The triangle under conjugation is inside a projective cube

When I get to Princeton John and I will not have been together for months and this is a good time to talk about what has been on my mind. This time two topics converged in an unexpected way and led to hours of intense conversation.

I wanted to talk about my own research, which I thought changed a fundamental topic in out book. Also my student Adam Morgan, now at Middlebury, had got me thinking about some topological questions. Now in the presence of the master, perhaps I could get those resolved. To my surprise these two conversations became one.

The sine function is usually represented by a periodic graph in the plane. A more natural space for it is to identify x = 0 with x = 2π, as topologists do, so that its graph fits on a cylinder. As it goes round the cylinder, the shape repeats over and over.

But one can do better. Identify x = 0 with x = π, representing half a cycle of a sine curve. But this time do the Möbius twist, creating a Möbius strip rather than a cylinder. Again as we go around this space, we trace out the repeating nature of the sine function, but we have to go around twice to complete a full cycle of the function. I had gone through these things with my Seminar class and we had demonstrated this by cutting them out on transparencies. What Adam wanted to do, and did, was to figure out if this could be done for a quarter cycle of the sine function. I told him that a reflection was probably needed in addition to these topological identifications. Adam struggled through this in a paper, but my brain exploded every time I thought about it.

So I went through this for John and several hangers on in the math department commons room. Surprisingly he had never heard of representing the sine function on a Möbius strip. It's very nice when I can tell John anything! He commented that I always come to Princeton with interesting topics. When we got to Adam's conjecture John at first thought it couldn't be done, but then I said maybe it could if we put a reflection line in. John immediately said yes; this is an orbifold, a form of space which contains topological gluing like the above but also can have mirror lines. At this point I was satisfied; I had figured it was this kind of object, but they are difficult and I did not want to get into a long conversation about it so I let the conversation move on the other things.

I had some new work to talk about. I had realized that the conception of conjugate in our book needed to be expanded. I had found an family of self-conjugate curves that helped define the position of the conjugates, and in this connection I wanted to talk about the strange behavior of a point compared to its conjugate. Strangely it seemed our previous conversation was related. Topologically unusual things were in the air!

There was unusual behavior as the defining point moved through vertex. If one put several points side by side, they flipped over as one went through a vertex. This happens from two directions, so it is as though each vertex had a double Mobius twist or cross-cap. To make matters even worse, as one went through the infinite point along a bisector we flip sides again just as in the projective plane.

I motivated this consideration of the topology of the triangle but would never had the courage to take it so far as we did. Here is the picture we came up with. The most important lines in the triangle plane are not the triangle edges, as one would think, but the bisectors, which form a projective cube of 4 vertices, six lines, and three faces (this is the part I would never have thought of). A cube usually has 8 vertices, six edges, and 6 faces (as a die), but the projective space cuts it in half by identifying the front and back as the same. Yes, this is hard to picture because these are very strange shapes. The strangeness happens at a vertex, which is in the middle of the face of the cube. So the result is a projective cube in which each face is twisted in two direction, the type Mobius figures known as a cross-cap. Don't worry if you did not follow the details of this shape, I just want to convey the unusualness of our creation. If the plane of a triangle under conjugation is a projective cube, then the triangle itself is a projective octahedron inscribed in that cube, and even stranger thought.

My head spun for a week. Not so much with the confusion of it, but with the wonder of it. John lives in this world all the time, and because of my own research, I could live in it too, albeit at a much lower level than his.

The wonder of it all.