If only it were all so simple! If only there were evil people somewhere insidiously committing evil deeds, and it were necessary only to separate them from the rest of us and destroy them. But the line dividing good and evil cuts through the heart of every human being. And who is willing to destroy a piece of his own heart?
― Aleksandr Solzhenitsyn,
Solzhenitsyn states that the line dividing good and evil cuts through every human heart. This can’t literally be true – human hearts are somewhat evenly distributed around the globe.
The first question we might ask is: instead of a line, what’s the simplest surface we can draw that cuts through the heart of every human being? But simply cutting through is a boring interpolation question. If you’re an idealist like me, you’d think every human heart is exactly half good and half evil. So the real question is: what’s the simplest surface that bisects every human heart?
The Polynomial Ham Sandwich Theorem states that in -space, we can bisect measures with a degree variety. What’s the lowest degree polynomial surface in that bisects every human heart from this argument?
By the Ham Sandwich Theorem, we need surprisingly few! With , the smallest for which is . This is as expected a cubic reduction in dimension (hidden underneath is the fact that a polynomial of degree in -space actually has degrees of freedom, so it’s not exactly surprising), so there is a degree algebraic variety that not only passes through, but cuts every human heart exactly in two.
That’s all well and good, but clearly Solzhenitsyn envisioned his problem on the -sphere, where “line” actually means line or curve and not plane or surface. So really the right question to ask is: what is the smallest degree polynomial curve embedded in the unit sphere that bisects heart-shaped regions? We get a nice answer to this modified question by simply intersecting the degree variety above with the sphere. Phew!
It’s fun to think about what this degree algebraic curve on the sphere looks like. It cuts the sphere into a bunch of connected components, some components are good, some components are evil. But it’s a line between good and evil, so good components are only next to evil components, and evil components must be next to good components! Does this present problems? Happily, if you draw a generic smooth curve on the plane or sphere without lifting your pen, the map of the regions is -colorable (exercise: the graph will have only even cycles for faces), so this checks out. Phew!
The number of connected components will in general be quite large. In real algebraic geometry, these components are called nodal domains, and the maximum (and generic) number is essentially . We will thus expect on the order of 10 million connected components on the sphere cut by our single curve of degree .
I will henceforth call this map Solzhenitsyn’s Yin Yang – an algebraic checkerboard of black and white regions cut out by a single continuous curve that passes through every human heart. What an interesting development over the traditional Yin Yang! Well, maybe more cliché than interesting. If you tried to draw Solzhenitsyn’s Yin Yang there would be so many tiny black and white regions everything would look gray from space.