There is a shape. There are of course infinitely many shapes, the universe being the wildly complicated place that it is, but this is a particular shape.
This shape has shadows of circle, triangle and square in three perpendicular directions, when illuminated by a bright light. It’s a simple shape, akin to a wedge or a door stop.
The circle, triangle and square are important shapes in some theories of aikido as basic shapes that we use as a scaffold for our movements, but that exploration is for another day. I mention this here because this is where I first encountered this shape, and this shape is the starting point for the thought I’ve set myself to start exploring.
Let’s start with an obvious statement. Most shapes, when illuminated in three different directions, have three different shadows, and so having a shape with three different shadows is not a surprise.
However, even here, there are mathematical questions lurking in the shadows, as it were. For instance, suppose X is a shape whose shadow in every direction is the same; what can we then say about X? A lot of work has been done on this question, and perhaps one day I’ll gather myself sufficiently to produce a summary. But as a teaser, while circles and spheres have this property, other shapes do as well.
We can also shift our perspective and view this question from a different direction, and this is what I would like to explore here. Our starting point is the observation that the question we’ve set out above has a direction to it. We start with a shape, we shine a light on it from different directions, and we see what shapes emerge as shadows.
But we could also ask, take three shadows. Does there exist a shape whose shadows in three (perpendicular) directions are these three shapes? Just asking this question, we are taking our original question and we’re inverting its direction.
My intention here is not to answer this inverse question, as I haven’t yet done the work I would like to do to understand the answer well enough to explain it here. That is for another day.
But looking back, I can see echoes of this question ringing though earlier days. This is in some sense the basic question of Rashamon: the testimonies of the different witnesses are the shadows, and we wish to understand the original shape, the event witnessed.
This is also a basic question that underpins the foundation of my mathematical life. Looking back, there are several questions that I have explored through a number of different papers. For one, the limit set intersection question is one that I started exploring in my doctoral thesis and is a question I’ve kept coming back to. Over time, the question has continued to grow into broader contexts, but I still have the feeling that all of the work done provides different shadows, and it isn’t clear to me that we yet have a clear view of the shape at the center.
And in the writing I’m doing, which is mostly half finished versions of stories at this point, I can see the same phenomenon at work. There are a few basic shapes that are lurking, and all of those half finished stories are the shadows.
If I am honest, I can say that part of what stands between me and finishing are the desire to see the shapes rather than merely the shadows they are casting. But I realized something during a long drive today.
I have been dancing with my unfinished stories for too long. The only way I’ll come to see these shapes is to first come to see clearly their shadows. And that is my task for the rest of the day.
