many mathematicians whose names begin with Q. All the same, I think I’ve found one with a reasonably accessible theorem to his name.
What is Qvist’s Theorem?
Qvist’s theorem states that, given any oval in a finite projective plane of order $n$:
- If $n$ is even, there is a node
- If $n$ is odd, then every point that isn’t on the oval lies on either no tangents or two.
I say “reasonably accessible.” I have at least four things I need to explain there.
What is a finite projective plane?
Imagine a 7-by-7 square of points lying in a plane. Or rather, on a torus: we’re going to wrap it up top-to-bottom and left-to-right so that when you go off of one side, you come back on the other.
I’m going to state, without proof, that if you draw a line through any pair of points from the original grid, it passes through five other points as well. So far, so uncontroversial.
Now, if I draw another line through another pair of points, there are two situations: either the lines intersect at exactly one of the grid points (and possibly at places in between, but those aren’t points in this set-up). Alternatively, the lines are parallel and don’t meet at one of our points.
That’s awkward. Wouldn’t it be nice if every pair of lines intersected?
Well now. In projective geometry, I add some points “at infinity” for exactly that purpose. In our 7-by-7 case, we add eight of them.
Why eight? Think about the top-left square. It lies on a line through its neighbour to the right, and on a different line through each of the squares in the row below. Each of those directions corresponds to a point at infinity.
We end up with 57 points and 57 lines (including the one that links the eight points at infinity). It’s not a coincidence that those numbers are the same, but we won’t go into it here.
Suffice to say, this set of points and lines is a finite projective plane of order 7. A similar construction always works for prime numbers.
What’s an oval?
In this context, an oval isn’t an ellipse. It’s a selection of $n+1$ points – the same number as a line – with the constraint that no three of them lie on the same line. It’s a fun exercise to try and find one on an order-7 plane, although there are many.
It’s easier to see on an order-3 plane, which has 13 points (four of which are at infinity). For example, no three of these points lie on the same straight line:
x - x x x - - - -
It’s an oval, whether it looks like what you think an oval is or not.
What’s a tangent?
In traditional geometry, a tangent is a line that meets a shape at exactly one point. It’s the same in projective geometry.
On our order-3 oval, there are four points (1,3,4 and 5), which between them define six lines.
There are 4 tangents:
- the vertical lines 2-5-8-| and 3-6-9-|
- the backward diagonal lines 1-6-8-/ and 2-4-9-/
There are 3 passants, lines that miss the oval altogether:
- the horizontal line 7-8-9–
- the forward diagonal 2-6-7-\
- the line at infinity /-|---
What’s a node?
It’s a point through which every tangent passes. For example, in the order-2 projective plane:
x x x -
Is an oval. The plane has seven lines – three passing through a pair of points in the oval, three tangents (1-4-\, 2-4-| and 3-4–) and a passant at infinity.
Qvist’s theorem predicts a node - here, it’s 4, which lies on all three tangents.
What about the odd case?
Let’s go back to the order-3 case. We can check the seven non-oval points:
- 2, 6, 8, 9, | and / lie on two tangents;
- 7, \ and - lie on no tangents.
It works! In this case, at least.
Why is it interesting?
I honestly don’t know, but it’s nice. I’ve spent several minutes trying to come up with an application to Dobble (which relies on an order-7 projective plane), but have come up short.
Who was Bertil Qvist?
He’s one of the first mathematicians in the DOME who doesn’t have an English wikipedia page (yet). Fortunately, I speak enough German to make up for that. Perhaps my Finnish fan club can extend this – although he doesn’t have a Finnish page either!
Qvist was born in Vaasa, Finland in 1920. He studied maths, astronomy and physics at the University of Helsinki, graduating in 1945. He became a professor of mathematics in 1962 and retired in 1983.
His work focussed on stellar models, general relativity and geometry.
He died in 1991.
A selection of other posts
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