Five words: “broken symmetry” and “unstable against perturbation.” There — wasn’t that easy?

An algebraic topologist could probably do it rigorously in a few lines. Of course, it wouldn’t be quantitative, and I gather you astro/ME types are compulsive about that.

Tom, here’s what I remember from my astronomy classes:
It’s largely to do with conservation of angular momentum, though not all galaxies are planar, and not all of the parts of a “typical” solar system are either. Spiral galaxies and solar systems are mostly planar because as a large gravitationally-bound cloud of gas (i.e. one in which the gas can be considered contained by its own mass) collapses under its own weight, even a slight initial rotation of the cloud will cause the cloud to flatten. The true answer is, of course, much more complicated.

There are parts of our galaxy that are non-planar (including the spherically-distributed globular clusters, and the much less-dense halo of relatively dark objects surrounding the galactic disk) that are thought to be “remnants” of an earlier stage of galactic evolution.

Our solar system has a vast constellation of spherically-distributed (we think) cometary cores called the Oort Cloud. These icy bodies are the source of most of the comets we see in the sky. One explanation for their distribution is that they are the result of countless interactions between large planets like Jupiter or Saturn and the many trillions of planetessimals that formed in the first few million years of the universe. When the planetessimals came near the gas giants, they were very nearly ejected from the solar system, but instead ended up in solar orbits at distances of as much as a light-year away.

Also, it is now believed that elliptical-type galaxies form when two spiral galaxies collide. These galaxies are described with the shape “oblate-spheroid”, meaning like a football, because of the relatively random distributions of angular momentum after the collision takes place.

Exploited to the max by Kim Stanley Robinson’s _Red Mars_, in which the falling/burning space elevator reveals its double-helical core of synthetic diamond, glowing white-hot in the sky. I’m not sure it made even hand-waving sense as materials science or engineering, but it sure delivered the symbolism.